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FwûMBó–éöͶ&B\ÒC§;3óAíƃù]´AA¢–^­nA	wGï""Aã$`u„Í@`DwQ:i@Pž2ÿÿÿÿÿÿÿÿ úîÂ_p¥ìí?ª¸ÿ? ì2libopenblas_v0.3.27--3aa239bc726cfb0bd8e5330d8d4c15c6.dll_err_test_functioninvalid value for nellip_harminvalid value for pinvalid signm or signnfailed to allocate memoryIVinvalid condition on `p - 1` while calling a Python objectNULL result without error in PyObject_Call%.200s() takes no arguments (%zd given)%.200s() takes exactly one argument (%zd given)Bad call flags for CyFunction%.200s() takes no keyword arguments<cyfunction %U at %p>__int__ returned non-int (type %.200s).  The ability to return an instance of a strict subclass of int is deprecated, and may be removed in a future version of Python.__%.4s__ returned non-%.4s (type %.200s)__pyx_capi__%.200s does not export expected C function %.200sC function %.200s.%.200s has wrong signature (expected %.500s, got %.500s)Interpreter change detected - this module can only be loaded into one interpreter per process.name__loader__loader__file__origin__package__parent__path__submodule_search_locationspolynomial only defined for nonnegative neval_hermitenormspherical_knpolynomial defined only for alpha > -1eval_genlaguerrespherical_jnhyperu%.200s() needs an argumentkeywords must be stringsunbound method %.200S() needs an argument%.200s does not export expected C variable %.200svoid *C variable %.200s.%.200s has wrong signature (expected %.500s, got %.500s)%.200s.%.200s is not a type object%.200s.%.200s size changed, may indicate binary incompatibility. Expected %zd from C header, got %zd from PyObject%s.%s size changed, may indicate binary incompatibility. Expected %zd from C header, got %zd from PyObjectcannot fit '%.200s' into an index-sized integernon-integer arg n is deprecated, removed in SciPy 1.7.xspherical_incalling %R should have returned an instance of BaseException, not %Rraise: exception class must be a subclass of BaseExceptionhyp2f1siciðÿøpqdfInput parameter %s is out of rangestdtritAnswer appears to be lower than lowest search bound (%g)Answer appears to be higher than highest search bound (%g)Two internal parameters that should sum to 1.0 do not.Computational errorUnknown error.eval_hermitencchndtrixfdfndfdncfdtrfloating point number truncated to an integer__annotations__ must be set to a dict object__name__ must be set to a string object__qualname__ must be set to a string object__kwdefaults__ must be set to a dict objectchanges to cyfunction.__kwdefaults__ will not currently affect the values used in function calls__defaults__ must be set to a tuple objectchanges to cyfunction.__defaults__ will not currently affect the values used in function callsfunction's dictionary may not be deletedsetting function's dictionary to a non-dictm should not be greater than nsph_harmshichiintan integer is requiredcan't convert negative value to sf_action_twright_besselcan't convert negative value to sf_error_tfloat divisionscipy.special._hyp0f1._hyp0f1_cmplxscipy.special._boxcox.boxcoxscipy.special._boxcox.boxcox1pscipy.special._exprel.expreløøøxchdtrivxlampdtrikname '%U' is not definedchndtrchndtrinctnctdtrchndtridfstdtridfs

Cumulative distribution function (CDF) of the cosine distribution::

             {             0,              x < -pi
    cdf(x) = { (pi + x + sin(x))/(2*pi),   -pi <= x <= pi
             {             1,              x > pi

Parameters
----------
x : array_like
    `x` must contain real numbers.

Returns
-------
scalar or ndarray
    The cosine distribution CDF evaluated at `x`._cosine_invcdf_cosine_invcdf(p)

Inverse of the cumulative distribution function (CDF) of the cosine
distribution.

The CDF of the cosine distribution is::

    cdf(x) = (pi + x + sin(x))/(2*pi)

This function computes the inverse of cdf(x).

Parameters
----------
p : array_like
    `p` must contain real numbers in the interval ``0 <= p <= 1``.
    `nan` is returned for values of `p` outside the interval [0, 1].

Returns
-------
scalar or ndarray
    The inverse of the cosine distribution CDF evaluated at `p`._cospiInternal function, do not use._ellip_harmInternal function, use `ellip_harm` instead._factorial_igam_fac_kolmogc_kolmogci_kolmogp_lambertwInternal function, use `lambertw` instead._lanczos_sum_expg_scaled_lgam1p_log1pmx_riemann_zetaInternal function, use `zeta` instead._scaled_exp1_scaled_exp1(x, out=None):

Compute the scaled exponential integral.

This is a private function, subject to change or removal with no
deprecation.

This function computes F(x), where F is the factor remaining in E_1(x)
when exp(-x)/x is factored out.  That is,::

    E_1(x) = exp(-x)/x * F(x)

or

    F(x) = x * exp(x) * E_1(x)

The function is defined for real x >= 0.  For x < 0, nan is returned.

F has the properties:

* F(0) = 0
* F(x) is increasing on [0, inf).
* The limit as x goes to infinity of F(x) is 1.

Parameters
----------
x: array_like
    The input values. Must be real.  The implementation is limited to
    double precision floating point, so other types will be cast to
    to double precision.
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    Values of the scaled exponential integral.

See Also
--------
exp1 : exponential integral E_1

Examples
--------
>>> from scipy.special import _scaled_exp1
>>> _scaled_exp1([0, 0.1, 1, 10, 100])_sf_error_test_functionPrivate function; do not use._sinpi_smirnovc_smirnovc(n, d)
 Internal function, do not use._smirnovci_smirnovp_smirnovp(n, p)
 Internal function, do not use._spherical_inInternal function, use `spherical_in` instead._spherical_in_d_spherical_jnInternal function, use `spherical_jn` instead._spherical_jn_d_spherical_knInternal function, use `spherical_kn` instead._spherical_kn_d_spherical_ynInternal function, use `spherical_yn` instead._spherical_yn_d_stirling2_inexact_struve_asymp_large_z_struve_asymp_large_z(v, z, is_h)

Internal function for testing `struve` & `modstruve`

Evaluates using asymptotic expansion

Returns
-------
v, err_struve_bessel_series_struve_bessel_series(v, z, is_h)

Internal function for testing `struve` & `modstruve`

Evaluates using Bessel function series

Returns
-------
v, err_struve_power_series_struve_power_series(v, z, is_h)

Internal function for testing `struve` & `modstruve`

Evaluates using power series

Returns
-------
v, err_zeta_zeta(x, q)

Internal function, Hurwitz zeta.agmagm(a, b, out=None)

Compute the arithmetic-geometric mean of `a` and `b`.

Start with a_0 = a and b_0 = b and iteratively compute::

    a_{n+1} = (a_n + b_n)/2
    b_{n+1} = sqrt(a_n*b_n)

a_n and b_n converge to the same limit as n increases; their common
limit is agm(a, b).

Parameters
----------
a, b : array_like
    Real values only. If the values are both negative, the result
    is negative. If one value is negative and the other is positive,
    `nan` is returned.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
scalar or ndarray
    The arithmetic-geometric mean of `a` and `b`.

Examples
--------
>>> import numpy as np
>>> from scipy.special import agm
>>> a, b = 24.0, 6.0
>>> agm(a, b)
13.458171481725614

Compare that result to the iteration:

>>> while a != b:
...     a, b = (a + b)/2, np.sqrt(a*b)
...     print("a = %19.16f  b=%19.16f" % (a, b))
...
a = 15.0000000000000000  b=12.0000000000000000
a = 13.5000000000000000  b=13.4164078649987388
a = 13.4582039324993694  b=13.4581390309909850
a = 13.4581714817451772  b=13.4581714817060547
a = 13.4581714817256159  b=13.4581714817256159

When array-like arguments are given, broadcasting applies:

>>> a = np.array([[1.5], [3], [6]])  # a has shape (3, 1).
>>> b = np.array([6, 12, 24, 48])    # b has shape (4,).
>>> agm(a, b)
array([[  3.36454287,   5.42363427,   9.05798751,  15.53650756],
       [  4.37037309,   6.72908574,  10.84726853,  18.11597502],
       [  6.        ,   8.74074619,  13.45817148,  21.69453707]])airyairy(z, out=None)

Airy functions and their derivatives.

Parameters
----------
z : array_like
    Real or complex argument.
out : tuple of ndarray, optional
    Optional output arrays for the function values

Returns
-------
Ai, Aip, Bi, Bip : 4-tuple of scalar or ndarray
    Airy functions Ai and Bi, and their derivatives Aip and Bip.

See Also
--------
airye : exponentially scaled Airy functions.

Notes
-----
The Airy functions Ai and Bi are two independent solutions of

.. math:: y''(x) = x y(x).

For real `z` in [-10, 10], the computation is carried out by calling
the Cephes [1]_ `airy` routine, which uses power series summation
for small `z` and rational minimax approximations for large `z`.

Outside this range, the AMOS [2]_ `zairy` and `zbiry` routines are
employed.  They are computed using power series for :math:`|z| < 1` and
the following relations to modified Bessel functions for larger `z`
(where :math:`t \equiv 2 z^{3/2}/3`):

.. math::

    Ai(z) = \frac{1}{\pi \sqrt{3}} K_{1/3}(t)

    Ai'(z) = -\frac{z}{\pi \sqrt{3}} K_{2/3}(t)

    Bi(z) = \sqrt{\frac{z}{3}} \left(I_{-1/3}(t) + I_{1/3}(t) \right)

    Bi'(z) = \frac{z}{\sqrt{3}} \left(I_{-2/3}(t) + I_{2/3}(t)\right)

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/
.. [2] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
       of a Complex Argument and Nonnegative Order",
       http://netlib.org/amos/

Examples
--------
Compute the Airy functions on the interval [-15, 5].

>>> import numpy as np
>>> from scipy import special
>>> x = np.linspace(-15, 5, 201)
>>> ai, aip, bi, bip = special.airy(x)

Plot Ai(x) and Bi(x).

>>> import matplotlib.pyplot as plt
>>> plt.plot(x, ai, 'r', label='Ai(x)')
>>> plt.plot(x, bi, 'b--', label='Bi(x)')
>>> plt.ylim(-0.5, 1.0)
>>> plt.grid()
>>> plt.legend(loc='upper left')
>>> plt.show()airyeairye(z, out=None)

Exponentially scaled Airy functions and their derivatives.

Scaling::

    eAi  = Ai  * exp(2.0/3.0*z*sqrt(z))
    eAip = Aip * exp(2.0/3.0*z*sqrt(z))
    eBi  = Bi  * exp(-abs(2.0/3.0*(z*sqrt(z)).real))
    eBip = Bip * exp(-abs(2.0/3.0*(z*sqrt(z)).real))

Parameters
----------
z : array_like
    Real or complex argument.
out : tuple of ndarray, optional
    Optional output arrays for the function values

Returns
-------
eAi, eAip, eBi, eBip : 4-tuple of scalar or ndarray
    Exponentially scaled Airy functions eAi and eBi, and their derivatives
    eAip and eBip

See Also
--------
airy

Notes
-----
Wrapper for the AMOS [1]_ routines `zairy` and `zbiry`.

References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
       of a Complex Argument and Nonnegative Order",
       http://netlib.org/amos/

Examples
--------
We can compute exponentially scaled Airy functions and their derivatives:

>>> import numpy as np
>>> from scipy.special import airye
>>> import matplotlib.pyplot as plt
>>> z = np.linspace(0, 50, 500)
>>> eAi, eAip, eBi, eBip = airye(z)
>>> f, ax = plt.subplots(2, 1, sharex=True)
>>> for ind, data in enumerate([[eAi, eAip, ["eAi", "eAip"]],
...                             [eBi, eBip, ["eBi", "eBip"]]]):
...     ax[ind].plot(z, data[0], "-r", z, data[1], "-b")
...     ax[ind].legend(data[2])
...     ax[ind].grid(True)
>>> plt.show()

We can compute these using usual non-scaled Airy functions by:

>>> from scipy.special import airy
>>> Ai, Aip, Bi, Bip = airy(z)
>>> np.allclose(eAi, Ai * np.exp(2.0 / 3.0 * z * np.sqrt(z)))
True
>>> np.allclose(eAip, Aip * np.exp(2.0 / 3.0 * z * np.sqrt(z)))
True
>>> np.allclose(eBi, Bi * np.exp(-abs(np.real(2.0 / 3.0 * z * np.sqrt(z)))))
True
>>> np.allclose(eBip, Bip * np.exp(-abs(np.real(2.0 / 3.0 * z * np.sqrt(z)))))
True

Comparing non-scaled and exponentially scaled ones, the usual non-scaled
function quickly underflows for large values, whereas the exponentially
scaled function does not.

>>> airy(200)
(0.0, 0.0, nan, nan)
>>> airye(200)
(0.07501041684381093, -1.0609012305109042, 0.15003188417418148, 2.1215836725571093)bdtrbdtr(k, n, p, out=None)

Binomial distribution cumulative distribution function.

Sum of the terms 0 through `floor(k)` of the Binomial probability density.

.. math::
    \mathrm{bdtr}(k, n, p) =
    \sum_{j=0}^{\lfloor k \rfloor} {{n}\choose{j}} p^j (1-p)^{n-j}

Parameters
----------
k : array_like
    Number of successes (double), rounded down to the nearest integer.
n : array_like
    Number of events (int).
p : array_like
    Probability of success in a single event (float).
out : ndarray, optional
    Optional output array for the function values

Returns
-------
y : scalar or ndarray
    Probability of `floor(k)` or fewer successes in `n` independent events with
    success probabilities of `p`.

Notes
-----
The terms are not summed directly; instead the regularized incomplete beta
function is employed, according to the formula,

.. math::
    \mathrm{bdtr}(k, n, p) =
    I_{1 - p}(n - \lfloor k \rfloor, \lfloor k \rfloor + 1).

Wrapper for the Cephes [1]_ routine `bdtr`.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/bdtrcbdtrc(k, n, p, out=None)

Binomial distribution survival function.

Sum of the terms `floor(k) + 1` through `n` of the binomial probability
density,

.. math::
    \mathrm{bdtrc}(k, n, p) =
    \sum_{j=\lfloor k \rfloor +1}^n {{n}\choose{j}} p^j (1-p)^{n-j}

Parameters
----------
k : array_like
    Number of successes (double), rounded down to nearest integer.
n : array_like
    Number of events (int)
p : array_like
    Probability of success in a single event.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
y : scalar or ndarray
    Probability of `floor(k) + 1` or more successes in `n` independent
    events with success probabilities of `p`.

See Also
--------
bdtr
betainc

Notes
-----
The terms are not summed directly; instead the regularized incomplete beta
function is employed, according to the formula,

.. math::
    \mathrm{bdtrc}(k, n, p) = I_{p}(\lfloor k \rfloor + 1, n - \lfloor k \rfloor).

Wrapper for the Cephes [1]_ routine `bdtrc`.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/bdtribdtri(k, n, y, out=None)

Inverse function to `bdtr` with respect to `p`.

Finds the event probability `p` such that the sum of the terms 0 through
`k` of the binomial probability density is equal to the given cumulative
probability `y`.

Parameters
----------
k : array_like
    Number of successes (float), rounded down to the nearest integer.
n : array_like
    Number of events (float)
y : array_like
    Cumulative probability (probability of `k` or fewer successes in `n`
    events).
out : ndarray, optional
    Optional output array for the function values

Returns
-------
p : scalar or ndarray
    The event probability such that `bdtr(\lfloor k \rfloor, n, p) = y`.

See Also
--------
bdtr
betaincinv

Notes
-----
The computation is carried out using the inverse beta integral function
and the relation,::

    1 - p = betaincinv(n - k, k + 1, y).

Wrapper for the Cephes [1]_ routine `bdtri`.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/bdtrikbdtrik(y, n, p, out=None)

Inverse function to `bdtr` with respect to `k`.

Finds the number of successes `k` such that the sum of the terms 0 through
`k` of the Binomial probability density for `n` events with probability
`p` is equal to the given cumulative probability `y`.

Parameters
----------
y : array_like
    Cumulative probability (probability of `k` or fewer successes in `n`
    events).
n : array_like
    Number of events (float).
p : array_like
    Success probability (float).
out : ndarray, optional
    Optional output array for the function values

Returns
-------
k : scalar or ndarray
    The number of successes `k` such that `bdtr(k, n, p) = y`.

See Also
--------
bdtr

Notes
-----
Formula 26.5.24 of [1]_ is used to reduce the binomial distribution to the
cumulative incomplete beta distribution.

Computation of `k` involves a search for a value that produces the desired
value of `y`. The search relies on the monotonicity of `y` with `k`.

Wrapper for the CDFLIB [2]_ Fortran routine `cdfbin`.

References
----------
.. [1] Milton Abramowitz and Irene A. Stegun, eds.
       Handbook of Mathematical Functions with Formulas,
       Graphs, and Mathematical Tables. New York: Dover, 1972.
.. [2] Barry Brown, James Lovato, and Kathy Russell,
       CDFLIB: Library of Fortran Routines for Cumulative Distribution
       Functions, Inverses, and Other Parameters.bdtrinbdtrin(k, y, p, out=None)

Inverse function to `bdtr` with respect to `n`.

Finds the number of events `n` such that the sum of the terms 0 through
`k` of the Binomial probability density for events with probability `p` is
equal to the given cumulative probability `y`.

Parameters
----------
k : array_like
    Number of successes (float).
y : array_like
    Cumulative probability (probability of `k` or fewer successes in `n`
    events).
p : array_like
    Success probability (float).
out : ndarray, optional
    Optional output array for the function values

Returns
-------
n : scalar or ndarray
    The number of events `n` such that `bdtr(k, n, p) = y`.

See Also
--------
bdtr

Notes
-----
Formula 26.5.24 of [1]_ is used to reduce the binomial distribution to the
cumulative incomplete beta distribution.

Computation of `n` involves a search for a value that produces the desired
value of `y`. The search relies on the monotonicity of `y` with `n`.

Wrapper for the CDFLIB [2]_ Fortran routine `cdfbin`.

References
----------
.. [1] Milton Abramowitz and Irene A. Stegun, eds.
       Handbook of Mathematical Functions with Formulas,
       Graphs, and Mathematical Tables. New York: Dover, 1972.
.. [2] Barry Brown, James Lovato, and Kathy Russell,
       CDFLIB: Library of Fortran Routines for Cumulative Distribution
       Functions, Inverses, and Other Parameters.beibei(x, out=None)

Kelvin function bei.

Defined as

.. math::

    \mathrm{bei}(x) = \Im[J_0(x e^{3 \pi i / 4})]

where :math:`J_0` is the Bessel function of the first kind of
order zero (see `jv`). See [dlmf]_ for more details.

Parameters
----------
x : array_like
    Real argument.
out : ndarray, optional
    Optional output array for the function results.

Returns
-------
scalar or ndarray
    Values of the Kelvin function.

See Also
--------
ber : the corresponding real part
beip : the derivative of bei
jv : Bessel function of the first kind

References
----------
.. [dlmf] NIST, Digital Library of Mathematical Functions,
    https://dlmf.nist.gov/10.61

Examples
--------
It can be expressed using Bessel functions.

>>> import numpy as np
>>> import scipy.special as sc
>>> x = np.array([1.0, 2.0, 3.0, 4.0])
>>> sc.jv(0, x * np.exp(3 * np.pi * 1j / 4)).imag
array([0.24956604, 0.97229163, 1.93758679, 2.29269032])
>>> sc.bei(x)
array([0.24956604, 0.97229163, 1.93758679, 2.29269032])beipbeip(x, out=None)

Derivative of the Kelvin function bei.

Parameters
----------
x : array_like
    Real argument.
out : ndarray, optional
    Optional output array for the function results.

Returns
-------
scalar or ndarray
    The values of the derivative of bei.

See Also
--------
bei

References
----------
.. [dlmf] NIST, Digital Library of Mathematical Functions,
    https://dlmf.nist.gov/10#PT5berber(x, out=None)

Kelvin function ber.

Defined as

.. math::

    \mathrm{ber}(x) = \Re[J_0(x e^{3 \pi i / 4})]

where :math:`J_0` is the Bessel function of the first kind of
order zero (see `jv`). See [dlmf]_ for more details.

Parameters
----------
x : array_like
    Real argument.
out : ndarray, optional
    Optional output array for the function results.

Returns
-------
scalar or ndarray
    Values of the Kelvin function.

See Also
--------
bei : the corresponding real part
berp : the derivative of bei
jv : Bessel function of the first kind

References
----------
.. [dlmf] NIST, Digital Library of Mathematical Functions,
    https://dlmf.nist.gov/10.61

Examples
--------
It can be expressed using Bessel functions.

>>> import numpy as np
>>> import scipy.special as sc
>>> x = np.array([1.0, 2.0, 3.0, 4.0])
>>> sc.jv(0, x * np.exp(3 * np.pi * 1j / 4)).real
array([ 0.98438178,  0.75173418, -0.22138025, -2.56341656])
>>> sc.ber(x)
array([ 0.98438178,  0.75173418, -0.22138025, -2.56341656])berpberp(x, out=None)

Derivative of the Kelvin function ber.

Parameters
----------
x : array_like
    Real argument.
out : ndarray, optional
    Optional output array for the function results.

Returns
-------
scalar or ndarray
    The values of the derivative of ber.

See Also
--------
ber

References
----------
.. [dlmf] NIST, Digital Library of Mathematical Functions,
    https://dlmf.nist.gov/10#PT5besselpolybesselpoly(a, lmb, nu, out=None)

Weighted integral of the Bessel function of the first kind.

Computes

.. math::

   \int_0^1 x^\lambda J_\nu(2 a x) \, dx

where :math:`J_\nu` is a Bessel function and :math:`\lambda=lmb`,
:math:`\nu=nu`.

Parameters
----------
a : array_like
    Scale factor inside the Bessel function.
lmb : array_like
    Power of `x`
nu : array_like
    Order of the Bessel function.
out : ndarray, optional
    Optional output array for the function results.

Returns
-------
scalar or ndarray
    Value of the integral.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/

Examples
--------
Evaluate the function for one parameter set.

>>> from scipy.special import besselpoly
>>> besselpoly(1, 1, 1)
0.24449718372863877

Evaluate the function for different scale factors.

>>> import numpy as np
>>> factors = np.array([0., 3., 6.])
>>> besselpoly(factors, 1, 1)
array([ 0.        , -0.00549029,  0.00140174])

Plot the function for varying powers, orders and scales.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> powers = np.linspace(0, 10, 100)
>>> orders = [1, 2, 3]
>>> scales = [1, 2]
>>> all_combinations = [(order, scale) for order in orders
...                     for scale in scales]
>>> for order, scale in all_combinations:
...     ax.plot(powers, besselpoly(scale, powers, order),
...             label=rf"$\nu={order}, a={scale}$")
>>> ax.legend()
>>> ax.set_xlabel(r"$\lambda$")
>>> ax.set_ylabel(r"$\int_0^1 x^{\lambda} J_{\nu}(2ax)\,dx$")
>>> plt.show()betabeta(a, b, out=None)

Beta function.

This function is defined in [1]_ as

.. math::

    B(a, b) = \int_0^1 t^{a-1}(1-t)^{b-1}dt
            = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)},

where :math:`\Gamma` is the gamma function.

Parameters
----------
a, b : array_like
    Real-valued arguments
out : ndarray, optional
    Optional output array for the function result

Returns
-------
scalar or ndarray
    Value of the beta function

See Also
--------
gamma : the gamma function
betainc :  the regularized incomplete beta function
betaln : the natural logarithm of the absolute
         value of the beta function

References
----------
.. [1] NIST Digital Library of Mathematical Functions,
       Eq. 5.12.1. https://dlmf.nist.gov/5.12

Examples
--------
>>> import scipy.special as sc

The beta function relates to the gamma function by the
definition given above:

>>> sc.beta(2, 3)
0.08333333333333333
>>> sc.gamma(2)*sc.gamma(3)/sc.gamma(2 + 3)
0.08333333333333333

As this relationship demonstrates, the beta function
is symmetric:

>>> sc.beta(1.7, 2.4)
0.16567527689031739
>>> sc.beta(2.4, 1.7)
0.16567527689031739

This function satisfies :math:`B(1, b) = 1/b`:

>>> sc.beta(1, 4)
0.25betaincbetainc(a, b, x, out=None)

Regularized incomplete beta function.

Computes the regularized incomplete beta function, defined as [1]_:

.. math::

    I_x(a, b) = \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)} \int_0^x
    t^{a-1}(1-t)^{b-1}dt,

for :math:`0 \leq x \leq 1`.

This function is the cumulative distribution function for the beta
distribution; its range is [0, 1].

Parameters
----------
a, b : array_like
       Positive, real-valued parameters
x : array_like
    Real-valued such that :math:`0 \leq x \leq 1`,
    the upper limit of integration
out : ndarray, optional
    Optional output array for the function values

Returns
-------
scalar or ndarray
    Value of the regularized incomplete beta function

See Also
--------
beta : beta function
betaincinv : inverse of the regularized incomplete beta function
betaincc : complement of the regularized incomplete beta function
scipy.stats.beta : beta distribution

Notes
-----
The term *regularized* in the name of this function refers to the
scaling of the function by the gamma function terms shown in the
formula.  When not qualified as *regularized*, the name *incomplete
beta function* often refers to just the integral expression,
without the gamma terms.  One can use the function `beta` from
`scipy.special` to get this "nonregularized" incomplete beta
function by multiplying the result of ``betainc(a, b, x)`` by
``beta(a, b)``.

References
----------
.. [1] NIST Digital Library of Mathematical Functions
       https://dlmf.nist.gov/8.17

Examples
--------

Let :math:`B(a, b)` be the `beta` function.

>>> import scipy.special as sc

The coefficient in terms of `gamma` is equal to
:math:`1/B(a, b)`. Also, when :math:`x=1`
the integral is equal to :math:`B(a, b)`.
Therefore, :math:`I_{x=1}(a, b) = 1` for any :math:`a, b`.

>>> sc.betainc(0.2, 3.5, 1.0)
1.0

It satisfies
:math:`I_x(a, b) = x^a F(a, 1-b, a+1, x)/ (aB(a, b))`,
where :math:`F` is the hypergeometric function `hyp2f1`:

>>> a, b, x = 1.4, 3.1, 0.5
>>> x**a * sc.hyp2f1(a, 1 - b, a + 1, x)/(a * sc.beta(a, b))
0.8148904036225295
>>> sc.betainc(a, b, x)
0.8148904036225296

This functions satisfies the relationship
:math:`I_x(a, b) = 1 - I_{1-x}(b, a)`:

>>> sc.betainc(2.2, 3.1, 0.4)
0.49339638807619446
>>> 1 - sc.betainc(3.1, 2.2, 1 - 0.4)
0.49339638807619446betainccbetaincc(a, b, x, out=None)

Complement of the regularized incomplete beta function.

Computes the complement of the regularized incomplete beta function,
defined as [1]_:

.. math::

    \bar{I}_x(a, b) = 1 - I_x(a, b)
                    = 1 - \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)} \int_0^x
                              t^{a-1}(1-t)^{b-1}dt,

for :math:`0 \leq x \leq 1`.

Parameters
----------
a, b : array_like
       Positive, real-valued parameters
x : array_like
    Real-valued such that :math:`0 \leq x \leq 1`,
    the upper limit of integration
out : ndarray, optional
    Optional output array for the function values

Returns
-------
scalar or ndarray
    Value of the regularized incomplete beta function

See Also
--------
betainc : regularized incomplete beta function
betaincinv : inverse of the regularized incomplete beta function
betainccinv :
    inverse of the complement of the regularized incomplete beta function
beta : beta function
scipy.stats.beta : beta distribution

Notes
-----
.. versionadded:: 1.11.0

References
----------
.. [1] NIST Digital Library of Mathematical Functions
       https://dlmf.nist.gov/8.17

Examples
--------
>>> from scipy.special import betaincc, betainc

The naive calculation ``1 - betainc(a, b, x)`` loses precision when
the values of ``betainc(a, b, x)`` are close to 1:

>>> 1 - betainc(0.5, 8, [0.9, 0.99, 0.999])
array([2.0574632e-09, 0.0000000e+00, 0.0000000e+00])

By using ``betaincc``, we get the correct values:

>>> betaincc(0.5, 8, [0.9, 0.99, 0.999])
array([2.05746321e-09, 1.97259354e-17, 1.96467954e-25])betainccinvbetainccinv(a, b, y, out=None)

Inverse of the complemented regularized incomplete beta function.

Computes :math:`x` such that:

.. math::

    y = 1 - I_x(a, b) = 1 - \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}
    \int_0^x t^{a-1}(1-t)^{b-1}dt,

where :math:`I_x` is the normalized incomplete beta function `betainc`
and :math:`\Gamma` is the `gamma` function [1]_.

Parameters
----------
a, b : array_like
    Positive, real-valued parameters
y : array_like
    Real-valued input
out : ndarray, optional
    Optional output array for function values

Returns
-------
scalar or ndarray
    Value of the inverse of the regularized incomplete beta function

See Also
--------
betainc : regularized incomplete beta function
betaincc : complement of the regularized incomplete beta function

Notes
-----
.. versionadded:: 1.11.0

References
----------
.. [1] NIST Digital Library of Mathematical Functions
       https://dlmf.nist.gov/8.17

Examples
--------
>>> from scipy.special import betainccinv, betaincc

This function is the inverse of `betaincc` for fixed
values of :math:`a` and :math:`b`.

>>> a, b = 1.2, 3.1
>>> y = betaincc(a, b, 0.2)
>>> betainccinv(a, b, y)
0.2

>>> a, b = 7, 2.5
>>> x = betainccinv(a, b, 0.875)
>>> betaincc(a, b, x)
0.875betaincinvbetaincinv(a, b, y, out=None)

Inverse of the regularized incomplete beta function.

Computes :math:`x` such that:

.. math::

    y = I_x(a, b) = \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}
    \int_0^x t^{a-1}(1-t)^{b-1}dt,

where :math:`I_x` is the normalized incomplete beta function `betainc`
and :math:`\Gamma` is the `gamma` function [1]_.

Parameters
----------
a, b : array_like
    Positive, real-valued parameters
y : array_like
    Real-valued input
out : ndarray, optional
    Optional output array for function values

Returns
-------
scalar or ndarray
    Value of the inverse of the regularized incomplete beta function

See Also
--------
betainc : regularized incomplete beta function
gamma : gamma function

References
----------
.. [1] NIST Digital Library of Mathematical Functions
       https://dlmf.nist.gov/8.17

Examples
--------
>>> import scipy.special as sc

This function is the inverse of `betainc` for fixed
values of :math:`a` and :math:`b`.

>>> a, b = 1.2, 3.1
>>> y = sc.betainc(a, b, 0.2)
>>> sc.betaincinv(a, b, y)
0.2
>>>
>>> a, b = 7.5, 0.4
>>> x = sc.betaincinv(a, b, 0.5)
>>> sc.betainc(a, b, x)
0.5betalnbetaln(a, b, out=None)

Natural logarithm of absolute value of beta function.

Computes ``ln(abs(beta(a, b)))``.

Parameters
----------
a, b : array_like
    Positive, real-valued parameters
out : ndarray, optional
    Optional output array for function values

Returns
-------
scalar or ndarray
    Value of the betaln function

See Also
--------
gamma : the gamma function
betainc :  the regularized incomplete beta function
beta : the beta function

Examples
--------
>>> import numpy as np
>>> from scipy.special import betaln, beta

Verify that, for moderate values of ``a`` and ``b``, ``betaln(a, b)``
is the same as ``log(beta(a, b))``:

>>> betaln(3, 4)
-4.0943445622221

>>> np.log(beta(3, 4))
-4.0943445622221

In the following ``beta(a, b)`` underflows to 0, so we can't compute
the logarithm of the actual value.

>>> a = 400
>>> b = 900
>>> beta(a, b)
0.0

We can compute the logarithm of ``beta(a, b)`` by using `betaln`:

>>> betaln(a, b)
-804.3069951764146binombinom(x, y, out=None)

Binomial coefficient considered as a function of two real variables.

For real arguments, the binomial coefficient is defined as

.. math::

    \binom{x}{y} = \frac{\Gamma(x + 1)}{\Gamma(y + 1)\Gamma(x - y + 1)} =
        \frac{1}{(x + 1)\mathrm{B}(x - y + 1, y + 1)}

Where :math:`\Gamma` is the Gamma function (`gamma`) and :math:`\mathrm{B}`
is the Beta function (`beta`) [1]_.

Parameters
----------
x, y: array_like
   Real arguments to :math:`\binom{x}{y}`.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
scalar or ndarray
    Value of binomial coefficient.

See Also
--------
comb : The number of combinations of N things taken k at a time.

Notes
-----
The Gamma function has poles at non-positive integers and tends to either
positive or negative infinity depending on the direction on the real line
from which a pole is approached. When considered as a function of two real
variables, :math:`\binom{x}{y}` is thus undefined when `x` is a negative
integer.  `binom` returns ``nan`` when ``x`` is a negative integer. This
is the case even when ``x`` is a negative integer and ``y`` an integer,
contrary to the usual convention for defining :math:`\binom{n}{k}` when it
is considered as a function of two integer variables.

References
----------
.. [1] https://en.wikipedia.org/wiki/Binomial_coefficient

Examples
--------
The following examples illustrate the ways in which `binom` differs from
the function `comb`.

>>> from scipy.special import binom, comb

When ``exact=False`` and ``x`` and ``y`` are both positive, `comb` calls
`binom` internally.

>>> x, y = 3, 2
>>> (binom(x, y), comb(x, y), comb(x, y, exact=True))
(3.0, 3.0, 3)

For larger values, `comb` with ``exact=True`` no longer agrees
with `binom`.

>>> x, y = 43, 23
>>> (binom(x, y), comb(x, y), comb(x, y, exact=True))
(960566918219.9999, 960566918219.9999, 960566918220)

`binom` returns ``nan`` when ``x`` is a negative integer, but is otherwise
defined for negative arguments. `comb` returns 0 whenever one of ``x`` or
``y`` is negative or ``x`` is less than ``y``.

>>> x, y = -3, 2
>>> (binom(x, y), comb(x, y), comb(x, y, exact=True))
(nan, 0.0, 0)

>>> x, y = -3.1, 2.2
>>> (binom(x, y), comb(x, y), comb(x, y, exact=True))
(18.714147876804432, 0.0, 0)

>>> x, y = 2.2, 3.1
>>> (binom(x, y), comb(x, y), comb(x, y, exact=True))
(0.037399983365134115, 0.0, 0)boxcoxboxcox(x, lmbda, out=None)

Compute the Box-Cox transformation.

The Box-Cox transformation is::

    y = (x**lmbda - 1) / lmbda  if lmbda != 0
        log(x)                  if lmbda == 0

Returns `nan` if ``x < 0``.
Returns `-inf` if ``x == 0`` and ``lmbda < 0``.

Parameters
----------
x : array_like
    Data to be transformed.
lmbda : array_like
    Power parameter of the Box-Cox transform.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
y : scalar or ndarray
    Transformed data.

Notes
-----

.. versionadded:: 0.14.0

Examples
--------
>>> from scipy.special import boxcox
>>> boxcox([1, 4, 10], 2.5)
array([   0.        ,   12.4       ,  126.09110641])
>>> boxcox(2, [0, 1, 2])
array([ 0.69314718,  1.        ,  1.5       ])boxcox1pboxcox1p(x, lmbda, out=None)

Compute the Box-Cox transformation of 1 + `x`.

The Box-Cox transformation computed by `boxcox1p` is::

    y = ((1+x)**lmbda - 1) / lmbda  if lmbda != 0
        log(1+x)                    if lmbda == 0

Returns `nan` if ``x < -1``.
Returns `-inf` if ``x == -1`` and ``lmbda < 0``.

Parameters
----------
x : array_like
    Data to be transformed.
lmbda : array_like
    Power parameter of the Box-Cox transform.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
y : scalar or ndarray
    Transformed data.

Notes
-----

.. versionadded:: 0.14.0

Examples
--------
>>> from scipy.special import boxcox1p
>>> boxcox1p(1e-4, [0, 0.5, 1])
array([  9.99950003e-05,   9.99975001e-05,   1.00000000e-04])
>>> boxcox1p([0.01, 0.1], 0.25)
array([ 0.00996272,  0.09645476])btdtrbtdtr(a, b, x, out=None)

Cumulative distribution function of the beta distribution.

Returns the integral from zero to `x` of the beta probability density
function,

.. math::
    I = \int_0^x \frac{\Gamma(a + b)}{\Gamma(a)\Gamma(b)} t^{a-1} (1-t)^{b-1}\,dt

where :math:`\Gamma` is the gamma function.

.. deprecated:: 1.12.0
    This function is deprecated and will be removed from SciPy 1.14.0.
    Use `scipy.special.betainc` instead.

Parameters
----------
a : array_like
    Shape parameter (a > 0).
b : array_like
    Shape parameter (b > 0).
x : array_like
    Upper limit of integration, in [0, 1].
out : ndarray, optional
    Optional output array for the function values

Returns
-------
I : scalar or ndarray
    Cumulative distribution function of the beta distribution with
    parameters `a` and `b` at `x`.

See Also
--------
betainc

Notes
-----
This function is identical to the incomplete beta integral function
`betainc`.

Wrapper for the Cephes [1]_ routine `btdtr`.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/btdtribtdtri(a, b, p, out=None)

The `p`-th quantile of the beta distribution.

This function is the inverse of the beta cumulative distribution function,
`btdtr`, returning the value of `x` for which `btdtr(a, b, x) = p`, or

.. math::
    p = \int_0^x \frac{\Gamma(a + b)}{\Gamma(a)\Gamma(b)} t^{a-1} (1-t)^{b-1}\,dt

.. deprecated:: 1.12.0
    This function is deprecated and will be removed from SciPy 1.14.0.
    Use `scipy.special.betaincinv` instead.

Parameters
----------
a : array_like
    Shape parameter (`a` > 0).
b : array_like
    Shape parameter (`b` > 0).
p : array_like
    Cumulative probability, in [0, 1].
out : ndarray, optional
    Optional output array for the function values

Returns
-------
x : scalar or ndarray
    The quantile corresponding to `p`.

See Also
--------
betaincinv
btdtr

Notes
-----
The value of `x` is found by interval halving or Newton iterations.

Wrapper for the Cephes [1]_ routine `incbi`, which solves the equivalent
problem of finding the inverse of the incomplete beta integral.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/btdtria(p, b, x, out=None)

Inverse of `btdtr` with respect to `a`.

This is the inverse of the beta cumulative distribution function, `btdtr`,
considered as a function of `a`, returning the value of `a` for which
`btdtr(a, b, x) = p`, or

.. math::
    p = \int_0^x \frac{\Gamma(a + b)}{\Gamma(a)\Gamma(b)} t^{a-1} (1-t)^{b-1}\,dt

Parameters
----------
p : array_like
    Cumulative probability, in [0, 1].
b : array_like
    Shape parameter (`b` > 0).
x : array_like
    The quantile, in [0, 1].
out : ndarray, optional
    Optional output array for the function values

Returns
-------
a : scalar or ndarray
    The value of the shape parameter `a` such that `btdtr(a, b, x) = p`.

See Also
--------
btdtr : Cumulative distribution function of the beta distribution.
btdtri : Inverse with respect to `x`.
btdtrib : Inverse with respect to `b`.

Notes
-----
Wrapper for the CDFLIB [1]_ Fortran routine `cdfbet`.

The cumulative distribution function `p` is computed using a routine by
DiDinato and Morris [2]_. Computation of `a` involves a search for a value
that produces the desired value of `p`. The search relies on the
monotonicity of `p` with `a`.

References
----------
.. [1] Barry Brown, James Lovato, and Kathy Russell,
       CDFLIB: Library of Fortran Routines for Cumulative Distribution
       Functions, Inverses, and Other Parameters.
.. [2] DiDinato, A. R. and Morris, A. H.,
       Algorithm 708: Significant Digit Computation of the Incomplete Beta
       Function Ratios. ACM Trans. Math. Softw. 18 (1993), 360-373.btdtria(a, p, x, out=None)

Inverse of `btdtr` with respect to `b`.

This is the inverse of the beta cumulative distribution function, `btdtr`,
considered as a function of `b`, returning the value of `b` for which
`btdtr(a, b, x) = p`, or

.. math::
    p = \int_0^x \frac{\Gamma(a + b)}{\Gamma(a)\Gamma(b)} t^{a-1} (1-t)^{b-1}\,dt

Parameters
----------
a : array_like
    Shape parameter (`a` > 0).
p : array_like
    Cumulative probability, in [0, 1].
x : array_like
    The quantile, in [0, 1].
out : ndarray, optional
    Optional output array for the function values

Returns
-------
b : scalar or ndarray
    The value of the shape parameter `b` such that `btdtr(a, b, x) = p`.

See Also
--------
btdtr : Cumulative distribution function of the beta distribution.
btdtri : Inverse with respect to `x`.
btdtria : Inverse with respect to `a`.

Notes
-----
Wrapper for the CDFLIB [1]_ Fortran routine `cdfbet`.

The cumulative distribution function `p` is computed using a routine by
DiDinato and Morris [2]_. Computation of `b` involves a search for a value
that produces the desired value of `p`. The search relies on the
monotonicity of `p` with `b`.

References
----------
.. [1] Barry Brown, James Lovato, and Kathy Russell,
       CDFLIB: Library of Fortran Routines for Cumulative Distribution
       Functions, Inverses, and Other Parameters.
.. [2] DiDinato, A. R. and Morris, A. H.,
       Algorithm 708: Significant Digit Computation of the Incomplete Beta
       Function Ratios. ACM Trans. Math. Softw. 18 (1993), 360-373.cbrtcbrt(x, out=None)

Element-wise cube root of `x`.

Parameters
----------
x : array_like
    `x` must contain real numbers.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
scalar or ndarray
    The cube root of each value in `x`.

Examples
--------
>>> from scipy.special import cbrt

>>> cbrt(8)
2.0
>>> cbrt([-8, -3, 0.125, 1.331])
array([-2.        , -1.44224957,  0.5       ,  1.1       ])chdtrchdtr(v, x, out=None)

Chi square cumulative distribution function.

Returns the area under the left tail (from 0 to `x`) of the Chi
square probability density function with `v` degrees of freedom:

.. math::

    \frac{1}{2^{v/2} \Gamma(v/2)} \int_0^x t^{v/2 - 1} e^{-t/2} dt

Here :math:`\Gamma` is the Gamma function; see `gamma`. This
integral can be expressed in terms of the regularized lower
incomplete gamma function `gammainc` as
``gammainc(v / 2, x / 2)``. [1]_

Parameters
----------
v : array_like
    Degrees of freedom.
x : array_like
    Upper bound of the integral.
out : ndarray, optional
    Optional output array for the function results.

Returns
-------
scalar or ndarray
    Values of the cumulative distribution function.

See Also
--------
chdtrc, chdtri, chdtriv, gammainc

References
----------
.. [1] Chi-Square distribution,
    https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm

Examples
--------
>>> import numpy as np
>>> import scipy.special as sc

It can be expressed in terms of the regularized lower incomplete
gamma function.

>>> v = 1
>>> x = np.arange(4)
>>> sc.chdtr(v, x)
array([0.        , 0.68268949, 0.84270079, 0.91673548])
>>> sc.gammainc(v / 2, x / 2)
array([0.        , 0.68268949, 0.84270079, 0.91673548])chdtrcchdtrc(v, x, out=None)

Chi square survival function.

Returns the area under the right hand tail (from `x` to infinity)
of the Chi square probability density function with `v` degrees of
freedom:

.. math::

    \frac{1}{2^{v/2} \Gamma(v/2)} \int_x^\infty t^{v/2 - 1} e^{-t/2} dt

Here :math:`\Gamma` is the Gamma function; see `gamma`. This
integral can be expressed in terms of the regularized upper
incomplete gamma function `gammaincc` as
``gammaincc(v / 2, x / 2)``. [1]_

Parameters
----------
v : array_like
    Degrees of freedom.
x : array_like
    Lower bound of the integral.
out : ndarray, optional
    Optional output array for the function results.

Returns
-------
scalar or ndarray
    Values of the survival function.

See Also
--------
chdtr, chdtri, chdtriv, gammaincc

References
----------
.. [1] Chi-Square distribution,
    https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm

Examples
--------
>>> import numpy as np
>>> import scipy.special as sc

It can be expressed in terms of the regularized upper incomplete
gamma function.

>>> v = 1
>>> x = np.arange(4)
>>> sc.chdtrc(v, x)
array([1.        , 0.31731051, 0.15729921, 0.08326452])
>>> sc.gammaincc(v / 2, x / 2)
array([1.        , 0.31731051, 0.15729921, 0.08326452])chdtrichdtri(v, p, out=None)

Inverse to `chdtrc` with respect to `x`.

Returns `x` such that ``chdtrc(v, x) == p``.

Parameters
----------
v : array_like
    Degrees of freedom.
p : array_like
    Probability.
out : ndarray, optional
    Optional output array for the function results.

Returns
-------
x : scalar or ndarray
    Value so that the probability a Chi square random variable
    with `v` degrees of freedom is greater than `x` equals `p`.

See Also
--------
chdtrc, chdtr, chdtriv

References
----------
.. [1] Chi-Square distribution,
    https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm

Examples
--------
>>> import scipy.special as sc

It inverts `chdtrc`.

>>> v, p = 1, 0.3
>>> sc.chdtrc(v, sc.chdtri(v, p))
0.3
>>> x = 1
>>> sc.chdtri(v, sc.chdtrc(v, x))
1.0chdtriv(p, x, out=None)

Inverse to `chdtr` with respect to `v`.

Returns `v` such that ``chdtr(v, x) == p``.

Parameters
----------
p : array_like
    Probability that the Chi square random variable is less than
    or equal to `x`.
x : array_like
    Nonnegative input.
out : ndarray, optional
    Optional output array for the function results.

Returns
-------
scalar or ndarray
    Degrees of freedom.

See Also
--------
chdtr, chdtrc, chdtri

References
----------
.. [1] Chi-Square distribution,
    https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm

Examples
--------
>>> import scipy.special as sc

It inverts `chdtr`.

>>> p, x = 0.5, 1
>>> sc.chdtr(sc.chdtriv(p, x), x)
0.5000000000202172
>>> v = 1
>>> sc.chdtriv(sc.chdtr(v, x), v)
1.0000000000000013chndtr(x, df, nc, out=None)

Non-central chi square cumulative distribution function

The cumulative distribution function is given by:

.. math::

    P(\chi^{\prime 2} \vert \nu, \lambda) =\sum_{j=0}^{\infty}
    e^{-\lambda /2}
    \frac{(\lambda /2)^j}{j!} P(\chi^{\prime 2} \vert \nu + 2j),

where :math:`\nu > 0` is the degrees of freedom (``df``) and
:math:`\lambda \geq 0` is the non-centrality parameter (``nc``).

Parameters
----------
x : array_like
    Upper bound of the integral; must satisfy ``x >= 0``
df : array_like
    Degrees of freedom; must satisfy ``df > 0``
nc : array_like
    Non-centrality parameter; must satisfy ``nc >= 0``
out : ndarray, optional
    Optional output array for the function results

Returns
-------
x : scalar or ndarray
    Value of the non-central chi square cumulative distribution function.

See Also
--------
chndtrix, chndtridf, chndtrincchndtridf(x, p, nc, out=None)

Inverse to `chndtr` vs `df`

Calculated using a search to find a value for `df` that produces the
desired value of `p`.

Parameters
----------
x : array_like
    Upper bound of the integral; must satisfy ``x >= 0``
p : array_like
    Probability; must satisfy ``0 <= p < 1``
nc : array_like
    Non-centrality parameter; must satisfy ``nc >= 0``
out : ndarray, optional
    Optional output array for the function results

Returns
-------
df : scalar or ndarray
    Degrees of freedom

See Also
--------
chndtr, chndtrix, chndtrincchndtrinc(x, df, p, out=None)

Inverse to `chndtr` vs `nc`

Calculated using a search to find a value for `df` that produces the
desired value of `p`.

Parameters
----------
x : array_like
    Upper bound of the integral; must satisfy ``x >= 0``
df : array_like
    Degrees of freedom; must satisfy ``df > 0``
p : array_like
    Probability; must satisfy ``0 <= p < 1``
out : ndarray, optional
    Optional output array for the function results

Returns
-------
nc : scalar or ndarray
    Non-centrality

See Also
--------
chndtr, chndtrix, chndtrincchndtrix(p, df, nc, out=None)

Inverse to `chndtr` vs `x`

Calculated using a search to find a value for `x` that produces the
desired value of `p`.

Parameters
----------
p : array_like
    Probability; must satisfy ``0 <= p < 1``
df : array_like
    Degrees of freedom; must satisfy ``df > 0``
nc : array_like
    Non-centrality parameter; must satisfy ``nc >= 0``
out : ndarray, optional
    Optional output array for the function results

Returns
-------
x : scalar or ndarray
    Value so that the probability a non-central Chi square random variable
    with `df` degrees of freedom and non-centrality, `nc`, is greater than
    `x` equals `p`.

See Also
--------
chndtr, chndtridf, chndtrinccosdgcosdg(x, out=None)

Cosine of the angle `x` given in degrees.

Parameters
----------
x : array_like
    Angle, given in degrees.
out : ndarray, optional
    Optional output array for the function results.

Returns
-------
scalar or ndarray
    Cosine of the input.

See Also
--------
sindg, tandg, cotdg

Examples
--------
>>> import numpy as np
>>> import scipy.special as sc

It is more accurate than using cosine directly.

>>> x = 90 + 180 * np.arange(3)
>>> sc.cosdg(x)
array([-0.,  0., -0.])
>>> np.cos(x * np.pi / 180)
array([ 6.1232340e-17, -1.8369702e-16,  3.0616170e-16])cosm1cosm1(x, out=None)

cos(x) - 1 for use when `x` is near zero.

Parameters
----------
x : array_like
    Real valued argument.
out : ndarray, optional
    Optional output array for the function results.

Returns
-------
scalar or ndarray
    Values of ``cos(x) - 1``.

See Also
--------
expm1, log1p

Examples
--------
>>> import numpy as np
>>> import scipy.special as sc

It is more accurate than computing ``cos(x) - 1`` directly for
``x`` around 0.

>>> x = 1e-30
>>> np.cos(x) - 1
0.0
>>> sc.cosm1(x)
-5.0000000000000005e-61cotdgcotdg(x, out=None)

Cotangent of the angle `x` given in degrees.

Parameters
----------
x : array_like
    Angle, given in degrees.
out : ndarray, optional
    Optional output array for the function results.

Returns
-------
scalar or ndarray
    Cotangent at the input.

See Also
--------
sindg, cosdg, tandg

Examples
--------
>>> import numpy as np
>>> import scipy.special as sc

It is more accurate than using cotangent directly.

>>> x = 90 + 180 * np.arange(3)
>>> sc.cotdg(x)
array([0., 0., 0.])
>>> 1 / np.tan(x * np.pi / 180)
array([6.1232340e-17, 1.8369702e-16, 3.0616170e-16])dawsndawsn(x, out=None)

Dawson's integral.

Computes::

    exp(-x**2) * integral(exp(t**2), t=0..x).

Parameters
----------
x : array_like
    Function parameter.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
y : scalar or ndarray
    Value of the integral.

See Also
--------
wofz, erf, erfc, erfcx, erfi

References
----------
.. [1] Steven G. Johnson, Faddeeva W function implementation.
   http://ab-initio.mit.edu/Faddeeva

Examples
--------
>>> import numpy as np
>>> from scipy import special
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-15, 15, num=1000)
>>> plt.plot(x, special.dawsn(x))
>>> plt.xlabel('$x$')
>>> plt.ylabel('$dawsn(x)$')
>>> plt.show()ellipeellipe(m, out=None)

Complete elliptic integral of the second kind

This function is defined as

.. math:: E(m) = \int_0^{\pi/2} [1 - m \sin(t)^2]^{1/2} dt

Parameters
----------
m : array_like
    Defines the parameter of the elliptic integral.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
E : scalar or ndarray
    Value of the elliptic integral.

See Also
--------
ellipkm1 : Complete elliptic integral of the first kind, near `m` = 1
ellipk : Complete elliptic integral of the first kind
ellipkinc : Incomplete elliptic integral of the first kind
ellipeinc : Incomplete elliptic integral of the second kind
elliprd : Symmetric elliptic integral of the second kind.
elliprg : Completely-symmetric elliptic integral of the second kind.

Notes
-----
Wrapper for the Cephes [1]_ routine `ellpe`.

For `m > 0` the computation uses the approximation,

.. math:: E(m) \approx P(1-m) - (1-m) \log(1-m) Q(1-m),

where :math:`P` and :math:`Q` are tenth-order polynomials.  For
`m < 0`, the relation

.. math:: E(m) = E(m/(m - 1)) \sqrt(1-m)

is used.

The parameterization in terms of :math:`m` follows that of section
17.2 in [2]_. Other parameterizations in terms of the
complementary parameter :math:`1 - m`, modular angle
:math:`\sin^2(\alpha) = m`, or modulus :math:`k^2 = m` are also
used, so be careful that you choose the correct parameter.

The Legendre E integral is related to Carlson's symmetric R_D or R_G
functions in multiple ways [3]_. For example,

.. math:: E(m) = 2 R_G(0, 1-k^2, 1) .

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/
.. [2] Milton Abramowitz and Irene A. Stegun, eds.
       Handbook of Mathematical Functions with Formulas,
       Graphs, and Mathematical Tables. New York: Dover, 1972.
.. [3] NIST Digital Library of Mathematical
       Functions. http://dlmf.nist.gov/, Release 1.0.28 of
       2020-09-15. See Sec. 19.25(i) https://dlmf.nist.gov/19.25#i

Examples
--------
This function is used in finding the circumference of an
ellipse with semi-major axis `a` and semi-minor axis `b`.

>>> import numpy as np
>>> from scipy import special

>>> a = 3.5
>>> b = 2.1
>>> e_sq = 1.0 - b**2/a**2  # eccentricity squared

Then the circumference is found using the following:

>>> C = 4*a*special.ellipe(e_sq)  # circumference formula
>>> C
17.868899204378693

When `a` and `b` are the same (meaning eccentricity is 0),
this reduces to the circumference of a circle.

>>> 4*a*special.ellipe(0.0)  # formula for ellipse with a = b
21.991148575128552
>>> 2*np.pi*a  # formula for circle of radius a
21.991148575128552ellipeincellipeinc(phi, m, out=None)

Incomplete elliptic integral of the second kind

This function is defined as

.. math:: E(\phi, m) = \int_0^{\phi} [1 - m \sin(t)^2]^{1/2} dt

Parameters
----------
phi : array_like
    amplitude of the elliptic integral.
m : array_like
    parameter of the elliptic integral.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
E : scalar or ndarray
    Value of the elliptic integral.

See Also
--------
ellipkm1 : Complete elliptic integral of the first kind, near `m` = 1
ellipk : Complete elliptic integral of the first kind
ellipkinc : Incomplete elliptic integral of the first kind
ellipe : Complete elliptic integral of the second kind
elliprd : Symmetric elliptic integral of the second kind.
elliprf : Completely-symmetric elliptic integral of the first kind.
elliprg : Completely-symmetric elliptic integral of the second kind.

Notes
-----
Wrapper for the Cephes [1]_ routine `ellie`.

Computation uses arithmetic-geometric means algorithm.

The parameterization in terms of :math:`m` follows that of section
17.2 in [2]_. Other parameterizations in terms of the
complementary parameter :math:`1 - m`, modular angle
:math:`\sin^2(\alpha) = m`, or modulus :math:`k^2 = m` are also
used, so be careful that you choose the correct parameter.

The Legendre E incomplete integral can be related to combinations
of Carlson's symmetric integrals R_D, R_F, and R_G in multiple
ways [3]_. For example, with :math:`c = \csc^2\phi`,

.. math::
  E(\phi, m) = R_F(c-1, c-k^2, c)
    - \frac{1}{3} k^2 R_D(c-1, c-k^2, c) .

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/
.. [2] Milton Abramowitz and Irene A. Stegun, eds.
       Handbook of Mathematical Functions with Formulas,
       Graphs, and Mathematical Tables. New York: Dover, 1972.
.. [3] NIST Digital Library of Mathematical
       Functions. http://dlmf.nist.gov/, Release 1.0.28 of
       2020-09-15. See Sec. 19.25(i) https://dlmf.nist.gov/19.25#iellipjellipj(u, m, out=None)

Jacobian elliptic functions

Calculates the Jacobian elliptic functions of parameter `m` between
0 and 1, and real argument `u`.

Parameters
----------
m : array_like
    Parameter.
u : array_like
    Argument.
out : tuple of ndarray, optional
    Optional output arrays for the function values

Returns
-------
sn, cn, dn, ph : 4-tuple of scalar or ndarray
    The returned functions::

        sn(u|m), cn(u|m), dn(u|m)

    The value `ph` is such that if `u = ellipkinc(ph, m)`,
    then `sn(u|m) = sin(ph)` and `cn(u|m) = cos(ph)`.

See Also
--------
ellipk : Complete elliptic integral of the first kind
ellipkinc : Incomplete elliptic integral of the first kind

Notes
-----
Wrapper for the Cephes [1]_ routine `ellpj`.

These functions are periodic, with quarter-period on the real axis
equal to the complete elliptic integral `ellipk(m)`.

Relation to incomplete elliptic integral: If `u = ellipkinc(phi,m)`, then
`sn(u|m) = sin(phi)`, and `cn(u|m) = cos(phi)`. The `phi` is called
the amplitude of `u`.

Computation is by means of the arithmetic-geometric mean algorithm,
except when `m` is within 1e-9 of 0 or 1. In the latter case with `m`
close to 1, the approximation applies only for `phi < pi/2`.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/ellipkellipk(m, out=None)

Complete elliptic integral of the first kind.

This function is defined as

.. math:: K(m) = \int_0^{\pi/2} [1 - m \sin(t)^2]^{-1/2} dt

Parameters
----------
m : array_like
    The parameter of the elliptic integral.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
K : scalar or ndarray
    Value of the elliptic integral.

See Also
--------
ellipkm1 : Complete elliptic integral of the first kind around m = 1
ellipkinc : Incomplete elliptic integral of the first kind
ellipe : Complete elliptic integral of the second kind
ellipeinc : Incomplete elliptic integral of the second kind
elliprf : Completely-symmetric elliptic integral of the first kind.

Notes
-----
For more precision around point m = 1, use `ellipkm1`, which this
function calls.

The parameterization in terms of :math:`m` follows that of section
17.2 in [1]_. Other parameterizations in terms of the
complementary parameter :math:`1 - m`, modular angle
:math:`\sin^2(\alpha) = m`, or modulus :math:`k^2 = m` are also
used, so be careful that you choose the correct parameter.

The Legendre K integral is related to Carlson's symmetric R_F
function by [2]_:

.. math:: K(m) = R_F(0, 1-k^2, 1) .

References
----------
.. [1] Milton Abramowitz and Irene A. Stegun, eds.
       Handbook of Mathematical Functions with Formulas,
       Graphs, and Mathematical Tables. New York: Dover, 1972.
.. [2] NIST Digital Library of Mathematical
       Functions. http://dlmf.nist.gov/, Release 1.0.28 of
       2020-09-15. See Sec. 19.25(i) https://dlmf.nist.gov/19.25#iellipkincellipkinc(phi, m, out=None)

Incomplete elliptic integral of the first kind

This function is defined as

.. math:: K(\phi, m) = \int_0^{\phi} [1 - m \sin(t)^2]^{-1/2} dt

This function is also called :math:`F(\phi, m)`.

Parameters
----------
phi : array_like
    amplitude of the elliptic integral
m : array_like
    parameter of the elliptic integral
out : ndarray, optional
    Optional output array for the function values

Returns
-------
K : scalar or ndarray
    Value of the elliptic integral

See Also
--------
ellipkm1 : Complete elliptic integral of the first kind, near `m` = 1
ellipk : Complete elliptic integral of the first kind
ellipe : Complete elliptic integral of the second kind
ellipeinc : Incomplete elliptic integral of the second kind
elliprf : Completely-symmetric elliptic integral of the first kind.

Notes
-----
Wrapper for the Cephes [1]_ routine `ellik`.  The computation is
carried out using the arithmetic-geometric mean algorithm.

The parameterization in terms of :math:`m` follows that of section
17.2 in [2]_. Other parameterizations in terms of the
complementary parameter :math:`1 - m`, modular angle
:math:`\sin^2(\alpha) = m`, or modulus :math:`k^2 = m` are also
used, so be careful that you choose the correct parameter.

The Legendre K incomplete integral (or F integral) is related to
Carlson's symmetric R_F function [3]_.
Setting :math:`c = \csc^2\phi`,

.. math:: F(\phi, m) = R_F(c-1, c-k^2, c) .

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/
.. [2] Milton Abramowitz and Irene A. Stegun, eds.
       Handbook of Mathematical Functions with Formulas,
       Graphs, and Mathematical Tables. New York: Dover, 1972.
.. [3] NIST Digital Library of Mathematical
       Functions. http://dlmf.nist.gov/, Release 1.0.28 of
       2020-09-15. See Sec. 19.25(i) https://dlmf.nist.gov/19.25#iellipkm1ellipkm1(p, out=None)

Complete elliptic integral of the first kind around `m` = 1

This function is defined as

.. math:: K(p) = \int_0^{\pi/2} [1 - m \sin(t)^2]^{-1/2} dt

where `m = 1 - p`.

Parameters
----------
p : array_like
    Defines the parameter of the elliptic integral as `m = 1 - p`.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
K : scalar or ndarray
    Value of the elliptic integral.

See Also
--------
ellipk : Complete elliptic integral of the first kind
ellipkinc : Incomplete elliptic integral of the first kind
ellipe : Complete elliptic integral of the second kind
ellipeinc : Incomplete elliptic integral of the second kind
elliprf : Completely-symmetric elliptic integral of the first kind.

Notes
-----
Wrapper for the Cephes [1]_ routine `ellpk`.

For `p <= 1`, computation uses the approximation,

.. math:: K(p) \approx P(p) - \log(p) Q(p),

where :math:`P` and :math:`Q` are tenth-order polynomials.  The
argument `p` is used internally rather than `m` so that the logarithmic
singularity at `m = 1` will be shifted to the origin; this preserves
maximum accuracy.  For `p > 1`, the identity

.. math:: K(p) = K(1/p)/\sqrt(p)

is used.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/elliprcelliprc(x, y, out=None)

Degenerate symmetric elliptic integral.

The function RC is defined as [1]_

.. math::

    R_{\mathrm{C}}(x, y) =
       \frac{1}{2} \int_0^{+\infty} (t + x)^{-1/2} (t + y)^{-1} dt
       = R_{\mathrm{F}}(x, y, y)

Parameters
----------
x, y : array_like
    Real or complex input parameters. `x` can be any number in the
    complex plane cut along the negative real axis. `y` must be non-zero.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
R : scalar or ndarray
    Value of the integral. If `y` is real and negative, the Cauchy
    principal value is returned. If both of `x` and `y` are real, the
    return value is real. Otherwise, the return value is complex.

See Also
--------
elliprf : Completely-symmetric elliptic integral of the first kind.
elliprd : Symmetric elliptic integral of the second kind.
elliprg : Completely-symmetric elliptic integral of the second kind.
elliprj : Symmetric elliptic integral of the third kind.

Notes
-----
RC is a degenerate case of the symmetric integral RF: ``elliprc(x, y) ==
elliprf(x, y, y)``. It is an elementary function rather than an elliptic
integral.

The code implements Carlson's algorithm based on the duplication theorems
and series expansion up to the 7th order. [2]_

.. versionadded:: 1.8.0

References
----------
.. [1] B. C. Carlson, ed., Chapter 19 in "Digital Library of Mathematical
       Functions," NIST, US Dept. of Commerce.
       https://dlmf.nist.gov/19.16.E6
.. [2] B. C. Carlson, "Numerical computation of real or complex elliptic
       integrals," Numer. Algorithm, vol. 10, no. 1, pp. 13-26, 1995.
       https://arxiv.org/abs/math/9409227
       https://doi.org/10.1007/BF02198293

Examples
--------
Basic homogeneity property:

>>> import numpy as np
>>> from scipy.special import elliprc

>>> x = 1.2 + 3.4j
>>> y = 5.
>>> scale = 0.3 + 0.4j
>>> elliprc(scale*x, scale*y)
(0.5484493976710874-0.4169557678995833j)

>>> elliprc(x, y)/np.sqrt(scale)
(0.5484493976710874-0.41695576789958333j)

When the two arguments coincide, the integral is particularly
simple:

>>> x = 1.2 + 3.4j
>>> elliprc(x, x)
(0.4299173120614631-0.3041729818745595j)

>>> 1/np.sqrt(x)
(0.4299173120614631-0.30417298187455954j)

Another simple case: the first argument vanishes:

>>> y = 1.2 + 3.4j
>>> elliprc(0, y)
(0.6753125346116815-0.47779380263880866j)

>>> np.pi/2/np.sqrt(y)
(0.6753125346116815-0.4777938026388088j)

When `x` and `y` are both positive, we can express
:math:`R_C(x,y)` in terms of more elementary functions.  For the
case :math:`0 \le x < y`,

>>> x = 3.2
>>> y = 6.
>>> elliprc(x, y)
0.44942991498453444

>>> np.arctan(np.sqrt((y-x)/x))/np.sqrt(y-x)
0.44942991498453433

And for the case :math:`0 \le y < x`,

>>> x = 6.
>>> y = 3.2
>>> elliprc(x,y)
0.4989837501576147

>>> np.log((np.sqrt(x)+np.sqrt(x-y))/np.sqrt(y))/np.sqrt(x-y)
0.49898375015761476elliprdelliprd(x, y, z, out=None)

Symmetric elliptic integral of the second kind.

The function RD is defined as [1]_

.. math::

    R_{\mathrm{D}}(x, y, z) =
       \frac{3}{2} \int_0^{+\infty} [(t + x) (t + y)]^{-1/2} (t + z)^{-3/2}
       dt

Parameters
----------
x, y, z : array_like
    Real or complex input parameters. `x` or `y` can be any number in the
    complex plane cut along the negative real axis, but at most one of them
    can be zero, while `z` must be non-zero.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
R : scalar or ndarray
    Value of the integral. If all of `x`, `y`, and `z` are real, the
    return value is real. Otherwise, the return value is complex.

See Also
--------
elliprc : Degenerate symmetric elliptic integral.
elliprf : Completely-symmetric elliptic integral of the first kind.
elliprg : Completely-symmetric elliptic integral of the second kind.
elliprj : Symmetric elliptic integral of the third kind.

Notes
-----
RD is a degenerate case of the elliptic integral RJ: ``elliprd(x, y, z) ==
elliprj(x, y, z, z)``.

The code implements Carlson's algorithm based on the duplication theorems
and series expansion up to the 7th order. [2]_

.. versionadded:: 1.8.0

References
----------
.. [1] B. C. Carlson, ed., Chapter 19 in "Digital Library of Mathematical
       Functions," NIST, US Dept. of Commerce.
       https://dlmf.nist.gov/19.16.E5
.. [2] B. C. Carlson, "Numerical computation of real or complex elliptic
       integrals," Numer. Algorithm, vol. 10, no. 1, pp. 13-26, 1995.
       https://arxiv.org/abs/math/9409227
       https://doi.org/10.1007/BF02198293

Examples
--------
Basic homogeneity property:

>>> import numpy as np
>>> from scipy.special import elliprd

>>> x = 1.2 + 3.4j
>>> y = 5.
>>> z = 6.
>>> scale = 0.3 + 0.4j
>>> elliprd(scale*x, scale*y, scale*z)
(-0.03703043835680379-0.24500934665683802j)

>>> elliprd(x, y, z)*np.power(scale, -1.5)
(-0.0370304383568038-0.24500934665683805j)

All three arguments coincide:

>>> x = 1.2 + 3.4j
>>> elliprd(x, x, x)
(-0.03986825876151896-0.14051741840449586j)

>>> np.power(x, -1.5)
(-0.03986825876151894-0.14051741840449583j)

The so-called "second lemniscate constant":

>>> elliprd(0, 2, 1)/3
0.5990701173677961

>>> from scipy.special import gamma
>>> gamma(0.75)**2/np.sqrt(2*np.pi)
0.5990701173677959elliprfelliprf(x, y, z, out=None)

Completely-symmetric elliptic integral of the first kind.

The function RF is defined as [1]_

.. math::

    R_{\mathrm{F}}(x, y, z) =
       \frac{1}{2} \int_0^{+\infty} [(t + x) (t + y) (t + z)]^{-1/2} dt

Parameters
----------
x, y, z : array_like
    Real or complex input parameters. `x`, `y`, or `z` can be any number in
    the complex plane cut along the negative real axis, but at most one of
    them can be zero.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
R : scalar or ndarray
    Value of the integral. If all of `x`, `y`, and `z` are real, the return
    value is real. Otherwise, the return value is complex.

See Also
--------
elliprc : Degenerate symmetric integral.
elliprd : Symmetric elliptic integral of the second kind.
elliprg : Completely-symmetric elliptic integral of the second kind.
elliprj : Symmetric elliptic integral of the third kind.

Notes
-----
The code implements Carlson's algorithm based on the duplication theorems
and series expansion up to the 7th order (cf.:
https://dlmf.nist.gov/19.36.i) and the AGM algorithm for the complete
integral. [2]_

.. versionadded:: 1.8.0

References
----------
.. [1] B. C. Carlson, ed., Chapter 19 in "Digital Library of Mathematical
       Functions," NIST, US Dept. of Commerce.
       https://dlmf.nist.gov/19.16.E1
.. [2] B. C. Carlson, "Numerical computation of real or complex elliptic
       integrals," Numer. Algorithm, vol. 10, no. 1, pp. 13-26, 1995.
       https://arxiv.org/abs/math/9409227
       https://doi.org/10.1007/BF02198293

Examples
--------
Basic homogeneity property:

>>> import numpy as np
>>> from scipy.special import elliprf

>>> x = 1.2 + 3.4j
>>> y = 5.
>>> z = 6.
>>> scale = 0.3 + 0.4j
>>> elliprf(scale*x, scale*y, scale*z)
(0.5328051227278146-0.4008623567957094j)

>>> elliprf(x, y, z)/np.sqrt(scale)
(0.5328051227278147-0.4008623567957095j)

All three arguments coincide:

>>> x = 1.2 + 3.4j
>>> elliprf(x, x, x)
(0.42991731206146316-0.30417298187455954j)

>>> 1/np.sqrt(x)
(0.4299173120614631-0.30417298187455954j)

The so-called "first lemniscate constant":

>>> elliprf(0, 1, 2)
1.3110287771460598

>>> from scipy.special import gamma
>>> gamma(0.25)**2/(4*np.sqrt(2*np.pi))
1.3110287771460598elliprgelliprg(x, y, z, out=None)

Completely-symmetric elliptic integral of the second kind.

The function RG is defined as [1]_

.. math::

    R_{\mathrm{G}}(x, y, z) =
       \frac{1}{4} \int_0^{+\infty} [(t + x) (t + y) (t + z)]^{-1/2}
       \left(\frac{x}{t + x} + \frac{y}{t + y} + \frac{z}{t + z}\right) t
       dt

Parameters
----------
x, y, z : array_like
    Real or complex input parameters. `x`, `y`, or `z` can be any number in
    the complex plane cut along the negative real axis.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
R : scalar or ndarray
    Value of the integral. If all of `x`, `y`, and `z` are real, the return
    value is real. Otherwise, the return value is complex.

See Also
--------
elliprc : Degenerate symmetric integral.
elliprd : Symmetric elliptic integral of the second kind.
elliprf : Completely-symmetric elliptic integral of the first kind.
elliprj : Symmetric elliptic integral of the third kind.

Notes
-----
The implementation uses the relation [1]_

.. math::

    2 R_{\mathrm{G}}(x, y, z) =
       z R_{\mathrm{F}}(x, y, z) -
       \frac{1}{3} (x - z) (y - z) R_{\mathrm{D}}(x, y, z) +
       \sqrt{\frac{x y}{z}}

and the symmetry of `x`, `y`, `z` when at least one non-zero parameter can
be chosen as the pivot. When one of the arguments is close to zero, the AGM
method is applied instead. Other special cases are computed following Ref.
[2]_

.. versionadded:: 1.8.0

References
----------
.. [1] B. C. Carlson, "Numerical computation of real or complex elliptic
       integrals," Numer. Algorithm, vol. 10, no. 1, pp. 13-26, 1995.
       https://arxiv.org/abs/math/9409227
       https://doi.org/10.1007/BF02198293
.. [2] B. C. Carlson, ed., Chapter 19 in "Digital Library of Mathematical
       Functions," NIST, US Dept. of Commerce.
       https://dlmf.nist.gov/19.16.E1
       https://dlmf.nist.gov/19.20.ii

Examples
--------
Basic homogeneity property:

>>> import numpy as np
>>> from scipy.special import elliprg

>>> x = 1.2 + 3.4j
>>> y = 5.
>>> z = 6.
>>> scale = 0.3 + 0.4j
>>> elliprg(scale*x, scale*y, scale*z)
(1.195936862005246+0.8470988320464167j)

>>> elliprg(x, y, z)*np.sqrt(scale)
(1.195936862005246+0.8470988320464165j)

Simplifications:

>>> elliprg(0, y, y)
1.756203682760182

>>> 0.25*np.pi*np.sqrt(y)
1.7562036827601817

>>> elliprg(0, 0, z)
1.224744871391589

>>> 0.5*np.sqrt(z)
1.224744871391589

The surface area of a triaxial ellipsoid with semiaxes ``a``, ``b``, and
``c`` is given by

.. math::

    S = 4 \pi a b c R_{\mathrm{G}}(1 / a^2, 1 / b^2, 1 / c^2).

>>> def ellipsoid_area(a, b, c):
...     r = 4.0 * np.pi * a * b * c
...     return r * elliprg(1.0 / (a * a), 1.0 / (b * b), 1.0 / (c * c))
>>> print(ellipsoid_area(1, 3, 5))
108.62688289491807elliprjelliprj(x, y, z, p, out=None)

Symmetric elliptic integral of the third kind.

The function RJ is defined as [1]_

.. math::

    R_{\mathrm{J}}(x, y, z, p) =
       \frac{3}{2} \int_0^{+\infty} [(t + x) (t + y) (t + z)]^{-1/2}
       (t + p)^{-1} dt

.. warning::
    This function should be considered experimental when the inputs are
    unbalanced.  Check correctness with another independent implementation.

Parameters
----------
x, y, z, p : array_like
    Real or complex input parameters. `x`, `y`, or `z` are numbers in
    the complex plane cut along the negative real axis (subject to further
    constraints, see Notes), and at most one of them can be zero. `p` must
    be non-zero.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
R : scalar or ndarray
    Value of the integral. If all of `x`, `y`, `z`, and `p` are real, the
    return value is real. Otherwise, the return value is complex.

    If `p` is real and negative, while `x`, `y`, and `z` are real,
    non-negative, and at most one of them is zero, the Cauchy principal
    value is returned. [1]_ [2]_

See Also
--------
elliprc : Degenerate symmetric integral.
elliprd : Symmetric elliptic integral of the second kind.
elliprf : Completely-symmetric elliptic integral of the first kind.
elliprg : Completely-symmetric elliptic integral of the second kind.

Notes
-----
The code implements Carlson's algorithm based on the duplication theorems
and series expansion up to the 7th order. [3]_ The algorithm is slightly
different from its earlier incarnation as it appears in [1]_, in that the
call to `elliprc` (or ``atan``/``atanh``, see [4]_) is no longer needed in
the inner loop. Asymptotic approximations are used where arguments differ
widely in the order of magnitude. [5]_

The input values are subject to certain sufficient but not necessary
constraints when input arguments are complex. Notably, ``x``, ``y``, and
``z`` must have non-negative real parts, unless two of them are
non-negative and complex-conjugates to each other while the other is a real
non-negative number. [1]_ If the inputs do not satisfy the sufficient
condition described in Ref. [1]_ they are rejected outright with the output
set to NaN.

In the case where one of ``x``, ``y``, and ``z`` is equal to ``p``, the
function ``elliprd`` should be preferred because of its less restrictive
domain.

.. versionadded:: 1.8.0

References
----------
.. [1] B. C. Carlson, "Numerical computation of real or complex elliptic
       integrals," Numer. Algorithm, vol. 10, no. 1, pp. 13-26, 1995.
       https://arxiv.org/abs/math/9409227
       https://doi.org/10.1007/BF02198293
.. [2] B. C. Carlson, ed., Chapter 19 in "Digital Library of Mathematical
       Functions," NIST, US Dept. of Commerce.
       https://dlmf.nist.gov/19.20.iii
.. [3] B. C. Carlson, J. FitzSimmons, "Reduction Theorems for Elliptic
       Integrands with the Square Root of Two Quadratic Factors," J.
       Comput. Appl. Math., vol. 118, nos. 1-2, pp. 71-85, 2000.
       https://doi.org/10.1016/S0377-0427(00)00282-X
.. [4] F. Johansson, "Numerical Evaluation of Elliptic Functions, Elliptic
       Integrals and Modular Forms," in J. Blumlein, C. Schneider, P.
       Paule, eds., "Elliptic Integrals, Elliptic Functions and Modular
       Forms in Quantum Field Theory," pp. 269-293, 2019 (Cham,
       Switzerland: Springer Nature Switzerland)
       https://arxiv.org/abs/1806.06725
       https://doi.org/10.1007/978-3-030-04480-0
.. [5] B. C. Carlson, J. L. Gustafson, "Asymptotic Approximations for
       Symmetric Elliptic Integrals," SIAM J. Math. Anls., vol. 25, no. 2,
       pp. 288-303, 1994.
       https://arxiv.org/abs/math/9310223
       https://doi.org/10.1137/S0036141092228477

Examples
--------
Basic homogeneity property:

>>> import numpy as np
>>> from scipy.special import elliprj

>>> x = 1.2 + 3.4j
>>> y = 5.
>>> z = 6.
>>> p = 7.
>>> scale = 0.3 - 0.4j
>>> elliprj(scale*x, scale*y, scale*z, scale*p)
(0.10834905565679157+0.19694950747103812j)

>>> elliprj(x, y, z, p)*np.power(scale, -1.5)
(0.10834905565679556+0.19694950747103854j)

Reduction to simpler elliptic integral:

>>> elliprj(x, y, z, z)
(0.08288462362195129-0.028376809745123258j)

>>> from scipy.special import elliprd
>>> elliprd(x, y, z)
(0.08288462362195136-0.028376809745123296j)

All arguments coincide:

>>> elliprj(x, x, x, x)
(-0.03986825876151896-0.14051741840449586j)

>>> np.power(x, -1.5)
(-0.03986825876151894-0.14051741840449583j)entrentr(x, out=None)

Elementwise function for computing entropy.

.. math:: \text{entr}(x) = \begin{cases} - x \log(x) & x > 0  \\ 0 & x = 0
          \\ -\infty & \text{otherwise} \end{cases}

Parameters
----------
x : ndarray
    Input array.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
res : scalar or ndarray
    The value of the elementwise entropy function at the given points `x`.

See Also
--------
kl_div, rel_entr, scipy.stats.entropy

Notes
-----
.. versionadded:: 0.15.0

This function is concave.

The origin of this function is in convex programming; see [1]_.
Given a probability distribution :math:`p_1, \ldots, p_n`,
the definition of entropy in the context of *information theory* is

.. math::

    \sum_{i = 1}^n \mathrm{entr}(p_i).

To compute the latter quantity, use `scipy.stats.entropy`.

References
----------
.. [1] Boyd, Stephen and Lieven Vandenberghe. *Convex optimization*.
       Cambridge University Press, 2004.
       :doi:`https://doi.org/10.1017/CBO9780511804441`erferf(z, out=None)

Returns the error function of complex argument.

It is defined as ``2/sqrt(pi)*integral(exp(-t**2), t=0..z)``.

Parameters
----------
x : ndarray
    Input array.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
res : scalar or ndarray
    The values of the error function at the given points `x`.

See Also
--------
erfc, erfinv, erfcinv, wofz, erfcx, erfi

Notes
-----
The cumulative of the unit normal distribution is given by
``Phi(z) = 1/2[1 + erf(z/sqrt(2))]``.

References
----------
.. [1] https://en.wikipedia.org/wiki/Error_function
.. [2] Milton Abramowitz and Irene A. Stegun, eds.
    Handbook of Mathematical Functions with Formulas,
    Graphs, and Mathematical Tables. New York: Dover,
    1972. http://www.math.sfu.ca/~cbm/aands/page_297.htm
.. [3] Steven G. Johnson, Faddeeva W function implementation.
   http://ab-initio.mit.edu/Faddeeva

Examples
--------
>>> import numpy as np
>>> from scipy import special
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-3, 3)
>>> plt.plot(x, special.erf(x))
>>> plt.xlabel('$x$')
>>> plt.ylabel('$erf(x)$')
>>> plt.show()erfcerfc(x, out=None)

Complementary error function, ``1 - erf(x)``.

Parameters
----------
x : array_like
    Real or complex valued argument
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    Values of the complementary error function

See Also
--------
erf, erfi, erfcx, dawsn, wofz

References
----------
.. [1] Steven G. Johnson, Faddeeva W function implementation.
   http://ab-initio.mit.edu/Faddeeva

Examples
--------
>>> import numpy as np
>>> from scipy import special
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-3, 3)
>>> plt.plot(x, special.erfc(x))
>>> plt.xlabel('$x$')
>>> plt.ylabel('$erfc(x)$')
>>> plt.show()erfcinverfcinv(y, out=None)

Inverse of the complementary error function.

Computes the inverse of the complementary error function.

In the complex domain, there is no unique complex number w satisfying
erfc(w)=z. This indicates a true inverse function would be multivalued.
When the domain restricts to the real, 0 < x < 2, there is a unique real
number satisfying erfc(erfcinv(x)) = erfcinv(erfc(x)).

It is related to inverse of the error function by erfcinv(1-x) = erfinv(x)

Parameters
----------
y : ndarray
    Argument at which to evaluate. Domain: [0, 2]
out : ndarray, optional
    Optional output array for the function values

Returns
-------
erfcinv : scalar or ndarray
    The inverse of erfc of y, element-wise

See Also
--------
erf : Error function of a complex argument
erfc : Complementary error function, ``1 - erf(x)``
erfinv : Inverse of the error function

Examples
--------
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.special import erfcinv

>>> erfcinv(0.5)
0.4769362762044699

>>> y = np.linspace(0.0, 2.0, num=11)
>>> erfcinv(y)
array([        inf,  0.9061938 ,  0.59511608,  0.37080716,  0.17914345,
       -0.        , -0.17914345, -0.37080716, -0.59511608, -0.9061938 ,
              -inf])

Plot the function:

>>> y = np.linspace(0, 2, 200)
>>> fig, ax = plt.subplots()
>>> ax.plot(y, erfcinv(y))
>>> ax.grid(True)
>>> ax.set_xlabel('y')
>>> ax.set_title('erfcinv(y)')
>>> plt.show()erfcxerfcx(x, out=None)

Scaled complementary error function, ``exp(x**2) * erfc(x)``.

Parameters
----------
x : array_like
    Real or complex valued argument
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    Values of the scaled complementary error function


See Also
--------
erf, erfc, erfi, dawsn, wofz

Notes
-----

.. versionadded:: 0.12.0

References
----------
.. [1] Steven G. Johnson, Faddeeva W function implementation.
   http://ab-initio.mit.edu/Faddeeva

Examples
--------
>>> import numpy as np
>>> from scipy import special
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-3, 3)
>>> plt.plot(x, special.erfcx(x))
>>> plt.xlabel('$x$')
>>> plt.ylabel('$erfcx(x)$')
>>> plt.show()erfierfi(z, out=None)

Imaginary error function, ``-i erf(i z)``.

Parameters
----------
z : array_like
    Real or complex valued argument
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    Values of the imaginary error function

See Also
--------
erf, erfc, erfcx, dawsn, wofz

Notes
-----

.. versionadded:: 0.12.0

References
----------
.. [1] Steven G. Johnson, Faddeeva W function implementation.
   http://ab-initio.mit.edu/Faddeeva

Examples
--------
>>> import numpy as np
>>> from scipy import special
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-3, 3)
>>> plt.plot(x, special.erfi(x))
>>> plt.xlabel('$x$')
>>> plt.ylabel('$erfi(x)$')
>>> plt.show()erfinverfinv(y, out=None)

Inverse of the error function.

Computes the inverse of the error function.

In the complex domain, there is no unique complex number w satisfying
erf(w)=z. This indicates a true inverse function would be multivalued.
When the domain restricts to the real, -1 < x < 1, there is a unique real
number satisfying erf(erfinv(x)) = x.

Parameters
----------
y : ndarray
    Argument at which to evaluate. Domain: [-1, 1]
out : ndarray, optional
    Optional output array for the function values

Returns
-------
erfinv : scalar or ndarray
    The inverse of erf of y, element-wise

See Also
--------
erf : Error function of a complex argument
erfc : Complementary error function, ``1 - erf(x)``
erfcinv : Inverse of the complementary error function

Examples
--------
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.special import erfinv, erf

>>> erfinv(0.5)
0.4769362762044699

>>> y = np.linspace(-1.0, 1.0, num=9)
>>> x = erfinv(y)
>>> x
array([       -inf, -0.81341985, -0.47693628, -0.22531206,  0.        ,
        0.22531206,  0.47693628,  0.81341985,         inf])

Verify that ``erf(erfinv(y))`` is ``y``.

>>> erf(x)
array([-1.  , -0.75, -0.5 , -0.25,  0.  ,  0.25,  0.5 ,  0.75,  1.  ])

Plot the function:

>>> y = np.linspace(-1, 1, 200)
>>> fig, ax = plt.subplots()
>>> ax.plot(y, erfinv(y))
>>> ax.grid(True)
>>> ax.set_xlabel('y')
>>> ax.set_title('erfinv(y)')
>>> plt.show()eval_chebyceval_chebyc(n, x, out=None)

Evaluate Chebyshev polynomial of the first kind on [-2, 2] at a
point.

These polynomials are defined as

.. math::

    C_n(x) = 2 T_n(x/2)

where :math:`T_n` is a Chebyshev polynomial of the first kind. See
22.5.11 in [AS]_ for details.

Parameters
----------
n : array_like
    Degree of the polynomial. If not an integer, the result is
    determined via the relation to `eval_chebyt`.
x : array_like
    Points at which to evaluate the Chebyshev polynomial
out : ndarray, optional
    Optional output array for the function values

Returns
-------
C : scalar or ndarray
    Values of the Chebyshev polynomial

See Also
--------
roots_chebyc : roots and quadrature weights of Chebyshev
               polynomials of the first kind on [-2, 2]
chebyc : Chebyshev polynomial object
numpy.polynomial.chebyshev.Chebyshev : Chebyshev series
eval_chebyt : evaluate Chebycshev polynomials of the first kind

References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
    Handbook of Mathematical Functions with Formulas,
    Graphs, and Mathematical Tables. New York: Dover, 1972.

Examples
--------
>>> import numpy as np
>>> import scipy.special as sc

They are a scaled version of the Chebyshev polynomials of the
first kind.

>>> x = np.linspace(-2, 2, 6)
>>> sc.eval_chebyc(3, x)
array([-2.   ,  1.872,  1.136, -1.136, -1.872,  2.   ])
>>> 2 * sc.eval_chebyt(3, x / 2)
array([-2.   ,  1.872,  1.136, -1.136, -1.872,  2.   ])eval_chebyseval_chebys(n, x, out=None)

Evaluate Chebyshev polynomial of the second kind on [-2, 2] at a
point.

These polynomials are defined as

.. math::

    S_n(x) = U_n(x/2)

where :math:`U_n` is a Chebyshev polynomial of the second
kind. See 22.5.13 in [AS]_ for details.

Parameters
----------
n : array_like
    Degree of the polynomial. If not an integer, the result is
    determined via the relation to `eval_chebyu`.
x : array_like
    Points at which to evaluate the Chebyshev polynomial
out : ndarray, optional
    Optional output array for the function values

Returns
-------
S : scalar or ndarray
    Values of the Chebyshev polynomial

See Also
--------
roots_chebys : roots and quadrature weights of Chebyshev
               polynomials of the second kind on [-2, 2]
chebys : Chebyshev polynomial object
eval_chebyu : evaluate Chebyshev polynomials of the second kind

References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
    Handbook of Mathematical Functions with Formulas,
    Graphs, and Mathematical Tables. New York: Dover, 1972.

Examples
--------
>>> import numpy as np
>>> import scipy.special as sc

They are a scaled version of the Chebyshev polynomials of the
second kind.

>>> x = np.linspace(-2, 2, 6)
>>> sc.eval_chebys(3, x)
array([-4.   ,  0.672,  0.736, -0.736, -0.672,  4.   ])
>>> sc.eval_chebyu(3, x / 2)
array([-4.   ,  0.672,  0.736, -0.736, -0.672,  4.   ])eval_chebyteval_chebyt(n, x, out=None)

Evaluate Chebyshev polynomial of the first kind at a point.

The Chebyshev polynomials of the first kind can be defined via the
Gauss hypergeometric function :math:`{}_2F_1` as

.. math::

    T_n(x) = {}_2F_1(n, -n; 1/2; (1 - x)/2).

When :math:`n` is an integer the result is a polynomial of degree
:math:`n`. See 22.5.47 in [AS]_ for details.

Parameters
----------
n : array_like
    Degree of the polynomial. If not an integer, the result is
    determined via the relation to the Gauss hypergeometric
    function.
x : array_like
    Points at which to evaluate the Chebyshev polynomial
out : ndarray, optional
    Optional output array for the function values

Returns
-------
T : scalar or ndarray
    Values of the Chebyshev polynomial

See Also
--------
roots_chebyt : roots and quadrature weights of Chebyshev
               polynomials of the first kind
chebyu : Chebychev polynomial object
eval_chebyu : evaluate Chebyshev polynomials of the second kind
hyp2f1 : Gauss hypergeometric function
numpy.polynomial.chebyshev.Chebyshev : Chebyshev series

Notes
-----
This routine is numerically stable for `x` in ``[-1, 1]`` at least
up to order ``10000``.

References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
    Handbook of Mathematical Functions with Formulas,
    Graphs, and Mathematical Tables. New York: Dover, 1972.eval_chebyueval_chebyu(n, x, out=None)

Evaluate Chebyshev polynomial of the second kind at a point.

The Chebyshev polynomials of the second kind can be defined via
the Gauss hypergeometric function :math:`{}_2F_1` as

.. math::

    U_n(x) = (n + 1) {}_2F_1(-n, n + 2; 3/2; (1 - x)/2).

When :math:`n` is an integer the result is a polynomial of degree
:math:`n`. See 22.5.48 in [AS]_ for details.

Parameters
----------
n : array_like
    Degree of the polynomial. If not an integer, the result is
    determined via the relation to the Gauss hypergeometric
    function.
x : array_like
    Points at which to evaluate the Chebyshev polynomial
out : ndarray, optional
    Optional output array for the function values

Returns
-------
U : scalar or ndarray
    Values of the Chebyshev polynomial

See Also
--------
roots_chebyu : roots and quadrature weights of Chebyshev
               polynomials of the second kind
chebyu : Chebyshev polynomial object
eval_chebyt : evaluate Chebyshev polynomials of the first kind
hyp2f1 : Gauss hypergeometric function

References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
    Handbook of Mathematical Functions with Formulas,
    Graphs, and Mathematical Tables. New York: Dover, 1972.eval_gegenbauereval_gegenbauer(n, alpha, x, out=None)

Evaluate Gegenbauer polynomial at a point.

The Gegenbauer polynomials can be defined via the Gauss
hypergeometric function :math:`{}_2F_1` as

.. math::

    C_n^{(\alpha)} = \frac{(2\alpha)_n}{\Gamma(n + 1)}
      {}_2F_1(-n, 2\alpha + n; \alpha + 1/2; (1 - z)/2).

When :math:`n` is an integer the result is a polynomial of degree
:math:`n`. See 22.5.46 in [AS]_ for details.

Parameters
----------
n : array_like
    Degree of the polynomial. If not an integer, the result is
    determined via the relation to the Gauss hypergeometric
    function.
alpha : array_like
    Parameter
x : array_like
    Points at which to evaluate the Gegenbauer polynomial
out : ndarray, optional
    Optional output array for the function values

Returns
-------
C : scalar or ndarray
    Values of the Gegenbauer polynomial

See Also
--------
roots_gegenbauer : roots and quadrature weights of Gegenbauer
                   polynomials
gegenbauer : Gegenbauer polynomial object
hyp2f1 : Gauss hypergeometric function

References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
    Handbook of Mathematical Functions with Formulas,
    Graphs, and Mathematical Tables. New York: Dover, 1972.eval_genlaguerre(n, alpha, x, out=None)

Evaluate generalized Laguerre polynomial at a point.

The generalized Laguerre polynomials can be defined via the
confluent hypergeometric function :math:`{}_1F_1` as

.. math::

    L_n^{(\alpha)}(x) = \binom{n + \alpha}{n}
      {}_1F_1(-n, \alpha + 1, x).

When :math:`n` is an integer the result is a polynomial of degree
:math:`n`. See 22.5.54 in [AS]_ for details. The Laguerre
polynomials are the special case where :math:`\alpha = 0`.

Parameters
----------
n : array_like
    Degree of the polynomial. If not an integer, the result is
    determined via the relation to the confluent hypergeometric
    function.
alpha : array_like
    Parameter; must have ``alpha > -1``
x : array_like
    Points at which to evaluate the generalized Laguerre
    polynomial
out : ndarray, optional
    Optional output array for the function values

Returns
-------
L : scalar or ndarray
    Values of the generalized Laguerre polynomial

See Also
--------
roots_genlaguerre : roots and quadrature weights of generalized
                    Laguerre polynomials
genlaguerre : generalized Laguerre polynomial object
hyp1f1 : confluent hypergeometric function
eval_laguerre : evaluate Laguerre polynomials

References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
    Handbook of Mathematical Functions with Formulas,
    Graphs, and Mathematical Tables. New York: Dover, 1972.eval_hermite(n, x, out=None)

Evaluate physicist's Hermite polynomial at a point.

Defined by

.. math::

    H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2};

:math:`H_n` is a polynomial of degree :math:`n`. See 22.11.7 in
[AS]_ for details.

Parameters
----------
n : array_like
    Degree of the polynomial
x : array_like
    Points at which to evaluate the Hermite polynomial
out : ndarray, optional
    Optional output array for the function values

Returns
-------
H : scalar or ndarray
    Values of the Hermite polynomial

See Also
--------
roots_hermite : roots and quadrature weights of physicist's
                Hermite polynomials
hermite : physicist's Hermite polynomial object
numpy.polynomial.hermite.Hermite : Physicist's Hermite series
eval_hermitenorm : evaluate Probabilist's Hermite polynomials

References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
    Handbook of Mathematical Functions with Formulas,
    Graphs, and Mathematical Tables. New York: Dover, 1972.eval_hermitenorm(n, x, out=None)

Evaluate probabilist's (normalized) Hermite polynomial at a
point.

Defined by

.. math::

    He_n(x) = (-1)^n e^{x^2/2} \frac{d^n}{dx^n} e^{-x^2/2};

:math:`He_n` is a polynomial of degree :math:`n`. See 22.11.8 in
[AS]_ for details.

Parameters
----------
n : array_like
    Degree of the polynomial
x : array_like
    Points at which to evaluate the Hermite polynomial
out : ndarray, optional
    Optional output array for the function values

Returns
-------
He : scalar or ndarray
    Values of the Hermite polynomial

See Also
--------
roots_hermitenorm : roots and quadrature weights of probabilist's
                    Hermite polynomials
hermitenorm : probabilist's Hermite polynomial object
numpy.polynomial.hermite_e.HermiteE : Probabilist's Hermite series
eval_hermite : evaluate physicist's Hermite polynomials

References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
    Handbook of Mathematical Functions with Formulas,
    Graphs, and Mathematical Tables. New York: Dover, 1972.eval_jacobieval_jacobi(n, alpha, beta, x, out=None)

Evaluate Jacobi polynomial at a point.

The Jacobi polynomials can be defined via the Gauss hypergeometric
function :math:`{}_2F_1` as

.. math::

    P_n^{(\alpha, \beta)}(x) = \frac{(\alpha + 1)_n}{\Gamma(n + 1)}
      {}_2F_1(-n, 1 + \alpha + \beta + n; \alpha + 1; (1 - z)/2)

where :math:`(\cdot)_n` is the Pochhammer symbol; see `poch`. When
:math:`n` is an integer the result is a polynomial of degree
:math:`n`. See 22.5.42 in [AS]_ for details.

Parameters
----------
n : array_like
    Degree of the polynomial. If not an integer the result is
    determined via the relation to the Gauss hypergeometric
    function.
alpha : array_like
    Parameter
beta : array_like
    Parameter
x : array_like
    Points at which to evaluate the polynomial
out : ndarray, optional
    Optional output array for the function values

Returns
-------
P : scalar or ndarray
    Values of the Jacobi polynomial

See Also
--------
roots_jacobi : roots and quadrature weights of Jacobi polynomials
jacobi : Jacobi polynomial object
hyp2f1 : Gauss hypergeometric function

References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
    Handbook of Mathematical Functions with Formulas,
    Graphs, and Mathematical Tables. New York: Dover, 1972.eval_laguerreeval_laguerre(n, x, out=None)

Evaluate Laguerre polynomial at a point.

The Laguerre polynomials can be defined via the confluent
hypergeometric function :math:`{}_1F_1` as

.. math::

    L_n(x) = {}_1F_1(-n, 1, x).

See 22.5.16 and 22.5.54 in [AS]_ for details. When :math:`n` is an
integer the result is a polynomial of degree :math:`n`.

Parameters
----------
n : array_like
    Degree of the polynomial. If not an integer the result is
    determined via the relation to the confluent hypergeometric
    function.
x : array_like
    Points at which to evaluate the Laguerre polynomial
out : ndarray, optional
    Optional output array for the function values

Returns
-------
L : scalar or ndarray
    Values of the Laguerre polynomial

See Also
--------
roots_laguerre : roots and quadrature weights of Laguerre
                 polynomials
laguerre : Laguerre polynomial object
numpy.polynomial.laguerre.Laguerre : Laguerre series
eval_genlaguerre : evaluate generalized Laguerre polynomials

References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
    Handbook of Mathematical Functions with Formulas,
    Graphs, and Mathematical Tables. New York: Dover, 1972.eval_legendreeval_legendre(n, x, out=None)

Evaluate Legendre polynomial at a point.

The Legendre polynomials can be defined via the Gauss
hypergeometric function :math:`{}_2F_1` as

.. math::

    P_n(x) = {}_2F_1(-n, n + 1; 1; (1 - x)/2).

When :math:`n` is an integer the result is a polynomial of degree
:math:`n`. See 22.5.49 in [AS]_ for details.

Parameters
----------
n : array_like
    Degree of the polynomial. If not an integer, the result is
    determined via the relation to the Gauss hypergeometric
    function.
x : array_like
    Points at which to evaluate the Legendre polynomial
out : ndarray, optional
    Optional output array for the function values

Returns
-------
P : scalar or ndarray
    Values of the Legendre polynomial

See Also
--------
roots_legendre : roots and quadrature weights of Legendre
                 polynomials
legendre : Legendre polynomial object
hyp2f1 : Gauss hypergeometric function
numpy.polynomial.legendre.Legendre : Legendre series

References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
    Handbook of Mathematical Functions with Formulas,
    Graphs, and Mathematical Tables. New York: Dover, 1972.

Examples
--------
>>> import numpy as np
>>> from scipy.special import eval_legendre

Evaluate the zero-order Legendre polynomial at x = 0

>>> eval_legendre(0, 0)
1.0

Evaluate the first-order Legendre polynomial between -1 and 1

>>> X = np.linspace(-1, 1, 5)  # Domain of Legendre polynomials
>>> eval_legendre(1, X)
array([-1. , -0.5,  0. ,  0.5,  1. ])

Evaluate Legendre polynomials of order 0 through 4 at x = 0

>>> N = range(0, 5)
>>> eval_legendre(N, 0)
array([ 1.   ,  0.   , -0.5  ,  0.   ,  0.375])

Plot Legendre polynomials of order 0 through 4

>>> X = np.linspace(-1, 1)

>>> import matplotlib.pyplot as plt
>>> for n in range(0, 5):
...     y = eval_legendre(n, X)
...     plt.plot(X, y, label=r'$P_{}(x)$'.format(n))

>>> plt.title("Legendre Polynomials")
>>> plt.xlabel("x")
>>> plt.ylabel(r'$P_n(x)$')
>>> plt.legend(loc='lower right')
>>> plt.show()eval_sh_chebyteval_sh_chebyt(n, x, out=None)

Evaluate shifted Chebyshev polynomial of the first kind at a
point.

These polynomials are defined as

.. math::

    T_n^*(x) = T_n(2x - 1)

where :math:`T_n` is a Chebyshev polynomial of the first kind. See
22.5.14 in [AS]_ for details.

Parameters
----------
n : array_like
    Degree of the polynomial. If not an integer, the result is
    determined via the relation to `eval_chebyt`.
x : array_like
    Points at which to evaluate the shifted Chebyshev polynomial
out : ndarray, optional
    Optional output array for the function values

Returns
-------
T : scalar or ndarray
    Values of the shifted Chebyshev polynomial

See Also
--------
roots_sh_chebyt : roots and quadrature weights of shifted
                  Chebyshev polynomials of the first kind
sh_chebyt : shifted Chebyshev polynomial object
eval_chebyt : evaluate Chebyshev polynomials of the first kind
numpy.polynomial.chebyshev.Chebyshev : Chebyshev series

References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
    Handbook of Mathematical Functions with Formulas,
    Graphs, and Mathematical Tables. New York: Dover, 1972.eval_sh_chebyueval_sh_chebyu(n, x, out=None)

Evaluate shifted Chebyshev polynomial of the second kind at a
point.

These polynomials are defined as

.. math::

    U_n^*(x) = U_n(2x - 1)

where :math:`U_n` is a Chebyshev polynomial of the first kind. See
22.5.15 in [AS]_ for details.

Parameters
----------
n : array_like
    Degree of the polynomial. If not an integer, the result is
    determined via the relation to `eval_chebyu`.
x : array_like
    Points at which to evaluate the shifted Chebyshev polynomial
out : ndarray, optional
    Optional output array for the function values

Returns
-------
U : scalar or ndarray
    Values of the shifted Chebyshev polynomial

See Also
--------
roots_sh_chebyu : roots and quadrature weights of shifted
                  Chebychev polynomials of the second kind
sh_chebyu : shifted Chebyshev polynomial object
eval_chebyu : evaluate Chebyshev polynomials of the second kind

References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
    Handbook of Mathematical Functions with Formulas,
    Graphs, and Mathematical Tables. New York: Dover, 1972.eval_sh_jacobieval_sh_jacobi(n, p, q, x, out=None)

Evaluate shifted Jacobi polynomial at a point.

Defined by

.. math::

    G_n^{(p, q)}(x)
      = \binom{2n + p - 1}{n}^{-1} P_n^{(p - q, q - 1)}(2x - 1),

where :math:`P_n^{(\cdot, \cdot)}` is the n-th Jacobi
polynomial. See 22.5.2 in [AS]_ for details.

Parameters
----------
n : int
    Degree of the polynomial. If not an integer, the result is
    determined via the relation to `binom` and `eval_jacobi`.
p : float
    Parameter
q : float
    Parameter
out : ndarray, optional
    Optional output array for the function values

Returns
-------
G : scalar or ndarray
    Values of the shifted Jacobi polynomial.

See Also
--------
roots_sh_jacobi : roots and quadrature weights of shifted Jacobi
                  polynomials
sh_jacobi : shifted Jacobi polynomial object
eval_jacobi : evaluate Jacobi polynomials

References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
    Handbook of Mathematical Functions with Formulas,
    Graphs, and Mathematical Tables. New York: Dover, 1972.eval_sh_legendreeval_sh_legendre(n, x, out=None)

Evaluate shifted Legendre polynomial at a point.

These polynomials are defined as

.. math::

    P_n^*(x) = P_n(2x - 1)

where :math:`P_n` is a Legendre polynomial. See 2.2.11 in [AS]_
for details.

Parameters
----------
n : array_like
    Degree of the polynomial. If not an integer, the value is
    determined via the relation to `eval_legendre`.
x : array_like
    Points at which to evaluate the shifted Legendre polynomial
out : ndarray, optional
    Optional output array for the function values

Returns
-------
P : scalar or ndarray
    Values of the shifted Legendre polynomial

See Also
--------
roots_sh_legendre : roots and quadrature weights of shifted
                    Legendre polynomials
sh_legendre : shifted Legendre polynomial object
eval_legendre : evaluate Legendre polynomials
numpy.polynomial.legendre.Legendre : Legendre series

References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
    Handbook of Mathematical Functions with Formulas,
    Graphs, and Mathematical Tables. New York: Dover, 1972.exp1exp1(z, out=None)

Exponential integral E1.

For complex :math:`z \ne 0` the exponential integral can be defined as
[1]_

.. math::

   E_1(z) = \int_z^\infty \frac{e^{-t}}{t} dt,

where the path of the integral does not cross the negative real
axis or pass through the origin.

Parameters
----------
z: array_like
    Real or complex argument.
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    Values of the exponential integral E1

See Also
--------
expi : exponential integral :math:`Ei`
expn : generalization of :math:`E_1`

Notes
-----
For :math:`x > 0` it is related to the exponential integral
:math:`Ei` (see `expi`) via the relation

.. math::

   E_1(x) = -Ei(-x).

References
----------
.. [1] Digital Library of Mathematical Functions, 6.2.1
       https://dlmf.nist.gov/6.2#E1

Examples
--------
>>> import numpy as np
>>> import scipy.special as sc

It has a pole at 0.

>>> sc.exp1(0)
inf

It has a branch cut on the negative real axis.

>>> sc.exp1(-1)
nan
>>> sc.exp1(complex(-1, 0))
(-1.8951178163559368-3.141592653589793j)
>>> sc.exp1(complex(-1, -0.0))
(-1.8951178163559368+3.141592653589793j)

It approaches 0 along the positive real axis.

>>> sc.exp1([1, 10, 100, 1000])
array([2.19383934e-01, 4.15696893e-06, 3.68359776e-46, 0.00000000e+00])

It is related to `expi`.

>>> x = np.array([1, 2, 3, 4])
>>> sc.exp1(x)
array([0.21938393, 0.04890051, 0.01304838, 0.00377935])
>>> -sc.expi(-x)
array([0.21938393, 0.04890051, 0.01304838, 0.00377935])exp10exp10(x, out=None)

Compute ``10**x`` element-wise.

Parameters
----------
x : array_like
    `x` must contain real numbers.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
scalar or ndarray
    ``10**x``, computed element-wise.

Examples
--------
>>> import numpy as np
>>> from scipy.special import exp10

>>> exp10(3)
1000.0
>>> x = np.array([[-1, -0.5, 0], [0.5, 1, 1.5]])
>>> exp10(x)
array([[  0.1       ,   0.31622777,   1.        ],
       [  3.16227766,  10.        ,  31.6227766 ]])exp2exp2(x, out=None)

Compute ``2**x`` element-wise.

Parameters
----------
x : array_like
    `x` must contain real numbers.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
scalar or ndarray
    ``2**x``, computed element-wise.

Examples
--------
>>> import numpy as np
>>> from scipy.special import exp2

>>> exp2(3)
8.0
>>> x = np.array([[-1, -0.5, 0], [0.5, 1, 1.5]])
>>> exp2(x)
array([[ 0.5       ,  0.70710678,  1.        ],
       [ 1.41421356,  2.        ,  2.82842712]])expiexpi(x, out=None)

Exponential integral Ei.

For real :math:`x`, the exponential integral is defined as [1]_

.. math::

    Ei(x) = \int_{-\infty}^x \frac{e^t}{t} dt.

For :math:`x > 0` the integral is understood as a Cauchy principal
value.

It is extended to the complex plane by analytic continuation of
the function on the interval :math:`(0, \infty)`. The complex
variant has a branch cut on the negative real axis.

Parameters
----------
x : array_like
    Real or complex valued argument
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    Values of the exponential integral

See Also
--------
exp1 : Exponential integral :math:`E_1`
expn : Generalized exponential integral :math:`E_n`

Notes
-----
The exponential integrals :math:`E_1` and :math:`Ei` satisfy the
relation

.. math::

    E_1(x) = -Ei(-x)

for :math:`x > 0`.

References
----------
.. [1] Digital Library of Mathematical Functions, 6.2.5
       https://dlmf.nist.gov/6.2#E5

Examples
--------
>>> import numpy as np
>>> import scipy.special as sc

It is related to `exp1`.

>>> x = np.array([1, 2, 3, 4])
>>> -sc.expi(-x)
array([0.21938393, 0.04890051, 0.01304838, 0.00377935])
>>> sc.exp1(x)
array([0.21938393, 0.04890051, 0.01304838, 0.00377935])

The complex variant has a branch cut on the negative real axis.

>>> sc.expi(-1 + 1e-12j)
(-0.21938393439552062+3.1415926535894254j)
>>> sc.expi(-1 - 1e-12j)
(-0.21938393439552062-3.1415926535894254j)

As the complex variant approaches the branch cut, the real parts
approach the value of the real variant.

>>> sc.expi(-1)
-0.21938393439552062

The SciPy implementation returns the real variant for complex
values on the branch cut.

>>> sc.expi(complex(-1, 0.0))
(-0.21938393439552062-0j)
>>> sc.expi(complex(-1, -0.0))
(-0.21938393439552062-0j)expitexpit(x, out=None)

Expit (a.k.a. logistic sigmoid) ufunc for ndarrays.

The expit function, also known as the logistic sigmoid function, is
defined as ``expit(x) = 1/(1+exp(-x))``.  It is the inverse of the
logit function.

Parameters
----------
x : ndarray
    The ndarray to apply expit to element-wise.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
scalar or ndarray
    An ndarray of the same shape as x. Its entries
    are `expit` of the corresponding entry of x.

See Also
--------
logit

Notes
-----
As a ufunc expit takes a number of optional
keyword arguments. For more information
see `ufuncs <https://docs.scipy.org/doc/numpy/reference/ufuncs.html>`_

.. versionadded:: 0.10.0

Examples
--------
>>> import numpy as np
>>> from scipy.special import expit, logit

>>> expit([-np.inf, -1.5, 0, 1.5, np.inf])
array([ 0.        ,  0.18242552,  0.5       ,  0.81757448,  1.        ])

`logit` is the inverse of `expit`:

>>> logit(expit([-2.5, 0, 3.1, 5.0]))
array([-2.5,  0. ,  3.1,  5. ])

Plot expit(x) for x in [-6, 6]:

>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-6, 6, 121)
>>> y = expit(x)
>>> plt.plot(x, y)
>>> plt.grid()
>>> plt.xlim(-6, 6)
>>> plt.xlabel('x')
>>> plt.title('expit(x)')
>>> plt.show()expm1expm1(x, out=None)

Compute ``exp(x) - 1``.

When `x` is near zero, ``exp(x)`` is near 1, so the numerical calculation
of ``exp(x) - 1`` can suffer from catastrophic loss of precision.
``expm1(x)`` is implemented to avoid the loss of precision that occurs when
`x` is near zero.

Parameters
----------
x : array_like
    `x` must contain real numbers.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
scalar or ndarray
    ``exp(x) - 1`` computed element-wise.

Examples
--------
>>> import numpy as np
>>> from scipy.special import expm1

>>> expm1(1.0)
1.7182818284590451
>>> expm1([-0.2, -0.1, 0, 0.1, 0.2])
array([-0.18126925, -0.09516258,  0.        ,  0.10517092,  0.22140276])

The exact value of ``exp(7.5e-13) - 1`` is::

    7.5000000000028125000000007031250000001318...*10**-13.

Here is what ``expm1(7.5e-13)`` gives:

>>> expm1(7.5e-13)
7.5000000000028135e-13

Compare that to ``exp(7.5e-13) - 1``, where the subtraction results in
a "catastrophic" loss of precision:

>>> np.exp(7.5e-13) - 1
7.5006667543675576e-13expnexpn(n, x, out=None)

Generalized exponential integral En.

For integer :math:`n \geq 0` and real :math:`x \geq 0` the
generalized exponential integral is defined as [dlmf]_

.. math::

    E_n(x) = x^{n - 1} \int_x^\infty \frac{e^{-t}}{t^n} dt.

Parameters
----------
n : array_like
    Non-negative integers
x : array_like
    Real argument
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    Values of the generalized exponential integral

See Also
--------
exp1 : special case of :math:`E_n` for :math:`n = 1`
expi : related to :math:`E_n` when :math:`n = 1`

References
----------
.. [dlmf] Digital Library of Mathematical Functions, 8.19.2
          https://dlmf.nist.gov/8.19#E2

Examples
--------
>>> import numpy as np
>>> import scipy.special as sc

Its domain is nonnegative n and x.

>>> sc.expn(-1, 1.0), sc.expn(1, -1.0)
(nan, nan)

It has a pole at ``x = 0`` for ``n = 1, 2``; for larger ``n`` it
is equal to ``1 / (n - 1)``.

>>> sc.expn([0, 1, 2, 3, 4], 0)
array([       inf,        inf, 1.        , 0.5       , 0.33333333])

For n equal to 0 it reduces to ``exp(-x) / x``.

>>> x = np.array([1, 2, 3, 4])
>>> sc.expn(0, x)
array([0.36787944, 0.06766764, 0.01659569, 0.00457891])
>>> np.exp(-x) / x
array([0.36787944, 0.06766764, 0.01659569, 0.00457891])

For n equal to 1 it reduces to `exp1`.

>>> sc.expn(1, x)
array([0.21938393, 0.04890051, 0.01304838, 0.00377935])
>>> sc.exp1(x)
array([0.21938393, 0.04890051, 0.01304838, 0.00377935])exprelexprel(x, out=None)

Relative error exponential, ``(exp(x) - 1)/x``.

When `x` is near zero, ``exp(x)`` is near 1, so the numerical calculation
of ``exp(x) - 1`` can suffer from catastrophic loss of precision.
``exprel(x)`` is implemented to avoid the loss of precision that occurs when
`x` is near zero.

Parameters
----------
x : ndarray
    Input array.  `x` must contain real numbers.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
scalar or ndarray
    ``(exp(x) - 1)/x``, computed element-wise.

See Also
--------
expm1

Notes
-----
.. versionadded:: 0.17.0

Examples
--------
>>> import numpy as np
>>> from scipy.special import exprel

>>> exprel(0.01)
1.0050167084168056
>>> exprel([-0.25, -0.1, 0, 0.1, 0.25])
array([ 0.88479687,  0.95162582,  1.        ,  1.05170918,  1.13610167])

Compare ``exprel(5e-9)`` to the naive calculation.  The exact value
is ``1.00000000250000000416...``.

>>> exprel(5e-9)
1.0000000025

>>> (np.exp(5e-9) - 1)/5e-9
0.99999999392252903fdtrfdtr(dfn, dfd, x, out=None)

F cumulative distribution function.

Returns the value of the cumulative distribution function of the
F-distribution, also known as Snedecor's F-distribution or the
Fisher-Snedecor distribution.

The F-distribution with parameters :math:`d_n` and :math:`d_d` is the
distribution of the random variable,

.. math::
    X = \frac{U_n/d_n}{U_d/d_d},

where :math:`U_n` and :math:`U_d` are random variables distributed
:math:`\chi^2`, with :math:`d_n` and :math:`d_d` degrees of freedom,
respectively.

Parameters
----------
dfn : array_like
    First parameter (positive float).
dfd : array_like
    Second parameter (positive float).
x : array_like
    Argument (nonnegative float).
out : ndarray, optional
    Optional output array for the function values

Returns
-------
y : scalar or ndarray
    The CDF of the F-distribution with parameters `dfn` and `dfd` at `x`.

See Also
--------
fdtrc : F distribution survival function
fdtri : F distribution inverse cumulative distribution
scipy.stats.f : F distribution

Notes
-----
The regularized incomplete beta function is used, according to the
formula,

.. math::
    F(d_n, d_d; x) = I_{xd_n/(d_d + xd_n)}(d_n/2, d_d/2).

Wrapper for the Cephes [1]_ routine `fdtr`. The F distribution is also
available as `scipy.stats.f`. Calling `fdtr` directly can improve
performance compared to the ``cdf`` method of `scipy.stats.f` (see last
example below).

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/

Examples
--------
Calculate the function for ``dfn=1`` and ``dfd=2`` at ``x=1``.

>>> import numpy as np
>>> from scipy.special import fdtr
>>> fdtr(1, 2, 1)
0.5773502691896258

Calculate the function at several points by providing a NumPy array for
`x`.

>>> x = np.array([0.5, 2., 3.])
>>> fdtr(1, 2, x)
array([0.4472136 , 0.70710678, 0.77459667])

Plot the function for several parameter sets.

>>> import matplotlib.pyplot as plt
>>> dfn_parameters = [1, 5, 10, 50]
>>> dfd_parameters = [1, 1, 2, 3]
>>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']
>>> parameters_list = list(zip(dfn_parameters, dfd_parameters,
...                            linestyles))
>>> x = np.linspace(0, 30, 1000)
>>> fig, ax = plt.subplots()
>>> for parameter_set in parameters_list:
...     dfn, dfd, style = parameter_set
...     fdtr_vals = fdtr(dfn, dfd, x)
...     ax.plot(x, fdtr_vals, label=rf"$d_n={dfn},\, d_d={dfd}$",
...             ls=style)
>>> ax.legend()
>>> ax.set_xlabel("$x$")
>>> ax.set_title("F distribution cumulative distribution function")
>>> plt.show()

The F distribution is also available as `scipy.stats.f`. Using `fdtr`
directly can be much faster than calling the ``cdf`` method of
`scipy.stats.f`, especially for small arrays or individual values.
To get the same results one must use the following parametrization:
``stats.f(dfn, dfd).cdf(x)=fdtr(dfn, dfd, x)``.

>>> from scipy.stats import f
>>> dfn, dfd = 1, 2
>>> x = 1
>>> fdtr_res = fdtr(dfn, dfd, x)  # this will often be faster than below
>>> f_dist_res = f(dfn, dfd).cdf(x)
>>> fdtr_res == f_dist_res  # test that results are equal
Truefdtrcfdtrc(dfn, dfd, x, out=None)

F survival function.

Returns the complemented F-distribution function (the integral of the
density from `x` to infinity).

Parameters
----------
dfn : array_like
    First parameter (positive float).
dfd : array_like
    Second parameter (positive float).
x : array_like
    Argument (nonnegative float).
out : ndarray, optional
    Optional output array for the function values

Returns
-------
y : scalar or ndarray
    The complemented F-distribution function with parameters `dfn` and
    `dfd` at `x`.

See Also
--------
fdtr : F distribution cumulative distribution function
fdtri : F distribution inverse cumulative distribution function
scipy.stats.f : F distribution

Notes
-----
The regularized incomplete beta function is used, according to the
formula,

.. math::
    F(d_n, d_d; x) = I_{d_d/(d_d + xd_n)}(d_d/2, d_n/2).

Wrapper for the Cephes [1]_ routine `fdtrc`. The F distribution is also
available as `scipy.stats.f`. Calling `fdtrc` directly can improve
performance compared to the ``sf`` method of `scipy.stats.f` (see last
example below).

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/

Examples
--------
Calculate the function for ``dfn=1`` and ``dfd=2`` at ``x=1``.

>>> import numpy as np
>>> from scipy.special import fdtrc
>>> fdtrc(1, 2, 1)
0.42264973081037427

Calculate the function at several points by providing a NumPy array for
`x`.

>>> x = np.array([0.5, 2., 3.])
>>> fdtrc(1, 2, x)
array([0.5527864 , 0.29289322, 0.22540333])

Plot the function for several parameter sets.

>>> import matplotlib.pyplot as plt
>>> dfn_parameters = [1, 5, 10, 50]
>>> dfd_parameters = [1, 1, 2, 3]
>>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']
>>> parameters_list = list(zip(dfn_parameters, dfd_parameters,
...                            linestyles))
>>> x = np.linspace(0, 30, 1000)
>>> fig, ax = plt.subplots()
>>> for parameter_set in parameters_list:
...     dfn, dfd, style = parameter_set
...     fdtrc_vals = fdtrc(dfn, dfd, x)
...     ax.plot(x, fdtrc_vals, label=rf"$d_n={dfn},\, d_d={dfd}$",
...             ls=style)
>>> ax.legend()
>>> ax.set_xlabel("$x$")
>>> ax.set_title("F distribution survival function")
>>> plt.show()

The F distribution is also available as `scipy.stats.f`. Using `fdtrc`
directly can be much faster than calling the ``sf`` method of
`scipy.stats.f`, especially for small arrays or individual values.
To get the same results one must use the following parametrization:
``stats.f(dfn, dfd).sf(x)=fdtrc(dfn, dfd, x)``.

>>> from scipy.stats import f
>>> dfn, dfd = 1, 2
>>> x = 1
>>> fdtrc_res = fdtrc(dfn, dfd, x)  # this will often be faster than below
>>> f_dist_res = f(dfn, dfd).sf(x)
>>> f_dist_res == fdtrc_res  # test that results are equal
Truefdtrifdtri(dfn, dfd, p, out=None)

The `p`-th quantile of the F-distribution.

This function is the inverse of the F-distribution CDF, `fdtr`, returning
the `x` such that `fdtr(dfn, dfd, x) = p`.

Parameters
----------
dfn : array_like
    First parameter (positive float).
dfd : array_like
    Second parameter (positive float).
p : array_like
    Cumulative probability, in [0, 1].
out : ndarray, optional
    Optional output array for the function values

Returns
-------
x : scalar or ndarray
    The quantile corresponding to `p`.

See Also
--------
fdtr : F distribution cumulative distribution function
fdtrc : F distribution survival function
scipy.stats.f : F distribution

Notes
-----
The computation is carried out using the relation to the inverse
regularized beta function, :math:`I^{-1}_x(a, b)`.  Let
:math:`z = I^{-1}_p(d_d/2, d_n/2).`  Then,

.. math::
    x = \frac{d_d (1 - z)}{d_n z}.

If `p` is such that :math:`x < 0.5`, the following relation is used
instead for improved stability: let
:math:`z' = I^{-1}_{1 - p}(d_n/2, d_d/2).` Then,

.. math::
    x = \frac{d_d z'}{d_n (1 - z')}.

Wrapper for the Cephes [1]_ routine `fdtri`.

The F distribution is also available as `scipy.stats.f`. Calling
`fdtri` directly can improve performance compared to the ``ppf``
method of `scipy.stats.f` (see last example below).

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/

Examples
--------
`fdtri` represents the inverse of the F distribution CDF which is
available as `fdtr`. Here, we calculate the CDF for ``df1=1``, ``df2=2``
at ``x=3``. `fdtri` then returns ``3`` given the same values for `df1`,
`df2` and the computed CDF value.

>>> import numpy as np
>>> from scipy.special import fdtri, fdtr
>>> df1, df2 = 1, 2
>>> x = 3
>>> cdf_value =  fdtr(df1, df2, x)
>>> fdtri(df1, df2, cdf_value)
3.000000000000006

Calculate the function at several points by providing a NumPy array for
`x`.

>>> x = np.array([0.1, 0.4, 0.7])
>>> fdtri(1, 2, x)
array([0.02020202, 0.38095238, 1.92156863])

Plot the function for several parameter sets.

>>> import matplotlib.pyplot as plt
>>> dfn_parameters = [50, 10, 1, 50]
>>> dfd_parameters = [0.5, 1, 1, 5]
>>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']
>>> parameters_list = list(zip(dfn_parameters, dfd_parameters,
...                            linestyles))
>>> x = np.linspace(0, 1, 1000)
>>> fig, ax = plt.subplots()
>>> for parameter_set in parameters_list:
...     dfn, dfd, style = parameter_set
...     fdtri_vals = fdtri(dfn, dfd, x)
...     ax.plot(x, fdtri_vals, label=rf"$d_n={dfn},\, d_d={dfd}$",
...             ls=style)
>>> ax.legend()
>>> ax.set_xlabel("$x$")
>>> title = "F distribution inverse cumulative distribution function"
>>> ax.set_title(title)
>>> ax.set_ylim(0, 30)
>>> plt.show()

The F distribution is also available as `scipy.stats.f`. Using `fdtri`
directly can be much faster than calling the ``ppf`` method of
`scipy.stats.f`, especially for small arrays or individual values.
To get the same results one must use the following parametrization:
``stats.f(dfn, dfd).ppf(x)=fdtri(dfn, dfd, x)``.

>>> from scipy.stats import f
>>> dfn, dfd = 1, 2
>>> x = 0.7
>>> fdtri_res = fdtri(dfn, dfd, x)  # this will often be faster than below
>>> f_dist_res = f(dfn, dfd).ppf(x)
>>> f_dist_res == fdtri_res  # test that results are equal
Truefdtridfd(dfn, p, x, out=None)

Inverse to `fdtr` vs dfd

Finds the F density argument dfd such that ``fdtr(dfn, dfd, x) == p``.

Parameters
----------
dfn : array_like
    First parameter (positive float).
p : array_like
    Cumulative probability, in [0, 1].
x : array_like
    Argument (nonnegative float).
out : ndarray, optional
    Optional output array for the function values

Returns
-------
dfd : scalar or ndarray
    `dfd` such that ``fdtr(dfn, dfd, x) == p``.

See Also
--------
fdtr : F distribution cumulative distribution function
fdtrc : F distribution survival function
fdtri : F distribution quantile function
scipy.stats.f : F distribution

Examples
--------
Compute the F distribution cumulative distribution function for one
parameter set.

>>> from scipy.special import fdtridfd, fdtr
>>> dfn, dfd, x = 10, 5, 2
>>> cdf_value = fdtr(dfn, dfd, x)
>>> cdf_value
0.7700248806501017

Verify that `fdtridfd` recovers the original value for `dfd`:

>>> fdtridfd(dfn, cdf_value, x)
5.0fresnelfresnel(z, out=None)

Fresnel integrals.

The Fresnel integrals are defined as

.. math::

   S(z) &= \int_0^z \sin(\pi t^2 /2) dt \\
   C(z) &= \int_0^z \cos(\pi t^2 /2) dt.

See [dlmf]_ for details.

Parameters
----------
z : array_like
    Real or complex valued argument
out : 2-tuple of ndarrays, optional
    Optional output arrays for the function results

Returns
-------
S, C : 2-tuple of scalar or ndarray
    Values of the Fresnel integrals

See Also
--------
fresnel_zeros : zeros of the Fresnel integrals

References
----------
.. [dlmf] NIST Digital Library of Mathematical Functions
          https://dlmf.nist.gov/7.2#iii

Examples
--------
>>> import numpy as np
>>> import scipy.special as sc

As z goes to infinity along the real axis, S and C converge to 0.5.

>>> S, C = sc.fresnel([0.1, 1, 10, 100, np.inf])
>>> S
array([0.00052359, 0.43825915, 0.46816998, 0.4968169 , 0.5       ])
>>> C
array([0.09999753, 0.7798934 , 0.49989869, 0.4999999 , 0.5       ])

They are related to the error function `erf`.

>>> z = np.array([1, 2, 3, 4])
>>> zeta = 0.5 * np.sqrt(np.pi) * (1 - 1j) * z
>>> S, C = sc.fresnel(z)
>>> C + 1j*S
array([0.7798934 +0.43825915j, 0.48825341+0.34341568j,
       0.60572079+0.496313j  , 0.49842603+0.42051575j])
>>> 0.5 * (1 + 1j) * sc.erf(zeta)
array([0.7798934 +0.43825915j, 0.48825341+0.34341568j,
       0.60572079+0.496313j  , 0.49842603+0.42051575j])gammagamma(z, out=None)

gamma function.

The gamma function is defined as

.. math::

   \Gamma(z) = \int_0^\infty t^{z-1} e^{-t} dt

for :math:`\Re(z) > 0` and is extended to the rest of the complex
plane by analytic continuation. See [dlmf]_ for more details.

Parameters
----------
z : array_like
    Real or complex valued argument
out : ndarray, optional
    Optional output array for the function values

Returns
-------
scalar or ndarray
    Values of the gamma function

Notes
-----
The gamma function is often referred to as the generalized
factorial since :math:`\Gamma(n + 1) = n!` for natural numbers
:math:`n`. More generally it satisfies the recurrence relation
:math:`\Gamma(z + 1) = z \cdot \Gamma(z)` for complex :math:`z`,
which, combined with the fact that :math:`\Gamma(1) = 1`, implies
the above identity for :math:`z = n`.

References
----------
.. [dlmf] NIST Digital Library of Mathematical Functions
          https://dlmf.nist.gov/5.2#E1

Examples
--------
>>> import numpy as np
>>> from scipy.special import gamma, factorial

>>> gamma([0, 0.5, 1, 5])
array([         inf,   1.77245385,   1.        ,  24.        ])

>>> z = 2.5 + 1j
>>> gamma(z)
(0.77476210455108352+0.70763120437959293j)
>>> gamma(z+1), z*gamma(z)  # Recurrence property
((1.2292740569981171+2.5438401155000685j),
 (1.2292740569981158+2.5438401155000658j))

>>> gamma(0.5)**2  # gamma(0.5) = sqrt(pi)
3.1415926535897927

Plot gamma(x) for real x

>>> x = np.linspace(-3.5, 5.5, 2251)
>>> y = gamma(x)

>>> import matplotlib.pyplot as plt
>>> plt.plot(x, y, 'b', alpha=0.6, label='gamma(x)')
>>> k = np.arange(1, 7)
>>> plt.plot(k, factorial(k-1), 'k*', alpha=0.6,
...          label='(x-1)!, x = 1, 2, ...')
>>> plt.xlim(-3.5, 5.5)
>>> plt.ylim(-10, 25)
>>> plt.grid()
>>> plt.xlabel('x')
>>> plt.legend(loc='lower right')
>>> plt.show()gammaincgammainc(a, x, out=None)

Regularized lower incomplete gamma function.

It is defined as

.. math::

    P(a, x) = \frac{1}{\Gamma(a)} \int_0^x t^{a - 1}e^{-t} dt

for :math:`a > 0` and :math:`x \geq 0`. See [dlmf]_ for details.

Parameters
----------
a : array_like
    Positive parameter
x : array_like
    Nonnegative argument
out : ndarray, optional
    Optional output array for the function values

Returns
-------
scalar or ndarray
    Values of the lower incomplete gamma function

See Also
--------
gammaincc : regularized upper incomplete gamma function
gammaincinv : inverse of the regularized lower incomplete gamma function
gammainccinv : inverse of the regularized upper incomplete gamma function

Notes
-----
The function satisfies the relation ``gammainc(a, x) +
gammaincc(a, x) = 1`` where `gammaincc` is the regularized upper
incomplete gamma function.

The implementation largely follows that of [boost]_.

References
----------
.. [dlmf] NIST Digital Library of Mathematical functions
          https://dlmf.nist.gov/8.2#E4
.. [boost] Maddock et. al., "Incomplete Gamma Functions",
   https://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html

Examples
--------
>>> import scipy.special as sc

It is the CDF of the gamma distribution, so it starts at 0 and
monotonically increases to 1.

>>> sc.gammainc(0.5, [0, 1, 10, 100])
array([0.        , 0.84270079, 0.99999226, 1.        ])

It is equal to one minus the upper incomplete gamma function.

>>> a, x = 0.5, 0.4
>>> sc.gammainc(a, x)
0.6289066304773024
>>> 1 - sc.gammaincc(a, x)
0.6289066304773024gammainccgammaincc(a, x, out=None)

Regularized upper incomplete gamma function.

It is defined as

.. math::

    Q(a, x) = \frac{1}{\Gamma(a)} \int_x^\infty t^{a - 1}e^{-t} dt

for :math:`a > 0` and :math:`x \geq 0`. See [dlmf]_ for details.

Parameters
----------
a : array_like
    Positive parameter
x : array_like
    Nonnegative argument
out : ndarray, optional
    Optional output array for the function values

Returns
-------
scalar or ndarray
    Values of the upper incomplete gamma function

See Also
--------
gammainc : regularized lower incomplete gamma function
gammaincinv : inverse of the regularized lower incomplete gamma function
gammainccinv : inverse of the regularized upper incomplete gamma function

Notes
-----
The function satisfies the relation ``gammainc(a, x) +
gammaincc(a, x) = 1`` where `gammainc` is the regularized lower
incomplete gamma function.

The implementation largely follows that of [boost]_.

References
----------
.. [dlmf] NIST Digital Library of Mathematical functions
          https://dlmf.nist.gov/8.2#E4
.. [boost] Maddock et. al., "Incomplete Gamma Functions",
   https://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html

Examples
--------
>>> import scipy.special as sc

It is the survival function of the gamma distribution, so it
starts at 1 and monotonically decreases to 0.

>>> sc.gammaincc(0.5, [0, 1, 10, 100, 1000])
array([1.00000000e+00, 1.57299207e-01, 7.74421643e-06, 2.08848758e-45,
       0.00000000e+00])

It is equal to one minus the lower incomplete gamma function.

>>> a, x = 0.5, 0.4
>>> sc.gammaincc(a, x)
0.37109336952269756
>>> 1 - sc.gammainc(a, x)
0.37109336952269756gammainccinvgammainccinv(a, y, out=None)

Inverse of the regularized upper incomplete gamma function.

Given an input :math:`y` between 0 and 1, returns :math:`x` such
that :math:`y = Q(a, x)`. Here :math:`Q` is the regularized upper
incomplete gamma function; see `gammaincc`. This is well-defined
because the upper incomplete gamma function is monotonic as can
be seen from its definition in [dlmf]_.

Parameters
----------
a : array_like
    Positive parameter
y : array_like
    Argument between 0 and 1, inclusive
out : ndarray, optional
    Optional output array for the function values

Returns
-------
scalar or ndarray
    Values of the inverse of the upper incomplete gamma function

See Also
--------
gammaincc : regularized upper incomplete gamma function
gammainc : regularized lower incomplete gamma function
gammaincinv : inverse of the regularized lower incomplete gamma function

References
----------
.. [dlmf] NIST Digital Library of Mathematical Functions
          https://dlmf.nist.gov/8.2#E4

Examples
--------
>>> import scipy.special as sc

It starts at infinity and monotonically decreases to 0.

>>> sc.gammainccinv(0.5, [0, 0.1, 0.5, 1])
array([       inf, 1.35277173, 0.22746821, 0.        ])

It inverts the upper incomplete gamma function.

>>> a, x = 0.5, [0, 0.1, 0.5, 1]
>>> sc.gammaincc(a, sc.gammainccinv(a, x))
array([0. , 0.1, 0.5, 1. ])

>>> a, x = 0.5, [0, 10, 50]
>>> sc.gammainccinv(a, sc.gammaincc(a, x))
array([ 0., 10., 50.])gammaincinvgammaincinv(a, y, out=None)

Inverse to the regularized lower incomplete gamma function.

Given an input :math:`y` between 0 and 1, returns :math:`x` such
that :math:`y = P(a, x)`. Here :math:`P` is the regularized lower
incomplete gamma function; see `gammainc`. This is well-defined
because the lower incomplete gamma function is monotonic as can be
seen from its definition in [dlmf]_.

Parameters
----------
a : array_like
    Positive parameter
y : array_like
    Parameter between 0 and 1, inclusive
out : ndarray, optional
    Optional output array for the function values

Returns
-------
scalar or ndarray
    Values of the inverse of the lower incomplete gamma function

See Also
--------
gammainc : regularized lower incomplete gamma function
gammaincc : regularized upper incomplete gamma function
gammainccinv : inverse of the regularized upper incomplete gamma function

References
----------
.. [dlmf] NIST Digital Library of Mathematical Functions
          https://dlmf.nist.gov/8.2#E4

Examples
--------
>>> import scipy.special as sc

It starts at 0 and monotonically increases to infinity.

>>> sc.gammaincinv(0.5, [0, 0.1 ,0.5, 1])
array([0.        , 0.00789539, 0.22746821,        inf])

It inverts the lower incomplete gamma function.

>>> a, x = 0.5, [0, 0.1, 0.5, 1]
>>> sc.gammainc(a, sc.gammaincinv(a, x))
array([0. , 0.1, 0.5, 1. ])

>>> a, x = 0.5, [0, 10, 25]
>>> sc.gammaincinv(a, sc.gammainc(a, x))
array([ 0.        , 10.        , 25.00001465])gammalngammaln(x, out=None)

Logarithm of the absolute value of the gamma function.

Defined as

.. math::

   \ln(\lvert\Gamma(x)\rvert)

where :math:`\Gamma` is the gamma function. For more details on
the gamma function, see [dlmf]_.

Parameters
----------
x : array_like
    Real argument
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    Values of the log of the absolute value of gamma

See Also
--------
gammasgn : sign of the gamma function
loggamma : principal branch of the logarithm of the gamma function

Notes
-----
It is the same function as the Python standard library function
:func:`math.lgamma`.

When used in conjunction with `gammasgn`, this function is useful
for working in logspace on the real axis without having to deal
with complex numbers via the relation ``exp(gammaln(x)) =
gammasgn(x) * gamma(x)``.

For complex-valued log-gamma, use `loggamma` instead of `gammaln`.

References
----------
.. [dlmf] NIST Digital Library of Mathematical Functions
          https://dlmf.nist.gov/5

Examples
--------
>>> import numpy as np
>>> import scipy.special as sc

It has two positive zeros.

>>> sc.gammaln([1, 2])
array([0., 0.])

It has poles at nonpositive integers.

>>> sc.gammaln([0, -1, -2, -3, -4])
array([inf, inf, inf, inf, inf])

It asymptotically approaches ``x * log(x)`` (Stirling's formula).

>>> x = np.array([1e10, 1e20, 1e40, 1e80])
>>> sc.gammaln(x)
array([2.20258509e+11, 4.50517019e+21, 9.11034037e+41, 1.83206807e+82])
>>> x * np.log(x)
array([2.30258509e+11, 4.60517019e+21, 9.21034037e+41, 1.84206807e+82])gammasgngammasgn(x, out=None)

Sign of the gamma function.

It is defined as

.. math::

   \text{gammasgn}(x) =
   \begin{cases}
     +1 & \Gamma(x) > 0 \\
     -1 & \Gamma(x) < 0
   \end{cases}

where :math:`\Gamma` is the gamma function; see `gamma`. This
definition is complete since the gamma function is never zero;
see the discussion after [dlmf]_.

Parameters
----------
x : array_like
    Real argument
out : ndarray, optional
    Optional output array for the function values

Returns
-------
scalar or ndarray
    Sign of the gamma function

See Also
--------
gamma : the gamma function
gammaln : log of the absolute value of the gamma function
loggamma : analytic continuation of the log of the gamma function

Notes
-----
The gamma function can be computed as ``gammasgn(x) *
np.exp(gammaln(x))``.

References
----------
.. [dlmf] NIST Digital Library of Mathematical Functions
          https://dlmf.nist.gov/5.2#E1

Examples
--------
>>> import numpy as np
>>> import scipy.special as sc

It is 1 for `x > 0`.

>>> sc.gammasgn([1, 2, 3, 4])
array([1., 1., 1., 1.])

It alternates between -1 and 1 for negative integers.

>>> sc.gammasgn([-0.5, -1.5, -2.5, -3.5])
array([-1.,  1., -1.,  1.])

It can be used to compute the gamma function.

>>> x = [1.5, 0.5, -0.5, -1.5]
>>> sc.gammasgn(x) * np.exp(sc.gammaln(x))
array([ 0.88622693,  1.77245385, -3.5449077 ,  2.3632718 ])
>>> sc.gamma(x)
array([ 0.88622693,  1.77245385, -3.5449077 ,  2.3632718 ])gdtrgdtr(a, b, x, out=None)

Gamma distribution cumulative distribution function.

Returns the integral from zero to `x` of the gamma probability density
function,

.. math::

    F = \int_0^x \frac{a^b}{\Gamma(b)} t^{b-1} e^{-at}\,dt,

where :math:`\Gamma` is the gamma function.

Parameters
----------
a : array_like
    The rate parameter of the gamma distribution, sometimes denoted
    :math:`\beta` (float).  It is also the reciprocal of the scale
    parameter :math:`\theta`.
b : array_like
    The shape parameter of the gamma distribution, sometimes denoted
    :math:`\alpha` (float).
x : array_like
    The quantile (upper limit of integration; float).
out : ndarray, optional
    Optional output array for the function values

Returns
-------
F : scalar or ndarray
    The CDF of the gamma distribution with parameters `a` and `b`
    evaluated at `x`.

See Also
--------
gdtrc : 1 - CDF of the gamma distribution.
scipy.stats.gamma: Gamma distribution

Notes
-----
The evaluation is carried out using the relation to the incomplete gamma
integral (regularized gamma function).

Wrapper for the Cephes [1]_ routine `gdtr`. Calling `gdtr` directly can
improve performance compared to the ``cdf`` method of `scipy.stats.gamma`
(see last example below).

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/

Examples
--------
Compute the function for ``a=1``, ``b=2`` at ``x=5``.

>>> import numpy as np
>>> from scipy.special import gdtr
>>> import matplotlib.pyplot as plt
>>> gdtr(1., 2., 5.)
0.9595723180054873

Compute the function for ``a=1`` and ``b=2`` at several points by
providing a NumPy array for `x`.

>>> xvalues = np.array([1., 2., 3., 4])
>>> gdtr(1., 1., xvalues)
array([0.63212056, 0.86466472, 0.95021293, 0.98168436])

`gdtr` can evaluate different parameter sets by providing arrays with
broadcasting compatible shapes for `a`, `b` and `x`. Here we compute the
function for three different `a` at four positions `x` and ``b=3``,
resulting in a 3x4 array.

>>> a = np.array([[0.5], [1.5], [2.5]])
>>> x = np.array([1., 2., 3., 4])
>>> a.shape, x.shape
((3, 1), (4,))

>>> gdtr(a, 3., x)
array([[0.01438768, 0.0803014 , 0.19115317, 0.32332358],
       [0.19115317, 0.57680992, 0.82642193, 0.9380312 ],
       [0.45618688, 0.87534798, 0.97974328, 0.9972306 ]])

Plot the function for four different parameter sets.

>>> a_parameters = [0.3, 1, 2, 6]
>>> b_parameters = [2, 10, 15, 20]
>>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']
>>> parameters_list = list(zip(a_parameters, b_parameters, linestyles))
>>> x = np.linspace(0, 30, 1000)
>>> fig, ax = plt.subplots()
>>> for parameter_set in parameters_list:
...     a, b, style = parameter_set
...     gdtr_vals = gdtr(a, b, x)
...     ax.plot(x, gdtr_vals, label=fr"$a= {a},\, b={b}$", ls=style)
>>> ax.legend()
>>> ax.set_xlabel("$x$")
>>> ax.set_title("Gamma distribution cumulative distribution function")
>>> plt.show()

The gamma distribution is also available as `scipy.stats.gamma`. Using
`gdtr` directly can be much faster than calling the ``cdf`` method of
`scipy.stats.gamma`, especially for small arrays or individual values.
To get the same results one must use the following parametrization:
``stats.gamma(b, scale=1/a).cdf(x)=gdtr(a, b, x)``.

>>> from scipy.stats import gamma
>>> a = 2.
>>> b = 3
>>> x = 1.
>>> gdtr_result = gdtr(a, b, x)  # this will often be faster than below
>>> gamma_dist_result = gamma(b, scale=1/a).cdf(x)
>>> gdtr_result == gamma_dist_result  # test that results are equal
Truegdtrcgdtrc(a, b, x, out=None)

Gamma distribution survival function.

Integral from `x` to infinity of the gamma probability density function,

.. math::

    F = \int_x^\infty \frac{a^b}{\Gamma(b)} t^{b-1} e^{-at}\,dt,

where :math:`\Gamma` is the gamma function.

Parameters
----------
a : array_like
    The rate parameter of the gamma distribution, sometimes denoted
    :math:`\beta` (float). It is also the reciprocal of the scale
    parameter :math:`\theta`.
b : array_like
    The shape parameter of the gamma distribution, sometimes denoted
    :math:`\alpha` (float).
x : array_like
    The quantile (lower limit of integration; float).
out : ndarray, optional
    Optional output array for the function values

Returns
-------
F : scalar or ndarray
    The survival function of the gamma distribution with parameters `a`
    and `b` evaluated at `x`.

See Also
--------
gdtr: Gamma distribution cumulative distribution function
scipy.stats.gamma: Gamma distribution
gdtrix

Notes
-----
The evaluation is carried out using the relation to the incomplete gamma
integral (regularized gamma function).

Wrapper for the Cephes [1]_ routine `gdtrc`. Calling `gdtrc` directly can
improve performance compared to the ``sf`` method of `scipy.stats.gamma`
(see last example below).

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/

Examples
--------
Compute the function for ``a=1`` and ``b=2`` at ``x=5``.

>>> import numpy as np
>>> from scipy.special import gdtrc
>>> import matplotlib.pyplot as plt
>>> gdtrc(1., 2., 5.)
0.04042768199451279

Compute the function for ``a=1``, ``b=2`` at several points by providing
a NumPy array for `x`.

>>> xvalues = np.array([1., 2., 3., 4])
>>> gdtrc(1., 1., xvalues)
array([0.36787944, 0.13533528, 0.04978707, 0.01831564])

`gdtrc` can evaluate different parameter sets by providing arrays with
broadcasting compatible shapes for `a`, `b` and `x`. Here we compute the
function for three different `a` at four positions `x` and ``b=3``,
resulting in a 3x4 array.

>>> a = np.array([[0.5], [1.5], [2.5]])
>>> x = np.array([1., 2., 3., 4])
>>> a.shape, x.shape
((3, 1), (4,))

>>> gdtrc(a, 3., x)
array([[0.98561232, 0.9196986 , 0.80884683, 0.67667642],
       [0.80884683, 0.42319008, 0.17357807, 0.0619688 ],
       [0.54381312, 0.12465202, 0.02025672, 0.0027694 ]])

Plot the function for four different parameter sets.

>>> a_parameters = [0.3, 1, 2, 6]
>>> b_parameters = [2, 10, 15, 20]
>>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']
>>> parameters_list = list(zip(a_parameters, b_parameters, linestyles))
>>> x = np.linspace(0, 30, 1000)
>>> fig, ax = plt.subplots()
>>> for parameter_set in parameters_list:
...     a, b, style = parameter_set
...     gdtrc_vals = gdtrc(a, b, x)
...     ax.plot(x, gdtrc_vals, label=fr"$a= {a},\, b={b}$", ls=style)
>>> ax.legend()
>>> ax.set_xlabel("$x$")
>>> ax.set_title("Gamma distribution survival function")
>>> plt.show()

The gamma distribution is also available as `scipy.stats.gamma`.
Using `gdtrc` directly can be much faster than calling the ``sf`` method
of `scipy.stats.gamma`, especially for small arrays or individual
values. To get the same results one must use the following parametrization:
``stats.gamma(b, scale=1/a).sf(x)=gdtrc(a, b, x)``.

>>> from scipy.stats import gamma
>>> a = 2
>>> b = 3
>>> x = 1.
>>> gdtrc_result = gdtrc(a, b, x)  # this will often be faster than below
>>> gamma_dist_result = gamma(b, scale=1/a).sf(x)
>>> gdtrc_result == gamma_dist_result  # test that results are equal
Truegdtria(p, b, x, out=None)

Inverse of `gdtr` vs a.

Returns the inverse with respect to the parameter `a` of ``p =
gdtr(a, b, x)``, the cumulative distribution function of the gamma
distribution.

Parameters
----------
p : array_like
    Probability values.
b : array_like
    `b` parameter values of `gdtr(a, b, x)`. `b` is the "shape" parameter
    of the gamma distribution.
x : array_like
    Nonnegative real values, from the domain of the gamma distribution.
out : ndarray, optional
    If a fourth argument is given, it must be a numpy.ndarray whose size
    matches the broadcast result of `a`, `b` and `x`.  `out` is then the
    array returned by the function.

Returns
-------
a : scalar or ndarray
    Values of the `a` parameter such that `p = gdtr(a, b, x)`.  `1/a`
    is the "scale" parameter of the gamma distribution.

See Also
--------
gdtr : CDF of the gamma distribution.
gdtrib : Inverse with respect to `b` of `gdtr(a, b, x)`.
gdtrix : Inverse with respect to `x` of `gdtr(a, b, x)`.

Notes
-----
Wrapper for the CDFLIB [1]_ Fortran routine `cdfgam`.

The cumulative distribution function `p` is computed using a routine by
DiDinato and Morris [2]_. Computation of `a` involves a search for a value
that produces the desired value of `p`. The search relies on the
monotonicity of `p` with `a`.

References
----------
.. [1] Barry Brown, James Lovato, and Kathy Russell,
       CDFLIB: Library of Fortran Routines for Cumulative Distribution
       Functions, Inverses, and Other Parameters.
.. [2] DiDinato, A. R. and Morris, A. H.,
       Computation of the incomplete gamma function ratios and their
       inverse.  ACM Trans. Math. Softw. 12 (1986), 377-393.

Examples
--------
First evaluate `gdtr`.

>>> from scipy.special import gdtr, gdtria
>>> p = gdtr(1.2, 3.4, 5.6)
>>> print(p)
0.94378087442

Verify the inverse.

>>> gdtria(p, 3.4, 5.6)
1.2gdtrib(a, p, x, out=None)

Inverse of `gdtr` vs b.

Returns the inverse with respect to the parameter `b` of ``p =
gdtr(a, b, x)``, the cumulative distribution function of the gamma
distribution.

Parameters
----------
a : array_like
    `a` parameter values of `gdtr(a, b, x)`. `1/a` is the "scale"
    parameter of the gamma distribution.
p : array_like
    Probability values.
x : array_like
    Nonnegative real values, from the domain of the gamma distribution.
out : ndarray, optional
    If a fourth argument is given, it must be a numpy.ndarray whose size
    matches the broadcast result of `a`, `b` and `x`.  `out` is then the
    array returned by the function.

Returns
-------
b : scalar or ndarray
    Values of the `b` parameter such that `p = gdtr(a, b, x)`.  `b` is
    the "shape" parameter of the gamma distribution.

See Also
--------
gdtr : CDF of the gamma distribution.
gdtria : Inverse with respect to `a` of `gdtr(a, b, x)`.
gdtrix : Inverse with respect to `x` of `gdtr(a, b, x)`.

Notes
-----
Wrapper for the CDFLIB [1]_ Fortran routine `cdfgam`.

The cumulative distribution function `p` is computed using a routine by
DiDinato and Morris [2]_. Computation of `b` involves a search for a value
that produces the desired value of `p`. The search relies on the
monotonicity of `p` with `b`.

References
----------
.. [1] Barry Brown, James Lovato, and Kathy Russell,
       CDFLIB: Library of Fortran Routines for Cumulative Distribution
       Functions, Inverses, and Other Parameters.
.. [2] DiDinato, A. R. and Morris, A. H.,
       Computation of the incomplete gamma function ratios and their
       inverse.  ACM Trans. Math. Softw. 12 (1986), 377-393.

Examples
--------
First evaluate `gdtr`.

>>> from scipy.special import gdtr, gdtrib
>>> p = gdtr(1.2, 3.4, 5.6)
>>> print(p)
0.94378087442

Verify the inverse.

>>> gdtrib(1.2, p, 5.6)
3.3999999999723882gdtrix(a, b, p, out=None)

Inverse of `gdtr` vs x.

Returns the inverse with respect to the parameter `x` of ``p =
gdtr(a, b, x)``, the cumulative distribution function of the gamma
distribution. This is also known as the pth quantile of the
distribution.

Parameters
----------
a : array_like
    `a` parameter values of `gdtr(a, b, x)`. `1/a` is the "scale"
    parameter of the gamma distribution.
b : array_like
    `b` parameter values of `gdtr(a, b, x)`. `b` is the "shape" parameter
    of the gamma distribution.
p : array_like
    Probability values.
out : ndarray, optional
    If a fourth argument is given, it must be a numpy.ndarray whose size
    matches the broadcast result of `a`, `b` and `x`. `out` is then the
    array returned by the function.

Returns
-------
x : scalar or ndarray
    Values of the `x` parameter such that `p = gdtr(a, b, x)`.

See Also
--------
gdtr : CDF of the gamma distribution.
gdtria : Inverse with respect to `a` of `gdtr(a, b, x)`.
gdtrib : Inverse with respect to `b` of `gdtr(a, b, x)`.

Notes
-----
Wrapper for the CDFLIB [1]_ Fortran routine `cdfgam`.

The cumulative distribution function `p` is computed using a routine by
DiDinato and Morris [2]_. Computation of `x` involves a search for a value
that produces the desired value of `p`. The search relies on the
monotonicity of `p` with `x`.

References
----------
.. [1] Barry Brown, James Lovato, and Kathy Russell,
       CDFLIB: Library of Fortran Routines for Cumulative Distribution
       Functions, Inverses, and Other Parameters.
.. [2] DiDinato, A. R. and Morris, A. H.,
       Computation of the incomplete gamma function ratios and their
       inverse.  ACM Trans. Math. Softw. 12 (1986), 377-393.

Examples
--------
First evaluate `gdtr`.

>>> from scipy.special import gdtr, gdtrix
>>> p = gdtr(1.2, 3.4, 5.6)
>>> print(p)
0.94378087442

Verify the inverse.

>>> gdtrix(1.2, 3.4, p)
5.5999999999999996hankel1hankel1(v, z, out=None)

Hankel function of the first kind

Parameters
----------
v : array_like
    Order (float).
z : array_like
    Argument (float or complex).
out : ndarray, optional
    Optional output array for the function values

Returns
-------
scalar or ndarray
    Values of the Hankel function of the first kind.

See Also
--------
hankel1e : ndarray
    This function with leading exponential behavior stripped off.

Notes
-----
A wrapper for the AMOS [1]_ routine `zbesh`, which carries out the
computation using the relation,

.. math:: H^{(1)}_v(z) =
          \frac{2}{\imath\pi} \exp(-\imath \pi v/2) K_v(z \exp(-\imath\pi/2))

where :math:`K_v` is the modified Bessel function of the second kind.
For negative orders, the relation

.. math:: H^{(1)}_{-v}(z) = H^{(1)}_v(z) \exp(\imath\pi v)

is used.

References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
       of a Complex Argument and Nonnegative Order",
       http://netlib.org/amos/hankel1ehankel1e(v, z, out=None)

Exponentially scaled Hankel function of the first kind

Defined as::

    hankel1e(v, z) = hankel1(v, z) * exp(-1j * z)

Parameters
----------
v : array_like
    Order (float).
z : array_like
    Argument (float or complex).
out : ndarray, optional
    Optional output array for the function values

Returns
-------
scalar or ndarray
    Values of the exponentially scaled Hankel function.

Notes
-----
A wrapper for the AMOS [1]_ routine `zbesh`, which carries out the
computation using the relation,

.. math:: H^{(1)}_v(z) =
          \frac{2}{\imath\pi} \exp(-\imath \pi v/2) K_v(z \exp(-\imath\pi/2))

where :math:`K_v` is the modified Bessel function of the second kind.
For negative orders, the relation

.. math:: H^{(1)}_{-v}(z) = H^{(1)}_v(z) \exp(\imath\pi v)

is used.

References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
       of a Complex Argument and Nonnegative Order",
       http://netlib.org/amos/hankel2hankel2(v, z, out=None)

Hankel function of the second kind

Parameters
----------
v : array_like
    Order (float).
z : array_like
    Argument (float or complex).
out : ndarray, optional
    Optional output array for the function values

Returns
-------
scalar or ndarray
    Values of the Hankel function of the second kind.

See Also
--------
hankel2e : this function with leading exponential behavior stripped off.

Notes
-----
A wrapper for the AMOS [1]_ routine `zbesh`, which carries out the
computation using the relation,

.. math:: H^{(2)}_v(z) =
          -\frac{2}{\imath\pi} \exp(\imath \pi v/2) K_v(z \exp(\imath\pi/2))

where :math:`K_v` is the modified Bessel function of the second kind.
For negative orders, the relation

.. math:: H^{(2)}_{-v}(z) = H^{(2)}_v(z) \exp(-\imath\pi v)

is used.

References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
       of a Complex Argument and Nonnegative Order",
       http://netlib.org/amos/hankel2ehankel2e(v, z, out=None)

Exponentially scaled Hankel function of the second kind

Defined as::

    hankel2e(v, z) = hankel2(v, z) * exp(1j * z)

Parameters
----------
v : array_like
    Order (float).
z : array_like
    Argument (float or complex).
out : ndarray, optional
    Optional output array for the function values

Returns
-------
scalar or ndarray
    Values of the exponentially scaled Hankel function of the second kind.

Notes
-----
A wrapper for the AMOS [1]_ routine `zbesh`, which carries out the
computation using the relation,

.. math:: H^{(2)}_v(z) = -\frac{2}{\imath\pi}
          \exp(\frac{\imath \pi v}{2}) K_v(z exp(\frac{\imath\pi}{2}))

where :math:`K_v` is the modified Bessel function of the second kind.
For negative orders, the relation

.. math:: H^{(2)}_{-v}(z) = H^{(2)}_v(z) \exp(-\imath\pi v)

is used.

References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
       of a Complex Argument and Nonnegative Order",
       http://netlib.org/amos/huberhuber(delta, r, out=None)

Huber loss function.

.. math:: \text{huber}(\delta, r) = \begin{cases} \infty & \delta < 0  \\
          \frac{1}{2}r^2 & 0 \le \delta, | r | \le \delta \\
          \delta ( |r| - \frac{1}{2}\delta ) & \text{otherwise} \end{cases}

Parameters
----------
delta : ndarray
    Input array, indicating the quadratic vs. linear loss changepoint.
r : ndarray
    Input array, possibly representing residuals.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
scalar or ndarray
    The computed Huber loss function values.

See Also
--------
pseudo_huber : smooth approximation of this function

Notes
-----
`huber` is useful as a loss function in robust statistics or machine
learning to reduce the influence of outliers as compared to the common
squared error loss, residuals with a magnitude higher than `delta` are
not squared [1]_.

Typically, `r` represents residuals, the difference
between a model prediction and data. Then, for :math:`|r|\leq\delta`,
`huber` resembles the squared error and for :math:`|r|>\delta` the
absolute error. This way, the Huber loss often achieves
a fast convergence in model fitting for small residuals like the squared
error loss function and still reduces the influence of outliers
(:math:`|r|>\delta`) like the absolute error loss. As :math:`\delta` is
the cutoff between squared and absolute error regimes, it has
to be tuned carefully for each problem. `huber` is also
convex, making it suitable for gradient based optimization.

.. versionadded:: 0.15.0

References
----------
.. [1] Peter Huber. "Robust Estimation of a Location Parameter",
       1964. Annals of Statistics. 53 (1): 73 - 101.

Examples
--------
Import all necessary modules.

>>> import numpy as np
>>> from scipy.special import huber
>>> import matplotlib.pyplot as plt

Compute the function for ``delta=1`` at ``r=2``

>>> huber(1., 2.)
1.5

Compute the function for different `delta` by providing a NumPy array or
list for `delta`.

>>> huber([1., 3., 5.], 4.)
array([3.5, 7.5, 8. ])

Compute the function at different points by providing a NumPy array or
list for `r`.

>>> huber(2., np.array([1., 1.5, 3.]))
array([0.5  , 1.125, 4.   ])

The function can be calculated for different `delta` and `r` by
providing arrays for both with compatible shapes for broadcasting.

>>> r = np.array([1., 2.5, 8., 10.])
>>> deltas = np.array([[1.], [5.], [9.]])
>>> print(r.shape, deltas.shape)
(4,) (3, 1)

>>> huber(deltas, r)
array([[ 0.5  ,  2.   ,  7.5  ,  9.5  ],
       [ 0.5  ,  3.125, 27.5  , 37.5  ],
       [ 0.5  ,  3.125, 32.   , 49.5  ]])

Plot the function for different `delta`.

>>> x = np.linspace(-4, 4, 500)
>>> deltas = [1, 2, 3]
>>> linestyles = ["dashed", "dotted", "dashdot"]
>>> fig, ax = plt.subplots()
>>> combined_plot_parameters = list(zip(deltas, linestyles))
>>> for delta, style in combined_plot_parameters:
...     ax.plot(x, huber(delta, x), label=fr"$\delta={delta}$", ls=style)
>>> ax.legend(loc="upper center")
>>> ax.set_xlabel("$x$")
>>> ax.set_title(r"Huber loss function $h_{\delta}(x)$")
>>> ax.set_xlim(-4, 4)
>>> ax.set_ylim(0, 8)
>>> plt.show()hyp0f1hyp0f1(v, z, out=None)

Confluent hypergeometric limit function 0F1.

Parameters
----------
v : array_like
    Real-valued parameter
z : array_like
    Real- or complex-valued argument
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    The confluent hypergeometric limit function

Notes
-----
This function is defined as:

.. math:: _0F_1(v, z) = \sum_{k=0}^{\infty}\frac{z^k}{(v)_k k!}.

It's also the limit as :math:`q \to \infty` of :math:`_1F_1(q; v; z/q)`,
and satisfies the differential equation :math:`f''(z) + vf'(z) =
f(z)`. See [1]_ for more information.

References
----------
.. [1] Wolfram MathWorld, "Confluent Hypergeometric Limit Function",
       http://mathworld.wolfram.com/ConfluentHypergeometricLimitFunction.html

Examples
--------
>>> import numpy as np
>>> import scipy.special as sc

It is one when `z` is zero.

>>> sc.hyp0f1(1, 0)
1.0

It is the limit of the confluent hypergeometric function as `q`
goes to infinity.

>>> q = np.array([1, 10, 100, 1000])
>>> v = 1
>>> z = 1
>>> sc.hyp1f1(q, v, z / q)
array([2.71828183, 2.31481985, 2.28303778, 2.27992985])
>>> sc.hyp0f1(v, z)
2.2795853023360673

It is related to Bessel functions.

>>> n = 1
>>> x = np.linspace(0, 1, 5)
>>> sc.jv(n, x)
array([0.        , 0.12402598, 0.24226846, 0.3492436 , 0.44005059])
>>> (0.5 * x)**n / sc.factorial(n) * sc.hyp0f1(n + 1, -0.25 * x**2)
array([0.        , 0.12402598, 0.24226846, 0.3492436 , 0.44005059])hyp1f1hyp1f1(a, b, x, out=None)

Confluent hypergeometric function 1F1.

The confluent hypergeometric function is defined by the series

.. math::

   {}_1F_1(a; b; x) = \sum_{k = 0}^\infty \frac{(a)_k}{(b)_k k!} x^k.

See [dlmf]_ for more details. Here :math:`(\cdot)_k` is the
Pochhammer symbol; see `poch`.

Parameters
----------
a, b : array_like
    Real parameters
x : array_like
    Real or complex argument
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    Values of the confluent hypergeometric function

See Also
--------
hyperu : another confluent hypergeometric function
hyp0f1 : confluent hypergeometric limit function
hyp2f1 : Gaussian hypergeometric function

References
----------
.. [dlmf] NIST Digital Library of Mathematical Functions
          https://dlmf.nist.gov/13.2#E2

Examples
--------
>>> import numpy as np
>>> import scipy.special as sc

It is one when `x` is zero:

>>> sc.hyp1f1(0.5, 0.5, 0)
1.0

It is singular when `b` is a nonpositive integer.

>>> sc.hyp1f1(0.5, -1, 0)
inf

It is a polynomial when `a` is a nonpositive integer.

>>> a, b, x = -1, 0.5, np.array([1.0, 2.0, 3.0, 4.0])
>>> sc.hyp1f1(a, b, x)
array([-1., -3., -5., -7.])
>>> 1 + (a / b) * x
array([-1., -3., -5., -7.])

It reduces to the exponential function when `a = b`.

>>> sc.hyp1f1(2, 2, [1, 2, 3, 4])
array([ 2.71828183,  7.3890561 , 20.08553692, 54.59815003])
>>> np.exp([1, 2, 3, 4])
array([ 2.71828183,  7.3890561 , 20.08553692, 54.59815003])hyp2f1(a, b, c, z, out=None)

Gauss hypergeometric function 2F1(a, b; c; z)

Parameters
----------
a, b, c : array_like
    Arguments, should be real-valued.
z : array_like
    Argument, real or complex.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
hyp2f1 : scalar or ndarray
    The values of the gaussian hypergeometric function.

See Also
--------
hyp0f1 : confluent hypergeometric limit function.
hyp1f1 : Kummer's (confluent hypergeometric) function.

Notes
-----
This function is defined for :math:`|z| < 1` as

.. math::

   \mathrm{hyp2f1}(a, b, c, z) = \sum_{n=0}^\infty
   \frac{(a)_n (b)_n}{(c)_n}\frac{z^n}{n!},

and defined on the rest of the complex z-plane by analytic
continuation [1]_.
Here :math:`(\cdot)_n` is the Pochhammer symbol; see `poch`. When
:math:`n` is an integer the result is a polynomial of degree :math:`n`.

The implementation for complex values of ``z`` is described in [2]_,
except for ``z`` in the region defined by

.. math::

     0.9 <= \left|z\right| < 1.1,
     \left|1 - z\right| >= 0.9,
     \mathrm{real}(z) >= 0

in which the implementation follows [4]_.

References
----------
.. [1] NIST Digital Library of Mathematical Functions
       https://dlmf.nist.gov/15.2
.. [2] S. Zhang and J.M. Jin, "Computation of Special Functions", Wiley 1996
.. [3] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/
.. [4] J.L. Lopez and N.M. Temme, "New series expansions of the Gauss
       hypergeometric function", Adv Comput Math 39, 349-365 (2013).
       https://doi.org/10.1007/s10444-012-9283-y

Examples
--------
>>> import numpy as np
>>> import scipy.special as sc

It has poles when `c` is a negative integer.

>>> sc.hyp2f1(1, 1, -2, 1)
inf

It is a polynomial when `a` or `b` is a negative integer.

>>> a, b, c = -1, 1, 1.5
>>> z = np.linspace(0, 1, 5)
>>> sc.hyp2f1(a, b, c, z)
array([1.        , 0.83333333, 0.66666667, 0.5       , 0.33333333])
>>> 1 + a * b * z / c
array([1.        , 0.83333333, 0.66666667, 0.5       , 0.33333333])

It is symmetric in `a` and `b`.

>>> a = np.linspace(0, 1, 5)
>>> b = np.linspace(0, 1, 5)
>>> sc.hyp2f1(a, b, 1, 0.5)
array([1.        , 1.03997334, 1.1803406 , 1.47074441, 2.        ])
>>> sc.hyp2f1(b, a, 1, 0.5)
array([1.        , 1.03997334, 1.1803406 , 1.47074441, 2.        ])

It contains many other functions as special cases.

>>> z = 0.5
>>> sc.hyp2f1(1, 1, 2, z)
1.3862943611198901
>>> -np.log(1 - z) / z
1.3862943611198906

>>> sc.hyp2f1(0.5, 1, 1.5, z**2)
1.098612288668109
>>> np.log((1 + z) / (1 - z)) / (2 * z)
1.0986122886681098

>>> sc.hyp2f1(0.5, 1, 1.5, -z**2)
0.9272952180016117
>>> np.arctan(z) / z
0.9272952180016122hyperu(a, b, x, out=None)

Confluent hypergeometric function U

It is defined as the solution to the equation

.. math::

   x \frac{d^2w}{dx^2} + (b - x) \frac{dw}{dx} - aw = 0

which satisfies the property

.. math::

   U(a, b, x) \sim x^{-a}

as :math:`x \to \infty`. See [dlmf]_ for more details.

Parameters
----------
a, b : array_like
    Real-valued parameters
x : array_like
    Real-valued argument
out : ndarray, optional
    Optional output array for the function values

Returns
-------
scalar or ndarray
    Values of `U`

References
----------
.. [dlmf] NIST Digital Library of Mathematics Functions
          https://dlmf.nist.gov/13.2#E6

Examples
--------
>>> import numpy as np
>>> import scipy.special as sc

It has a branch cut along the negative `x` axis.

>>> x = np.linspace(-0.1, -10, 5)
>>> sc.hyperu(1, 1, x)
array([nan, nan, nan, nan, nan])

It approaches zero as `x` goes to infinity.

>>> x = np.array([1, 10, 100])
>>> sc.hyperu(1, 1, x)
array([0.59634736, 0.09156333, 0.00990194])

It satisfies Kummer's transformation.

>>> a, b, x = 2, 1, 1
>>> sc.hyperu(a, b, x)
0.1926947246463881
>>> x**(1 - b) * sc.hyperu(a - b + 1, 2 - b, x)
0.1926947246463881i0i0(x, out=None)

Modified Bessel function of order 0.

Defined as,

.. math::
    I_0(x) = \sum_{k=0}^\infty \frac{(x^2/4)^k}{(k!)^2} = J_0(\imath x),

where :math:`J_0` is the Bessel function of the first kind of order 0.

Parameters
----------
x : array_like
    Argument (float)
out : ndarray, optional
    Optional output array for the function values

Returns
-------
I : scalar or ndarray
    Value of the modified Bessel function of order 0 at `x`.

See Also
--------
iv: Modified Bessel function of any order
i0e: Exponentially scaled modified Bessel function of order 0

Notes
-----
The range is partitioned into the two intervals [0, 8] and (8, infinity).
Chebyshev polynomial expansions are employed in each interval.

This function is a wrapper for the Cephes [1]_ routine `i0`.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/

Examples
--------
Calculate the function at one point:

>>> from scipy.special import i0
>>> i0(1.)
1.2660658777520082

Calculate at several points:

>>> import numpy as np
>>> i0(np.array([-2., 0., 3.5]))
array([2.2795853 , 1.        , 7.37820343])

Plot the function from -10 to 10.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-10., 10., 1000)
>>> y = i0(x)
>>> ax.plot(x, y)
>>> plt.show()i0ei0e(x, out=None)

Exponentially scaled modified Bessel function of order 0.

Defined as::

    i0e(x) = exp(-abs(x)) * i0(x).

Parameters
----------
x : array_like
    Argument (float)
out : ndarray, optional
    Optional output array for the function values

Returns
-------
I : scalar or ndarray
    Value of the exponentially scaled modified Bessel function of order 0
    at `x`.

See Also
--------
iv: Modified Bessel function of the first kind
i0: Modified Bessel function of order 0

Notes
-----
The range is partitioned into the two intervals [0, 8] and (8, infinity).
Chebyshev polynomial expansions are employed in each interval. The
polynomial expansions used are the same as those in `i0`, but
they are not multiplied by the dominant exponential factor.

This function is a wrapper for the Cephes [1]_ routine `i0e`. `i0e`
is useful for large arguments `x`: for these, `i0` quickly overflows.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/

Examples
--------
In the following example `i0` returns infinity whereas `i0e` still returns
a finite number.

>>> from scipy.special import i0, i0e
>>> i0(1000.), i0e(1000.)
(inf, 0.012617240455891257)

Calculate the function at several points by providing a NumPy array or
list for `x`:

>>> import numpy as np
>>> i0e(np.array([-2., 0., 3.]))
array([0.30850832, 1.        , 0.24300035])

Plot the function from -10 to 10.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-10., 10., 1000)
>>> y = i0e(x)
>>> ax.plot(x, y)
>>> plt.show()i1i1(x, out=None)

Modified Bessel function of order 1.

Defined as,

.. math::
    I_1(x) = \frac{1}{2}x \sum_{k=0}^\infty \frac{(x^2/4)^k}{k! (k + 1)!}
           = -\imath J_1(\imath x),

where :math:`J_1` is the Bessel function of the first kind of order 1.

Parameters
----------
x : array_like
    Argument (float)
out : ndarray, optional
    Optional output array for the function values

Returns
-------
I : scalar or ndarray
    Value of the modified Bessel function of order 1 at `x`.

See Also
--------
iv: Modified Bessel function of the first kind
i1e: Exponentially scaled modified Bessel function of order 1

Notes
-----
The range is partitioned into the two intervals [0, 8] and (8, infinity).
Chebyshev polynomial expansions are employed in each interval.

This function is a wrapper for the Cephes [1]_ routine `i1`.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/

Examples
--------
Calculate the function at one point:

>>> from scipy.special import i1
>>> i1(1.)
0.5651591039924851

Calculate the function at several points:

>>> import numpy as np
>>> i1(np.array([-2., 0., 6.]))
array([-1.59063685,  0.        , 61.34193678])

Plot the function between -10 and 10.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-10., 10., 1000)
>>> y = i1(x)
>>> ax.plot(x, y)
>>> plt.show()i1ei1e(x, out=None)

Exponentially scaled modified Bessel function of order 1.

Defined as::

    i1e(x) = exp(-abs(x)) * i1(x)

Parameters
----------
x : array_like
    Argument (float)
out : ndarray, optional
    Optional output array for the function values

Returns
-------
I : scalar or ndarray
    Value of the exponentially scaled modified Bessel function of order 1
    at `x`.

See Also
--------
iv: Modified Bessel function of the first kind
i1: Modified Bessel function of order 1

Notes
-----
The range is partitioned into the two intervals [0, 8] and (8, infinity).
Chebyshev polynomial expansions are employed in each interval. The
polynomial expansions used are the same as those in `i1`, but
they are not multiplied by the dominant exponential factor.

This function is a wrapper for the Cephes [1]_ routine `i1e`. `i1e`
is useful for large arguments `x`: for these, `i1` quickly overflows.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/

Examples
--------
In the following example `i1` returns infinity whereas `i1e` still returns
a finite number.

>>> from scipy.special import i1, i1e
>>> i1(1000.), i1e(1000.)
(inf, 0.01261093025692863)

Calculate the function at several points by providing a NumPy array or
list for `x`:

>>> import numpy as np
>>> i1e(np.array([-2., 0., 6.]))
array([-0.21526929,  0.        ,  0.15205146])

Plot the function between -10 and 10.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-10., 10., 1000)
>>> y = i1e(x)
>>> ax.plot(x, y)
>>> plt.show()inv_boxcoxinv_boxcox(y, lmbda, out=None)

Compute the inverse of the Box-Cox transformation.

Find ``x`` such that::

    y = (x**lmbda - 1) / lmbda  if lmbda != 0
        log(x)                  if lmbda == 0

Parameters
----------
y : array_like
    Data to be transformed.
lmbda : array_like
    Power parameter of the Box-Cox transform.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
x : scalar or ndarray
    Transformed data.

Notes
-----

.. versionadded:: 0.16.0

Examples
--------
>>> from scipy.special import boxcox, inv_boxcox
>>> y = boxcox([1, 4, 10], 2.5)
>>> inv_boxcox(y, 2.5)
array([1., 4., 10.])inv_boxcox1pinv_boxcox1p(y, lmbda, out=None)

Compute the inverse of the Box-Cox transformation.

Find ``x`` such that::

    y = ((1+x)**lmbda - 1) / lmbda  if lmbda != 0
        log(1+x)                    if lmbda == 0

Parameters
----------
y : array_like
    Data to be transformed.
lmbda : array_like
    Power parameter of the Box-Cox transform.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
x : scalar or ndarray
    Transformed data.

Notes
-----

.. versionadded:: 0.16.0

Examples
--------
>>> from scipy.special import boxcox1p, inv_boxcox1p
>>> y = boxcox1p([1, 4, 10], 2.5)
>>> inv_boxcox1p(y, 2.5)
array([1., 4., 10.])it2i0k0it2i0k0(x, out=None)

Integrals related to modified Bessel functions of order 0.

Computes the integrals

.. math::

    \int_0^x \frac{I_0(t) - 1}{t} dt \\
    \int_x^\infty \frac{K_0(t)}{t} dt.

Parameters
----------
x : array_like
    Values at which to evaluate the integrals.
out : tuple of ndarrays, optional
    Optional output arrays for the function results.

Returns
-------
ii0 : scalar or ndarray
    The integral for `i0`
ik0 : scalar or ndarray
    The integral for `k0`

References
----------
.. [1] S. Zhang and J.M. Jin, "Computation of Special Functions",
       Wiley 1996

Examples
--------
Evaluate the functions at one point.

>>> from scipy.special import it2i0k0
>>> int_i, int_k = it2i0k0(1.)
>>> int_i, int_k
(0.12897944249456852, 0.2085182909001295)

Evaluate the functions at several points.

>>> import numpy as np
>>> points = np.array([0.5, 1.5, 3.])
>>> int_i, int_k = it2i0k0(points)
>>> int_i, int_k
(array([0.03149527, 0.30187149, 1.50012461]),
 array([0.66575102, 0.0823715 , 0.00823631]))

Plot the functions from 0 to 5.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 5., 1000)
>>> int_i, int_k = it2i0k0(x)
>>> ax.plot(x, int_i, label=r"$\int_0^x \frac{I_0(t)-1}{t}\,dt$")
>>> ax.plot(x, int_k, label=r"$\int_x^{\infty} \frac{K_0(t)}{t}\,dt$")
>>> ax.legend()
>>> ax.set_ylim(0, 10)
>>> plt.show()it2j0y0it2j0y0(x, out=None)

Integrals related to Bessel functions of the first kind of order 0.

Computes the integrals

.. math::

    \int_0^x \frac{1 - J_0(t)}{t} dt \\
    \int_x^\infty \frac{Y_0(t)}{t} dt.

For more on :math:`J_0` and :math:`Y_0` see `j0` and `y0`.

Parameters
----------
x : array_like
    Values at which to evaluate the integrals.
out : tuple of ndarrays, optional
    Optional output arrays for the function results.

Returns
-------
ij0 : scalar or ndarray
    The integral for `j0`
iy0 : scalar or ndarray
    The integral for `y0`

References
----------
.. [1] S. Zhang and J.M. Jin, "Computation of Special Functions",
       Wiley 1996

Examples
--------
Evaluate the functions at one point.

>>> from scipy.special import it2j0y0
>>> int_j, int_y = it2j0y0(1.)
>>> int_j, int_y
(0.12116524699506871, 0.39527290169929336)

Evaluate the functions at several points.

>>> import numpy as np
>>> points = np.array([0.5, 1.5, 3.])
>>> int_j, int_y = it2j0y0(points)
>>> int_j, int_y
(array([0.03100699, 0.26227724, 0.85614669]),
 array([ 0.26968854,  0.29769696, -0.02987272]))

Plot the functions from 0 to 10.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 10., 1000)
>>> int_j, int_y = it2j0y0(x)
>>> ax.plot(x, int_j, label=r"$\int_0^x \frac{1-J_0(t)}{t}\,dt$")
>>> ax.plot(x, int_y, label=r"$\int_x^{\infty} \frac{Y_0(t)}{t}\,dt$")
>>> ax.legend()
>>> ax.set_ylim(-2.5, 2.5)
>>> plt.show()it2struve0it2struve0(x, out=None)

Integral related to the Struve function of order 0.

Returns the integral,

.. math::
    \int_x^\infty \frac{H_0(t)}{t}\,dt

where :math:`H_0` is the Struve function of order 0.

Parameters
----------
x : array_like
    Lower limit of integration.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
I : scalar or ndarray
    The value of the integral.

See Also
--------
struve

Notes
-----
Wrapper for a Fortran routine created by Shanjie Zhang and Jianming
Jin [1]_.

References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
       Functions", John Wiley and Sons, 1996.
       https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html

Examples
--------
Evaluate the function at one point.

>>> import numpy as np
>>> from scipy.special import it2struve0
>>> it2struve0(1.)
0.9571973506383524

Evaluate the function at several points by supplying
an array for `x`.

>>> points = np.array([1., 2., 3.5])
>>> it2struve0(points)
array([0.95719735, 0.46909296, 0.10366042])

Plot the function from -10 to 10.

>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-10., 10., 1000)
>>> it2struve0_values = it2struve0(x)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, it2struve0_values)
>>> ax.set_xlabel(r'$x$')
>>> ax.set_ylabel(r'$\int_x^{\infty}\frac{H_0(t)}{t}\,dt$')
>>> plt.show()itairyitairy(x, out=None)

Integrals of Airy functions

Calculates the integrals of Airy functions from 0 to `x`.

Parameters
----------

x : array_like
    Upper limit of integration (float).
out : tuple of ndarray, optional
    Optional output arrays for the function values

Returns
-------
Apt : scalar or ndarray
    Integral of Ai(t) from 0 to x.
Bpt : scalar or ndarray
    Integral of Bi(t) from 0 to x.
Ant : scalar or ndarray
    Integral of Ai(-t) from 0 to x.
Bnt : scalar or ndarray
    Integral of Bi(-t) from 0 to x.

Notes
-----

Wrapper for a Fortran routine created by Shanjie Zhang and Jianming
Jin [1]_.

References
----------

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
       Functions", John Wiley and Sons, 1996.
       https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html

Examples
--------
Compute the functions at ``x=1.``.

>>> import numpy as np
>>> from scipy.special import itairy
>>> import matplotlib.pyplot as plt
>>> apt, bpt, ant, bnt = itairy(1.)
>>> apt, bpt, ant, bnt
(0.23631734191710949,
 0.8727691167380077,
 0.46567398346706845,
 0.3730050096342943)

Compute the functions at several points by providing a NumPy array for `x`.

>>> x = np.array([1., 1.5, 2.5, 5])
>>> apt, bpt, ant, bnt = itairy(x)
>>> apt, bpt, ant, bnt
(array([0.23631734, 0.28678675, 0.324638  , 0.33328759]),
 array([  0.87276912,   1.62470809,   5.20906691, 321.47831857]),
 array([0.46567398, 0.72232876, 0.93187776, 0.7178822 ]),
 array([ 0.37300501,  0.35038814, -0.02812939,  0.15873094]))

Plot the functions from -10 to 10.

>>> x = np.linspace(-10, 10, 500)
>>> apt, bpt, ant, bnt = itairy(x)
>>> fig, ax = plt.subplots(figsize=(6, 5))
>>> ax.plot(x, apt, label=r"$\int_0^x\, Ai(t)\, dt$")
>>> ax.plot(x, bpt, ls="dashed", label=r"$\int_0^x\, Bi(t)\, dt$")
>>> ax.plot(x, ant, ls="dashdot", label=r"$\int_0^x\, Ai(-t)\, dt$")
>>> ax.plot(x, bnt, ls="dotted", label=r"$\int_0^x\, Bi(-t)\, dt$")
>>> ax.set_ylim(-2, 1.5)
>>> ax.legend(loc="lower right")
>>> plt.show()iti0k0iti0k0(x, out=None)

Integrals of modified Bessel functions of order 0.

Computes the integrals

.. math::

    \int_0^x I_0(t) dt \\
    \int_0^x K_0(t) dt.

For more on :math:`I_0` and :math:`K_0` see `i0` and `k0`.

Parameters
----------
x : array_like
    Values at which to evaluate the integrals.
out : tuple of ndarrays, optional
    Optional output arrays for the function results.

Returns
-------
ii0 : scalar or ndarray
    The integral for `i0`
ik0 : scalar or ndarray
    The integral for `k0`

References
----------
.. [1] S. Zhang and J.M. Jin, "Computation of Special Functions",
       Wiley 1996

Examples
--------
Evaluate the functions at one point.

>>> from scipy.special import iti0k0
>>> int_i, int_k = iti0k0(1.)
>>> int_i, int_k
(1.0865210970235892, 1.2425098486237771)

Evaluate the functions at several points.

>>> import numpy as np
>>> points = np.array([0., 1.5, 3.])
>>> int_i, int_k = iti0k0(points)
>>> int_i, int_k
(array([0.        , 1.80606937, 6.16096149]),
 array([0.        , 1.39458246, 1.53994809]))

Plot the functions from 0 to 5.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 5., 1000)
>>> int_i, int_k = iti0k0(x)
>>> ax.plot(x, int_i, label=r"$\int_0^x I_0(t)\,dt$")
>>> ax.plot(x, int_k, label=r"$\int_0^x K_0(t)\,dt$")
>>> ax.legend()
>>> plt.show()itj0y0itj0y0(x, out=None)

Integrals of Bessel functions of the first kind of order 0.

Computes the integrals

.. math::

    \int_0^x J_0(t) dt \\
    \int_0^x Y_0(t) dt.

For more on :math:`J_0` and :math:`Y_0` see `j0` and `y0`.

Parameters
----------
x : array_like
    Values at which to evaluate the integrals.
out : tuple of ndarrays, optional
    Optional output arrays for the function results.

Returns
-------
ij0 : scalar or ndarray
    The integral of `j0`
iy0 : scalar or ndarray
    The integral of `y0`

References
----------
.. [1] S. Zhang and J.M. Jin, "Computation of Special Functions",
       Wiley 1996

Examples
--------
Evaluate the functions at one point.

>>> from scipy.special import itj0y0
>>> int_j, int_y = itj0y0(1.)
>>> int_j, int_y
(0.9197304100897596, -0.637069376607422)

Evaluate the functions at several points.

>>> import numpy as np
>>> points = np.array([0., 1.5, 3.])
>>> int_j, int_y = itj0y0(points)
>>> int_j, int_y
(array([0.        , 1.24144951, 1.38756725]),
 array([ 0.        , -0.51175903,  0.19765826]))

Plot the functions from 0 to 10.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 10., 1000)
>>> int_j, int_y = itj0y0(x)
>>> ax.plot(x, int_j, label=r"$\int_0^x J_0(t)\,dt$")
>>> ax.plot(x, int_y, label=r"$\int_0^x Y_0(t)\,dt$")
>>> ax.legend()
>>> plt.show()itmodstruve0itmodstruve0(x, out=None)

Integral of the modified Struve function of order 0.

.. math::
    I = \int_0^x L_0(t)\,dt

Parameters
----------
x : array_like
    Upper limit of integration (float).
out : ndarray, optional
    Optional output array for the function values

Returns
-------
I : scalar or ndarray
    The integral of :math:`L_0` from 0 to `x`.

See Also
--------
modstruve: Modified Struve function which is integrated by this function

Notes
-----
Wrapper for a Fortran routine created by Shanjie Zhang and Jianming
Jin [1]_.

References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
       Functions", John Wiley and Sons, 1996.
       https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html

Examples
--------
Evaluate the function at one point.

>>> import numpy as np
>>> from scipy.special import itmodstruve0
>>> itmodstruve0(1.)
0.3364726286440384

Evaluate the function at several points by supplying
an array for `x`.

>>> points = np.array([1., 2., 3.5])
>>> itmodstruve0(points)
array([0.33647263, 1.588285  , 7.60382578])

Plot the function from -10 to 10.

>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-10., 10., 1000)
>>> itmodstruve0_values = itmodstruve0(x)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, itmodstruve0_values)
>>> ax.set_xlabel(r'$x$')
>>> ax.set_ylabel(r'$\int_0^xL_0(t)\,dt$')
>>> plt.show()itstruve0itstruve0(x, out=None)

Integral of the Struve function of order 0.

.. math::
    I = \int_0^x H_0(t)\,dt

Parameters
----------
x : array_like
    Upper limit of integration (float).
out : ndarray, optional
    Optional output array for the function values

Returns
-------
I : scalar or ndarray
    The integral of :math:`H_0` from 0 to `x`.

See Also
--------
struve: Function which is integrated by this function

Notes
-----
Wrapper for a Fortran routine created by Shanjie Zhang and Jianming
Jin [1]_.

References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
       Functions", John Wiley and Sons, 1996.
       https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html

Examples
--------
Evaluate the function at one point.

>>> import numpy as np
>>> from scipy.special import itstruve0
>>> itstruve0(1.)
0.30109042670805547

Evaluate the function at several points by supplying
an array for `x`.

>>> points = np.array([1., 2., 3.5])
>>> itstruve0(points)
array([0.30109043, 1.01870116, 1.96804581])

Plot the function from -20 to 20.

>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-20., 20., 1000)
>>> istruve0_values = itstruve0(x)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, istruve0_values)
>>> ax.set_xlabel(r'$x$')
>>> ax.set_ylabel(r'$\int_0^{x}H_0(t)\,dt$')
>>> plt.show()iviv(v, z, out=None)

Modified Bessel function of the first kind of real order.

Parameters
----------
v : array_like
    Order. If `z` is of real type and negative, `v` must be integer
    valued.
z : array_like of float or complex
    Argument.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
scalar or ndarray
    Values of the modified Bessel function.

See Also
--------
ive : This function with leading exponential behavior stripped off.
i0 : Faster version of this function for order 0.
i1 : Faster version of this function for order 1.

Notes
-----
For real `z` and :math:`v \in [-50, 50]`, the evaluation is carried out
using Temme's method [1]_.  For larger orders, uniform asymptotic
expansions are applied.

For complex `z` and positive `v`, the AMOS [2]_ `zbesi` routine is
called. It uses a power series for small `z`, the asymptotic expansion
for large `abs(z)`, the Miller algorithm normalized by the Wronskian
and a Neumann series for intermediate magnitudes, and the uniform
asymptotic expansions for :math:`I_v(z)` and :math:`J_v(z)` for large
orders. Backward recurrence is used to generate sequences or reduce
orders when necessary.

The calculations above are done in the right half plane and continued
into the left half plane by the formula,

.. math:: I_v(z \exp(\pm\imath\pi)) = \exp(\pm\pi v) I_v(z)

(valid when the real part of `z` is positive).  For negative `v`, the
formula

.. math:: I_{-v}(z) = I_v(z) + \frac{2}{\pi} \sin(\pi v) K_v(z)

is used, where :math:`K_v(z)` is the modified Bessel function of the
second kind, evaluated using the AMOS routine `zbesk`.

References
----------
.. [1] Temme, Journal of Computational Physics, vol 21, 343 (1976)
.. [2] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
       of a Complex Argument and Nonnegative Order",
       http://netlib.org/amos/

Examples
--------
Evaluate the function of order 0 at one point.

>>> from scipy.special import iv
>>> iv(0, 1.)
1.2660658777520084

Evaluate the function at one point for different orders.

>>> iv(0, 1.), iv(1, 1.), iv(1.5, 1.)
(1.2660658777520084, 0.565159103992485, 0.2935253263474798)

The evaluation for different orders can be carried out in one call by
providing a list or NumPy array as argument for the `v` parameter:

>>> iv([0, 1, 1.5], 1.)
array([1.26606588, 0.5651591 , 0.29352533])

Evaluate the function at several points for order 0 by providing an
array for `z`.

>>> import numpy as np
>>> points = np.array([-2., 0., 3.])
>>> iv(0, points)
array([2.2795853 , 1.        , 4.88079259])

If `z` is an array, the order parameter `v` must be broadcastable to
the correct shape if different orders shall be computed in one call.
To calculate the orders 0 and 1 for an 1D array:

>>> orders = np.array([[0], [1]])
>>> orders.shape
(2, 1)

>>> iv(orders, points)
array([[ 2.2795853 ,  1.        ,  4.88079259],
       [-1.59063685,  0.        ,  3.95337022]])

Plot the functions of order 0 to 3 from -5 to 5.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-5., 5., 1000)
>>> for i in range(4):
...     ax.plot(x, iv(i, x), label=f'$I_{i!r}$')
>>> ax.legend()
>>> plt.show()iveive(v, z, out=None)

Exponentially scaled modified Bessel function of the first kind.

Defined as::

    ive(v, z) = iv(v, z) * exp(-abs(z.real))

For imaginary numbers without a real part, returns the unscaled
Bessel function of the first kind `iv`.

Parameters
----------
v : array_like of float
    Order.
z : array_like of float or complex
    Argument.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
scalar or ndarray
    Values of the exponentially scaled modified Bessel function.

See Also
--------
iv: Modified Bessel function of the first kind
i0e: Faster implementation of this function for order 0
i1e: Faster implementation of this function for order 1

Notes
-----
For positive `v`, the AMOS [1]_ `zbesi` routine is called. It uses a
power series for small `z`, the asymptotic expansion for large
`abs(z)`, the Miller algorithm normalized by the Wronskian and a
Neumann series for intermediate magnitudes, and the uniform asymptotic
expansions for :math:`I_v(z)` and :math:`J_v(z)` for large orders.
Backward recurrence is used to generate sequences or reduce orders when
necessary.

The calculations above are done in the right half plane and continued
into the left half plane by the formula,

.. math:: I_v(z \exp(\pm\imath\pi)) = \exp(\pm\pi v) I_v(z)

(valid when the real part of `z` is positive).  For negative `v`, the
formula

.. math:: I_{-v}(z) = I_v(z) + \frac{2}{\pi} \sin(\pi v) K_v(z)

is used, where :math:`K_v(z)` is the modified Bessel function of the
second kind, evaluated using the AMOS routine `zbesk`.

`ive` is useful for large arguments `z`: for these, `iv` easily overflows,
while `ive` does not due to the exponential scaling.

References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
       of a Complex Argument and Nonnegative Order",
       http://netlib.org/amos/

Examples
--------
In the following example `iv` returns infinity whereas `ive` still returns
a finite number.

>>> from scipy.special import iv, ive
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> iv(3, 1000.), ive(3, 1000.)
(inf, 0.01256056218254712)

Evaluate the function at one point for different orders by
providing a list or NumPy array as argument for the `v` parameter:

>>> ive([0, 1, 1.5], 1.)
array([0.46575961, 0.20791042, 0.10798193])

Evaluate the function at several points for order 0 by providing an
array for `z`.

>>> points = np.array([-2., 0., 3.])
>>> ive(0, points)
array([0.30850832, 1.        , 0.24300035])

Evaluate the function at several points for different orders by
providing arrays for both `v` for `z`. Both arrays have to be
broadcastable to the correct shape. To calculate the orders 0, 1
and 2 for a 1D array of points:

>>> ive([[0], [1], [2]], points)
array([[ 0.30850832,  1.        ,  0.24300035],
       [-0.21526929,  0.        ,  0.19682671],
       [ 0.09323903,  0.        ,  0.11178255]])

Plot the functions of order 0 to 3 from -5 to 5.

>>> fig, ax = plt.subplots()
>>> x = np.linspace(-5., 5., 1000)
>>> for i in range(4):
...     ax.plot(x, ive(i, x), label=fr'$I_{i!r}(z)\cdot e^{{-|z|}}$')
>>> ax.legend()
>>> ax.set_xlabel(r"$z$")
>>> plt.show()j0j0(x, out=None)

Bessel function of the first kind of order 0.

Parameters
----------
x : array_like
    Argument (float).
out : ndarray, optional
    Optional output array for the function values

Returns
-------
J : scalar or ndarray
    Value of the Bessel function of the first kind of order 0 at `x`.

See Also
--------
jv : Bessel function of real order and complex argument.
spherical_jn : spherical Bessel functions.

Notes
-----
The domain is divided into the intervals [0, 5] and (5, infinity). In the
first interval the following rational approximation is used:

.. math::

    J_0(x) \approx (w - r_1^2)(w - r_2^2) \frac{P_3(w)}{Q_8(w)},

where :math:`w = x^2` and :math:`r_1`, :math:`r_2` are the zeros of
:math:`J_0`, and :math:`P_3` and :math:`Q_8` are polynomials of degrees 3
and 8, respectively.

In the second interval, the Hankel asymptotic expansion is employed with
two rational functions of degree 6/6 and 7/7.

This function is a wrapper for the Cephes [1]_ routine `j0`.
It should not be confused with the spherical Bessel functions (see
`spherical_jn`).

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/

Examples
--------
Calculate the function at one point:

>>> from scipy.special import j0
>>> j0(1.)
0.7651976865579665

Calculate the function at several points:

>>> import numpy as np
>>> j0(np.array([-2., 0., 4.]))
array([ 0.22389078,  1.        , -0.39714981])

Plot the function from -20 to 20.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-20., 20., 1000)
>>> y = j0(x)
>>> ax.plot(x, y)
>>> plt.show()j1j1(x, out=None)

Bessel function of the first kind of order 1.

Parameters
----------
x : array_like
    Argument (float).
out : ndarray, optional
    Optional output array for the function values

Returns
-------
J : scalar or ndarray
    Value of the Bessel function of the first kind of order 1 at `x`.

See Also
--------
jv: Bessel function of the first kind
spherical_jn: spherical Bessel functions.

Notes
-----
The domain is divided into the intervals [0, 8] and (8, infinity). In the
first interval a 24 term Chebyshev expansion is used. In the second, the
asymptotic trigonometric representation is employed using two rational
functions of degree 5/5.

This function is a wrapper for the Cephes [1]_ routine `j1`.
It should not be confused with the spherical Bessel functions (see
`spherical_jn`).

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/

Examples
--------
Calculate the function at one point:

>>> from scipy.special import j1
>>> j1(1.)
0.44005058574493355

Calculate the function at several points:

>>> import numpy as np
>>> j1(np.array([-2., 0., 4.]))
array([-0.57672481,  0.        , -0.06604333])

Plot the function from -20 to 20.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-20., 20., 1000)
>>> y = j1(x)
>>> ax.plot(x, y)
>>> plt.show()jvjv(v, z, out=None)

Bessel function of the first kind of real order and complex argument.

Parameters
----------
v : array_like
    Order (float).
z : array_like
    Argument (float or complex).
out : ndarray, optional
    Optional output array for the function values

Returns
-------
J : scalar or ndarray
    Value of the Bessel function, :math:`J_v(z)`.

See Also
--------
jve : :math:`J_v` with leading exponential behavior stripped off.
spherical_jn : spherical Bessel functions.
j0 : faster version of this function for order 0.
j1 : faster version of this function for order 1.

Notes
-----
For positive `v` values, the computation is carried out using the AMOS
[1]_ `zbesj` routine, which exploits the connection to the modified
Bessel function :math:`I_v`,

.. math::
    J_v(z) = \exp(v\pi\imath/2) I_v(-\imath z)\qquad (\Im z > 0)

    J_v(z) = \exp(-v\pi\imath/2) I_v(\imath z)\qquad (\Im z < 0)

For negative `v` values the formula,

.. math:: J_{-v}(z) = J_v(z) \cos(\pi v) - Y_v(z) \sin(\pi v)

is used, where :math:`Y_v(z)` is the Bessel function of the second
kind, computed using the AMOS routine `zbesy`.  Note that the second
term is exactly zero for integer `v`; to improve accuracy the second
term is explicitly omitted for `v` values such that `v = floor(v)`.

Not to be confused with the spherical Bessel functions (see `spherical_jn`).

References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
       of a Complex Argument and Nonnegative Order",
       http://netlib.org/amos/

Examples
--------
Evaluate the function of order 0 at one point.

>>> from scipy.special import jv
>>> jv(0, 1.)
0.7651976865579666

Evaluate the function at one point for different orders.

>>> jv(0, 1.), jv(1, 1.), jv(1.5, 1.)
(0.7651976865579666, 0.44005058574493355, 0.24029783912342725)

The evaluation for different orders can be carried out in one call by
providing a list or NumPy array as argument for the `v` parameter:

>>> jv([0, 1, 1.5], 1.)
array([0.76519769, 0.44005059, 0.24029784])

Evaluate the function at several points for order 0 by providing an
array for `z`.

>>> import numpy as np
>>> points = np.array([-2., 0., 3.])
>>> jv(0, points)
array([ 0.22389078,  1.        , -0.26005195])

If `z` is an array, the order parameter `v` must be broadcastable to
the correct shape if different orders shall be computed in one call.
To calculate the orders 0 and 1 for an 1D array:

>>> orders = np.array([[0], [1]])
>>> orders.shape
(2, 1)

>>> jv(orders, points)
array([[ 0.22389078,  1.        , -0.26005195],
       [-0.57672481,  0.        ,  0.33905896]])

Plot the functions of order 0 to 3 from -10 to 10.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-10., 10., 1000)
>>> for i in range(4):
...     ax.plot(x, jv(i, x), label=f'$J_{i!r}$')
>>> ax.legend()
>>> plt.show()jvejve(v, z, out=None)

Exponentially scaled Bessel function of the first kind of order `v`.

Defined as::

    jve(v, z) = jv(v, z) * exp(-abs(z.imag))

Parameters
----------
v : array_like
    Order (float).
z : array_like
    Argument (float or complex).
out : ndarray, optional
    Optional output array for the function values

Returns
-------
J : scalar or ndarray
    Value of the exponentially scaled Bessel function.

See Also
--------
jv: Unscaled Bessel function of the first kind

Notes
-----
For positive `v` values, the computation is carried out using the AMOS
[1]_ `zbesj` routine, which exploits the connection to the modified
Bessel function :math:`I_v`,

.. math::
    J_v(z) = \exp(v\pi\imath/2) I_v(-\imath z)\qquad (\Im z > 0)

    J_v(z) = \exp(-v\pi\imath/2) I_v(\imath z)\qquad (\Im z < 0)

For negative `v` values the formula,

.. math:: J_{-v}(z) = J_v(z) \cos(\pi v) - Y_v(z) \sin(\pi v)

is used, where :math:`Y_v(z)` is the Bessel function of the second
kind, computed using the AMOS routine `zbesy`.  Note that the second
term is exactly zero for integer `v`; to improve accuracy the second
term is explicitly omitted for `v` values such that `v = floor(v)`.

Exponentially scaled Bessel functions are useful for large arguments `z`:
for these, the unscaled Bessel functions can easily under-or overflow.

References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
       of a Complex Argument and Nonnegative Order",
       http://netlib.org/amos/

Examples
--------
Compare the output of `jv` and `jve` for large complex arguments for `z`
by computing their values for order ``v=1`` at ``z=1000j``. We see that
`jv` overflows but `jve` returns a finite number:

>>> import numpy as np
>>> from scipy.special import jv, jve
>>> v = 1
>>> z = 1000j
>>> jv(v, z), jve(v, z)
((inf+infj), (7.721967686709077e-19+0.012610930256928629j))

For real arguments for `z`, `jve` returns the same as `jv`.

>>> v, z = 1, 1000
>>> jv(v, z), jve(v, z)
(0.004728311907089523, 0.004728311907089523)

The function can be evaluated for several orders at the same time by
providing a list or NumPy array for `v`:

>>> jve([1, 3, 5], 1j)
array([1.27304208e-17+2.07910415e-01j, -4.99352086e-19-8.15530777e-03j,
       6.11480940e-21+9.98657141e-05j])

In the same way, the function can be evaluated at several points in one
call by providing a list or NumPy array for `z`:

>>> jve(1, np.array([1j, 2j, 3j]))
array([1.27308412e-17+0.20791042j, 1.31814423e-17+0.21526929j,
       1.20521602e-17+0.19682671j])

It is also possible to evaluate several orders at several points
at the same time by providing arrays for `v` and `z` with
compatible shapes for broadcasting. Compute `jve` for two different orders
`v` and three points `z` resulting in a 2x3 array.

>>> v = np.array([[1], [3]])
>>> z = np.array([1j, 2j, 3j])
>>> v.shape, z.shape
((2, 1), (3,))

>>> jve(v, z)
array([[1.27304208e-17+0.20791042j,  1.31810070e-17+0.21526929j,
        1.20517622e-17+0.19682671j],
       [-4.99352086e-19-0.00815531j, -1.76289571e-18-0.02879122j,
        -2.92578784e-18-0.04778332j]])k0k0(x, out=None)

Modified Bessel function of the second kind of order 0, :math:`K_0`.

This function is also sometimes referred to as the modified Bessel
function of the third kind of order 0.

Parameters
----------
x : array_like
    Argument (float).
out : ndarray, optional
    Optional output array for the function values

Returns
-------
K : scalar or ndarray
    Value of the modified Bessel function :math:`K_0` at `x`.

See Also
--------
kv: Modified Bessel function of the second kind of any order
k0e: Exponentially scaled modified Bessel function of the second kind

Notes
-----
The range is partitioned into the two intervals [0, 2] and (2, infinity).
Chebyshev polynomial expansions are employed in each interval.

This function is a wrapper for the Cephes [1]_ routine `k0`.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/

Examples
--------
Calculate the function at one point:

>>> from scipy.special import k0
>>> k0(1.)
0.42102443824070823

Calculate the function at several points:

>>> import numpy as np
>>> k0(np.array([0.5, 2., 3.]))
array([0.92441907, 0.11389387, 0.0347395 ])

Plot the function from 0 to 10.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 10., 1000)
>>> y = k0(x)
>>> ax.plot(x, y)
>>> plt.show()k0ek0e(x, out=None)

Exponentially scaled modified Bessel function K of order 0

Defined as::

    k0e(x) = exp(x) * k0(x).

Parameters
----------
x : array_like
    Argument (float)
out : ndarray, optional
    Optional output array for the function values

Returns
-------
K : scalar or ndarray
    Value of the exponentially scaled modified Bessel function K of order
    0 at `x`.

See Also
--------
kv: Modified Bessel function of the second kind of any order
k0: Modified Bessel function of the second kind

Notes
-----
The range is partitioned into the two intervals [0, 2] and (2, infinity).
Chebyshev polynomial expansions are employed in each interval.

This function is a wrapper for the Cephes [1]_ routine `k0e`. `k0e` is
useful for large arguments: for these, `k0` easily underflows.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/

Examples
--------
In the following example `k0` returns 0 whereas `k0e` still returns a
useful finite number:

>>> from scipy.special import k0, k0e
>>> k0(1000.), k0e(1000)
(0., 0.03962832160075422)

Calculate the function at several points by providing a NumPy array or
list for `x`:

>>> import numpy as np
>>> k0e(np.array([0.5, 2., 3.]))
array([1.52410939, 0.84156822, 0.6977616 ])

Plot the function from 0 to 10.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 10., 1000)
>>> y = k0e(x)
>>> ax.plot(x, y)
>>> plt.show()k1k1(x, out=None)

Modified Bessel function of the second kind of order 1, :math:`K_1(x)`.

Parameters
----------
x : array_like
    Argument (float)
out : ndarray, optional
    Optional output array for the function values

Returns
-------
K : scalar or ndarray
    Value of the modified Bessel function K of order 1 at `x`.

See Also
--------
kv: Modified Bessel function of the second kind of any order
k1e: Exponentially scaled modified Bessel function K of order 1

Notes
-----
The range is partitioned into the two intervals [0, 2] and (2, infinity).
Chebyshev polynomial expansions are employed in each interval.

This function is a wrapper for the Cephes [1]_ routine `k1`.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/

Examples
--------
Calculate the function at one point:

>>> from scipy.special import k1
>>> k1(1.)
0.6019072301972346

Calculate the function at several points:

>>> import numpy as np
>>> k1(np.array([0.5, 2., 3.]))
array([1.65644112, 0.13986588, 0.04015643])

Plot the function from 0 to 10.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 10., 1000)
>>> y = k1(x)
>>> ax.plot(x, y)
>>> plt.show()k1ek1e(x, out=None)

Exponentially scaled modified Bessel function K of order 1

Defined as::

    k1e(x) = exp(x) * k1(x)

Parameters
----------
x : array_like
    Argument (float)
out : ndarray, optional
    Optional output array for the function values

Returns
-------
K : scalar or ndarray
    Value of the exponentially scaled modified Bessel function K of order
    1 at `x`.

See Also
--------
kv: Modified Bessel function of the second kind of any order
k1: Modified Bessel function of the second kind of order 1

Notes
-----
The range is partitioned into the two intervals [0, 2] and (2, infinity).
Chebyshev polynomial expansions are employed in each interval.

This function is a wrapper for the Cephes [1]_ routine `k1e`.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/

Examples
--------
In the following example `k1` returns 0 whereas `k1e` still returns a
useful floating point number.

>>> from scipy.special import k1, k1e
>>> k1(1000.), k1e(1000.)
(0., 0.03964813081296021)

Calculate the function at several points by providing a NumPy array or
list for `x`:

>>> import numpy as np
>>> k1e(np.array([0.5, 2., 3.]))
array([2.73100971, 1.03347685, 0.80656348])

Plot the function from 0 to 10.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 10., 1000)
>>> y = k1e(x)
>>> ax.plot(x, y)
>>> plt.show()keikei(x, out=None)

Kelvin function kei.

Defined as

.. math::

    \mathrm{kei}(x) = \Im[K_0(x e^{\pi i / 4})]

where :math:`K_0` is the modified Bessel function of the second
kind (see `kv`). See [dlmf]_ for more details.

Parameters
----------
x : array_like
    Real argument.
out : ndarray, optional
    Optional output array for the function results.

Returns
-------
scalar or ndarray
    Values of the Kelvin function.

See Also
--------
ker : the corresponding real part
keip : the derivative of kei
kv : modified Bessel function of the second kind

References
----------
.. [dlmf] NIST, Digital Library of Mathematical Functions,
    https://dlmf.nist.gov/10.61

Examples
--------
It can be expressed using the modified Bessel function of the
second kind.

>>> import numpy as np
>>> import scipy.special as sc
>>> x = np.array([1.0, 2.0, 3.0, 4.0])
>>> sc.kv(0, x * np.exp(np.pi * 1j / 4)).imag
array([-0.49499464, -0.20240007, -0.05112188,  0.0021984 ])
>>> sc.kei(x)
array([-0.49499464, -0.20240007, -0.05112188,  0.0021984 ])keipkeip(x, out=None)

Derivative of the Kelvin function kei.

Parameters
----------
x : array_like
    Real argument.
out : ndarray, optional
    Optional output array for the function results.

Returns
-------
scalar or ndarray
    The values of the derivative of kei.

See Also
--------
kei

References
----------
.. [dlmf] NIST, Digital Library of Mathematical Functions,
    https://dlmf.nist.gov/10#PT5kelvinkelvin(x, out=None)

Kelvin functions as complex numbers

Parameters
----------
x : array_like
    Argument
out : tuple of ndarray, optional
    Optional output arrays for the function values

Returns
-------
Be, Ke, Bep, Kep : 4-tuple of scalar or ndarray
    The tuple (Be, Ke, Bep, Kep) contains complex numbers
    representing the real and imaginary Kelvin functions and their
    derivatives evaluated at `x`.  For example, kelvin(x)[0].real =
    ber x and kelvin(x)[0].imag = bei x with similar relationships
    for ker and kei.kerker(x, out=None)

Kelvin function ker.

Defined as

.. math::

    \mathrm{ker}(x) = \Re[K_0(x e^{\pi i / 4})]

Where :math:`K_0` is the modified Bessel function of the second
kind (see `kv`). See [dlmf]_ for more details.

Parameters
----------
x : array_like
    Real argument.
out : ndarray, optional
    Optional output array for the function results.

Returns
-------
scalar or ndarray
    Values of the Kelvin function.

See Also
--------
kei : the corresponding imaginary part
kerp : the derivative of ker
kv : modified Bessel function of the second kind

References
----------
.. [dlmf] NIST, Digital Library of Mathematical Functions,
    https://dlmf.nist.gov/10.61

Examples
--------
It can be expressed using the modified Bessel function of the
second kind.

>>> import numpy as np
>>> import scipy.special as sc
>>> x = np.array([1.0, 2.0, 3.0, 4.0])
>>> sc.kv(0, x * np.exp(np.pi * 1j / 4)).real
array([ 0.28670621, -0.04166451, -0.06702923, -0.03617885])
>>> sc.ker(x)
array([ 0.28670621, -0.04166451, -0.06702923, -0.03617885])kerpkerp(x, out=None)

Derivative of the Kelvin function ker.

Parameters
----------
x : array_like
    Real argument.
out : ndarray, optional
    Optional output array for the function results.

Returns
-------
scalar or ndarray
    Values of the derivative of ker.

See Also
--------
ker

References
----------
.. [dlmf] NIST, Digital Library of Mathematical Functions,
    https://dlmf.nist.gov/10#PT5kl_divkl_div(x, y, out=None)

Elementwise function for computing Kullback-Leibler divergence.

.. math::

    \mathrm{kl\_div}(x, y) =
      \begin{cases}
        x \log(x / y) - x + y & x > 0, y > 0 \\
        y & x = 0, y \ge 0 \\
        \infty & \text{otherwise}
      \end{cases}

Parameters
----------
x, y : array_like
    Real arguments
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    Values of the Kullback-Liebler divergence.

See Also
--------
entr, rel_entr, scipy.stats.entropy

Notes
-----
.. versionadded:: 0.15.0

This function is non-negative and is jointly convex in `x` and `y`.

The origin of this function is in convex programming; see [1]_ for
details. This is why the function contains the extra :math:`-x
+ y` terms over what might be expected from the Kullback-Leibler
divergence. For a version of the function without the extra terms,
see `rel_entr`.

References
----------
.. [1] Boyd, Stephen and Lieven Vandenberghe. *Convex optimization*.
       Cambridge University Press, 2004.
       :doi:`https://doi.org/10.1017/CBO9780511804441`knkn(n, x, out=None)

Modified Bessel function of the second kind of integer order `n`

Returns the modified Bessel function of the second kind for integer order
`n` at real `z`.

These are also sometimes called functions of the third kind, Basset
functions, or Macdonald functions.

Parameters
----------
n : array_like of int
    Order of Bessel functions (floats will truncate with a warning)
x : array_like of float
    Argument at which to evaluate the Bessel functions
out : ndarray, optional
    Optional output array for the function results.

Returns
-------
scalar or ndarray
    Value of the Modified Bessel function of the second kind,
    :math:`K_n(x)`.

See Also
--------
kv : Same function, but accepts real order and complex argument
kvp : Derivative of this function

Notes
-----
Wrapper for AMOS [1]_ routine `zbesk`.  For a discussion of the
algorithm used, see [2]_ and the references therein.

References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
       of a Complex Argument and Nonnegative Order",
       http://netlib.org/amos/
.. [2] Donald E. Amos, "Algorithm 644: A portable package for Bessel
       functions of a complex argument and nonnegative order", ACM
       TOMS Vol. 12 Issue 3, Sept. 1986, p. 265

Examples
--------
Plot the function of several orders for real input:

>>> import numpy as np
>>> from scipy.special import kn
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(0, 5, 1000)
>>> for N in range(6):
...     plt.plot(x, kn(N, x), label='$K_{}(x)$'.format(N))
>>> plt.ylim(0, 10)
>>> plt.legend()
>>> plt.title(r'Modified Bessel function of the second kind $K_n(x)$')
>>> plt.show()

Calculate for a single value at multiple orders:

>>> kn([4, 5, 6], 1)
array([   44.23241585,   360.9605896 ,  3653.83831186])kolmogikolmogi(p, out=None)

Inverse Survival Function of Kolmogorov distribution

It is the inverse function to `kolmogorov`.
Returns y such that ``kolmogorov(y) == p``.

Parameters
----------
p : float array_like
    Probability
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    The value(s) of kolmogi(p)

See Also
--------
kolmogorov : The Survival Function for the distribution
scipy.stats.kstwobign : Provides the functionality as a continuous distribution
smirnov, smirnovi : Functions for the one-sided distribution

Notes
-----
`kolmogorov` is used by `stats.kstest` in the application of the
Kolmogorov-Smirnov Goodness of Fit test. For historical reasons this
function is exposed in `scpy.special`, but the recommended way to achieve
the most accurate CDF/SF/PDF/PPF/ISF computations is to use the
`stats.kstwobign` distribution.

Examples
--------
>>> from scipy.special import kolmogi
>>> kolmogi([0, 0.1, 0.25, 0.5, 0.75, 0.9, 1.0])
array([        inf,  1.22384787,  1.01918472,  0.82757356,  0.67644769,
        0.57117327,  0.        ])kolmogorovkolmogorov(y, out=None)

Complementary cumulative distribution (Survival Function) function of
Kolmogorov distribution.

Returns the complementary cumulative distribution function of
Kolmogorov's limiting distribution (``D_n*\sqrt(n)`` as n goes to infinity)
of a two-sided test for equality between an empirical and a theoretical
distribution. It is equal to the (limit as n->infinity of the)
probability that ``sqrt(n) * max absolute deviation > y``.

Parameters
----------
y : float array_like
  Absolute deviation between the Empirical CDF (ECDF) and the target CDF,
  multiplied by sqrt(n).
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    The value(s) of kolmogorov(y)

See Also
--------
kolmogi : The Inverse Survival Function for the distribution
scipy.stats.kstwobign : Provides the functionality as a continuous distribution
smirnov, smirnovi : Functions for the one-sided distribution

Notes
-----
`kolmogorov` is used by `stats.kstest` in the application of the
Kolmogorov-Smirnov Goodness of Fit test. For historical reasons this
function is exposed in `scpy.special`, but the recommended way to achieve
the most accurate CDF/SF/PDF/PPF/ISF computations is to use the
`stats.kstwobign` distribution.

Examples
--------
Show the probability of a gap at least as big as 0, 0.5 and 1.0.

>>> import numpy as np
>>> from scipy.special import kolmogorov
>>> from scipy.stats import kstwobign
>>> kolmogorov([0, 0.5, 1.0])
array([ 1.        ,  0.96394524,  0.26999967])

Compare a sample of size 1000 drawn from a Laplace(0, 1) distribution against
the target distribution, a Normal(0, 1) distribution.

>>> from scipy.stats import norm, laplace
>>> rng = np.random.default_rng()
>>> n = 1000
>>> lap01 = laplace(0, 1)
>>> x = np.sort(lap01.rvs(n, random_state=rng))
>>> np.mean(x), np.std(x)
(-0.05841730131499543, 1.3968109101997568)

Construct the Empirical CDF and the K-S statistic Dn.

>>> target = norm(0,1)  # Normal mean 0, stddev 1
>>> cdfs = target.cdf(x)
>>> ecdfs = np.arange(n+1, dtype=float)/n
>>> gaps = np.column_stack([cdfs - ecdfs[:n], ecdfs[1:] - cdfs])
>>> Dn = np.max(gaps)
>>> Kn = np.sqrt(n) * Dn
>>> print('Dn=%f, sqrt(n)*Dn=%f' % (Dn, Kn))
Dn=0.043363, sqrt(n)*Dn=1.371265
>>> print(chr(10).join(['For a sample of size n drawn from a N(0, 1) distribution:',
...   ' the approximate Kolmogorov probability that sqrt(n)*Dn>=%f is %f' %
...    (Kn, kolmogorov(Kn)),
...   ' the approximate Kolmogorov probability that sqrt(n)*Dn<=%f is %f' %
...    (Kn, kstwobign.cdf(Kn))]))
For a sample of size n drawn from a N(0, 1) distribution:
 the approximate Kolmogorov probability that sqrt(n)*Dn>=1.371265 is 0.046533
 the approximate Kolmogorov probability that sqrt(n)*Dn<=1.371265 is 0.953467

Plot the Empirical CDF against the target N(0, 1) CDF.

>>> import matplotlib.pyplot as plt
>>> plt.step(np.concatenate([[-3], x]), ecdfs, where='post', label='Empirical CDF')
>>> x3 = np.linspace(-3, 3, 100)
>>> plt.plot(x3, target.cdf(x3), label='CDF for N(0, 1)')
>>> plt.ylim([0, 1]); plt.grid(True); plt.legend();
>>> # Add vertical lines marking Dn+ and Dn-
>>> iminus, iplus = np.argmax(gaps, axis=0)
>>> plt.vlines([x[iminus]], ecdfs[iminus], cdfs[iminus],
...            color='r', linestyle='dashed', lw=4)
>>> plt.vlines([x[iplus]], cdfs[iplus], ecdfs[iplus+1],
...            color='r', linestyle='dashed', lw=4)
>>> plt.show()kvkv(v, z, out=None)

Modified Bessel function of the second kind of real order `v`

Returns the modified Bessel function of the second kind for real order
`v` at complex `z`.

These are also sometimes called functions of the third kind, Basset
functions, or Macdonald functions.  They are defined as those solutions
of the modified Bessel equation for which,

.. math::
    K_v(x) \sim \sqrt{\pi/(2x)} \exp(-x)

as :math:`x \to \infty` [3]_.

Parameters
----------
v : array_like of float
    Order of Bessel functions
z : array_like of complex
    Argument at which to evaluate the Bessel functions
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    The results. Note that input must be of complex type to get complex
    output, e.g. ``kv(3, -2+0j)`` instead of ``kv(3, -2)``.

See Also
--------
kve : This function with leading exponential behavior stripped off.
kvp : Derivative of this function

Notes
-----
Wrapper for AMOS [1]_ routine `zbesk`.  For a discussion of the
algorithm used, see [2]_ and the references therein.

References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
       of a Complex Argument and Nonnegative Order",
       http://netlib.org/amos/
.. [2] Donald E. Amos, "Algorithm 644: A portable package for Bessel
       functions of a complex argument and nonnegative order", ACM
       TOMS Vol. 12 Issue 3, Sept. 1986, p. 265
.. [3] NIST Digital Library of Mathematical Functions,
       Eq. 10.25.E3. https://dlmf.nist.gov/10.25.E3

Examples
--------
Plot the function of several orders for real input:

>>> import numpy as np
>>> from scipy.special import kv
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(0, 5, 1000)
>>> for N in np.linspace(0, 6, 5):
...     plt.plot(x, kv(N, x), label='$K_{{{}}}(x)$'.format(N))
>>> plt.ylim(0, 10)
>>> plt.legend()
>>> plt.title(r'Modified Bessel function of the second kind $K_\nu(x)$')
>>> plt.show()

Calculate for a single value at multiple orders:

>>> kv([4, 4.5, 5], 1+2j)
array([ 0.1992+2.3892j,  2.3493+3.6j   ,  7.2827+3.8104j])kvekve(v, z, out=None)

Exponentially scaled modified Bessel function of the second kind.

Returns the exponentially scaled, modified Bessel function of the
second kind (sometimes called the third kind) for real order `v` at
complex `z`::

    kve(v, z) = kv(v, z) * exp(z)

Parameters
----------
v : array_like of float
    Order of Bessel functions
z : array_like of complex
    Argument at which to evaluate the Bessel functions
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    The exponentially scaled modified Bessel function of the second kind.

See Also
--------
kv : This function without exponential scaling.
k0e : Faster version of this function for order 0.
k1e : Faster version of this function for order 1.

Notes
-----
Wrapper for AMOS [1]_ routine `zbesk`.  For a discussion of the
algorithm used, see [2]_ and the references therein.

References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
       of a Complex Argument and Nonnegative Order",
       http://netlib.org/amos/
.. [2] Donald E. Amos, "Algorithm 644: A portable package for Bessel
       functions of a complex argument and nonnegative order", ACM
       TOMS Vol. 12 Issue 3, Sept. 1986, p. 265

Examples
--------
In the following example `kv` returns 0 whereas `kve` still returns
a useful finite number.

>>> import numpy as np
>>> from scipy.special import kv, kve
>>> import matplotlib.pyplot as plt
>>> kv(3, 1000.), kve(3, 1000.)
(0.0, 0.03980696128440973)

Evaluate the function at one point for different orders by
providing a list or NumPy array as argument for the `v` parameter:

>>> kve([0, 1, 1.5], 1.)
array([1.14446308, 1.63615349, 2.50662827])

Evaluate the function at several points for order 0 by providing an
array for `z`.

>>> points = np.array([1., 3., 10.])
>>> kve(0, points)
array([1.14446308, 0.6977616 , 0.39163193])

Evaluate the function at several points for different orders by
providing arrays for both `v` for `z`. Both arrays have to be
broadcastable to the correct shape. To calculate the orders 0, 1
and 2 for a 1D array of points:

>>> kve([[0], [1], [2]], points)
array([[1.14446308, 0.6977616 , 0.39163193],
       [1.63615349, 0.80656348, 0.41076657],
       [4.41677005, 1.23547058, 0.47378525]])

Plot the functions of order 0 to 3 from 0 to 5.

>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 5., 1000)
>>> for i in range(4):
...     ax.plot(x, kve(i, x), label=fr'$K_{i!r}(z)\cdot e^z$')
>>> ax.legend()
>>> ax.set_xlabel(r"$z$")
>>> ax.set_ylim(0, 4)
>>> ax.set_xlim(0, 5)
>>> plt.show()log1plog1p(x, out=None)

Calculates log(1 + x) for use when `x` is near zero.

Parameters
----------
x : array_like
    Real or complex valued input.
out : ndarray, optional
    Optional output array for the function results.

Returns
-------
scalar or ndarray
    Values of ``log(1 + x)``.

See Also
--------
expm1, cosm1

Examples
--------
>>> import numpy as np
>>> import scipy.special as sc

It is more accurate than using ``log(1 + x)`` directly for ``x``
near 0. Note that in the below example ``1 + 1e-17 == 1`` to
double precision.

>>> sc.log1p(1e-17)
1e-17
>>> np.log(1 + 1e-17)
0.0log_expitlog_expit(x, out=None)

Logarithm of the logistic sigmoid function.

The SciPy implementation of the logistic sigmoid function is
`scipy.special.expit`, so this function is called ``log_expit``.

The function is mathematically equivalent to ``log(expit(x))``, but
is formulated to avoid loss of precision for inputs with large
(positive or negative) magnitude.

Parameters
----------
x : array_like
    The values to apply ``log_expit`` to element-wise.
out : ndarray, optional
    Optional output array for the function results

Returns
-------
out : scalar or ndarray
    The computed values, an ndarray of the same shape as ``x``.

See Also
--------
expit

Notes
-----
As a ufunc, ``log_expit`` takes a number of optional keyword arguments.
For more information see
`ufuncs <https://docs.scipy.org/doc/numpy/reference/ufuncs.html>`_

.. versionadded:: 1.8.0

Examples
--------
>>> import numpy as np
>>> from scipy.special import log_expit, expit

>>> log_expit([-3.0, 0.25, 2.5, 5.0])
array([-3.04858735, -0.57593942, -0.07888973, -0.00671535])

Large negative values:

>>> log_expit([-100, -500, -1000])
array([ -100.,  -500., -1000.])

Note that ``expit(-1000)`` returns 0, so the naive implementation
``log(expit(-1000))`` return ``-inf``.

Large positive values:

>>> log_expit([29, 120, 400])
array([-2.54366565e-013, -7.66764807e-053, -1.91516960e-174])

Compare that to the naive implementation:

>>> np.log(expit([29, 120, 400]))
array([-2.54463117e-13,  0.00000000e+00,  0.00000000e+00])

The first value is accurate to only 3 digits, and the larger inputs
lose all precision and return 0.log_ndtrlog_ndtr(x, out=None)

Logarithm of Gaussian cumulative distribution function.

Returns the log of the area under the standard Gaussian probability
density function, integrated from minus infinity to `x`::

    log(1/sqrt(2*pi) * integral(exp(-t**2 / 2), t=-inf..x))

Parameters
----------
x : array_like, real or complex
    Argument
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    The value of the log of the normal CDF evaluated at `x`

See Also
--------
erf
erfc
scipy.stats.norm
ndtr

Examples
--------
>>> import numpy as np
>>> from scipy.special import log_ndtr, ndtr

The benefit of ``log_ndtr(x)`` over the naive implementation
``np.log(ndtr(x))`` is most evident with moderate to large positive
values of ``x``:

>>> x = np.array([6, 7, 9, 12, 15, 25])
>>> log_ndtr(x)
array([-9.86587646e-010, -1.27981254e-012, -1.12858841e-019,
       -1.77648211e-033, -3.67096620e-051, -3.05669671e-138])

The results of the naive calculation for the moderate ``x`` values
have only 5 or 6 correct significant digits. For values of ``x``
greater than approximately 8.3, the naive expression returns 0:

>>> np.log(ndtr(x))
array([-9.86587701e-10, -1.27986510e-12,  0.00000000e+00,
        0.00000000e+00,  0.00000000e+00,  0.00000000e+00])loggammaloggamma(z, out=None)

Principal branch of the logarithm of the gamma function.

Defined to be :math:`\log(\Gamma(x))` for :math:`x > 0` and
extended to the complex plane by analytic continuation. The
function has a single branch cut on the negative real axis.

.. versionadded:: 0.18.0

Parameters
----------
z : array_like
    Values in the complex plane at which to compute ``loggamma``
out : ndarray, optional
    Output array for computed values of ``loggamma``

Returns
-------
loggamma : scalar or ndarray
    Values of ``loggamma`` at z.

See Also
--------
gammaln : logarithm of the absolute value of the gamma function
gammasgn : sign of the gamma function

Notes
-----
It is not generally true that :math:`\log\Gamma(z) =
\log(\Gamma(z))`, though the real parts of the functions do
agree. The benefit of not defining `loggamma` as
:math:`\log(\Gamma(z))` is that the latter function has a
complicated branch cut structure whereas `loggamma` is analytic
except for on the negative real axis.

The identities

.. math::
  \exp(\log\Gamma(z)) &= \Gamma(z) \\
  \log\Gamma(z + 1) &= \log(z) + \log\Gamma(z)

make `loggamma` useful for working in complex logspace.

On the real line `loggamma` is related to `gammaln` via
``exp(loggamma(x + 0j)) = gammasgn(x)*exp(gammaln(x))``, up to
rounding error.

The implementation here is based on [hare1997]_.

References
----------
.. [hare1997] D.E.G. Hare,
  *Computing the Principal Branch of log-Gamma*,
  Journal of Algorithms, Volume 25, Issue 2, November 1997, pages 221-236.logitlogit(x, out=None)

Logit ufunc for ndarrays.

The logit function is defined as logit(p) = log(p/(1-p)).
Note that logit(0) = -inf, logit(1) = inf, and logit(p)
for p<0 or p>1 yields nan.

Parameters
----------
x : ndarray
    The ndarray to apply logit to element-wise.
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    An ndarray of the same shape as x. Its entries
    are logit of the corresponding entry of x.

See Also
--------
expit

Notes
-----
As a ufunc logit takes a number of optional
keyword arguments. For more information
see `ufuncs <https://docs.scipy.org/doc/numpy/reference/ufuncs.html>`_

.. versionadded:: 0.10.0

Examples
--------
>>> import numpy as np
>>> from scipy.special import logit, expit

>>> logit([0, 0.25, 0.5, 0.75, 1])
array([       -inf, -1.09861229,  0.        ,  1.09861229,         inf])

`expit` is the inverse of `logit`:

>>> expit(logit([0.1, 0.75, 0.999]))
array([ 0.1  ,  0.75 ,  0.999])

Plot logit(x) for x in [0, 1]:

>>> import matplotlib.pyplot as plt
>>> x = np.linspace(0, 1, 501)
>>> y = logit(x)
>>> plt.plot(x, y)
>>> plt.grid()
>>> plt.ylim(-6, 6)
>>> plt.xlabel('x')
>>> plt.title('logit(x)')
>>> plt.show()lpmvlpmv(m, v, x, out=None)

Associated Legendre function of integer order and real degree.

Defined as

.. math::

    P_v^m = (-1)^m (1 - x^2)^{m/2} \frac{d^m}{dx^m} P_v(x)

where

.. math::

    P_v = \sum_{k = 0}^\infty \frac{(-v)_k (v + 1)_k}{(k!)^2}
            \left(\frac{1 - x}{2}\right)^k

is the Legendre function of the first kind. Here :math:`(\cdot)_k`
is the Pochhammer symbol; see `poch`.

Parameters
----------
m : array_like
    Order (int or float). If passed a float not equal to an
    integer the function returns NaN.
v : array_like
    Degree (float).
x : array_like
    Argument (float). Must have ``|x| <= 1``.
out : ndarray, optional
    Optional output array for the function results

Returns
-------
pmv : scalar or ndarray
    Value of the associated Legendre function.

See Also
--------
lpmn : Compute the associated Legendre function for all orders
       ``0, ..., m`` and degrees ``0, ..., n``.
clpmn : Compute the associated Legendre function at complex
        arguments.

Notes
-----
Note that this implementation includes the Condon-Shortley phase.

References
----------
.. [1] Zhang, Jin, "Computation of Special Functions", John Wiley
       and Sons, Inc, 1996.mathieu_amathieu_a(m, q, out=None)

Characteristic value of even Mathieu functions

Parameters
----------
m : array_like
    Order of the function
q : array_like
    Parameter of the function
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    Characteristic value for the even solution, ``ce_m(z, q)``, of
    Mathieu's equation.

See Also
--------
mathieu_b, mathieu_cem, mathieu_semmathieu_bmathieu_b(m, q, out=None)

Characteristic value of odd Mathieu functions

Parameters
----------
m : array_like
    Order of the function
q : array_like
    Parameter of the function
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    Characteristic value for the odd solution, ``se_m(z, q)``, of Mathieu's
    equation.

See Also
--------
mathieu_a, mathieu_cem, mathieu_semmathieu_cemmathieu_cem(m, q, x, out=None)

Even Mathieu function and its derivative

Returns the even Mathieu function, ``ce_m(x, q)``, of order `m` and
parameter `q` evaluated at `x` (given in degrees).  Also returns the
derivative with respect to `x` of ce_m(x, q)

Parameters
----------
m : array_like
    Order of the function
q : array_like
    Parameter of the function
x : array_like
    Argument of the function, *given in degrees, not radians*
out : tuple of ndarray, optional
    Optional output arrays for the function results

Returns
-------
y : scalar or ndarray
    Value of the function
yp : scalar or ndarray
    Value of the derivative vs x

See Also
--------
mathieu_a, mathieu_b, mathieu_semmathieu_modcem1mathieu_modcem1(m, q, x, out=None)

Even modified Mathieu function of the first kind and its derivative

Evaluates the even modified Mathieu function of the first kind,
``Mc1m(x, q)``, and its derivative at `x` for order `m` and parameter
`q`.

Parameters
----------
m : array_like
    Order of the function
q : array_like
    Parameter of the function
x : array_like
    Argument of the function, *given in degrees, not radians*
out : tuple of ndarray, optional
    Optional output arrays for the function results

Returns
-------
y : scalar or ndarray
    Value of the function
yp : scalar or ndarray
    Value of the derivative vs x

See Also
--------
mathieu_modsem1mathieu_modcem2mathieu_modcem2(m, q, x, out=None)

Even modified Mathieu function of the second kind and its derivative

Evaluates the even modified Mathieu function of the second kind,
Mc2m(x, q), and its derivative at `x` (given in degrees) for order `m`
and parameter `q`.

Parameters
----------
m : array_like
    Order of the function
q : array_like
    Parameter of the function
x : array_like
    Argument of the function, *given in degrees, not radians*
out : tuple of ndarray, optional
    Optional output arrays for the function results

Returns
-------
y : scalar or ndarray
    Value of the function
yp : scalar or ndarray
    Value of the derivative vs x

See Also
--------
mathieu_modsem2mathieu_modsem1mathieu_modsem1(m, q, x, out=None)

Odd modified Mathieu function of the first kind and its derivative

Evaluates the odd modified Mathieu function of the first kind,
Ms1m(x, q), and its derivative at `x` (given in degrees) for order `m`
and parameter `q`.

Parameters
----------
m : array_like
    Order of the function
q : array_like
    Parameter of the function
x : array_like
    Argument of the function, *given in degrees, not radians*
out : tuple of ndarray, optional
    Optional output arrays for the function results

Returns
-------
y : scalar or ndarray
    Value of the function
yp : scalar or ndarray
    Value of the derivative vs x

See Also
--------
mathieu_modcem1mathieu_modsem2mathieu_modsem2(m, q, x, out=None)

Odd modified Mathieu function of the second kind and its derivative

Evaluates the odd modified Mathieu function of the second kind,
Ms2m(x, q), and its derivative at `x` (given in degrees) for order `m`
and parameter q.

Parameters
----------
m : array_like
    Order of the function
q : array_like
    Parameter of the function
x : array_like
    Argument of the function, *given in degrees, not radians*
out : tuple of ndarray, optional
    Optional output arrays for the function results

Returns
-------
y : scalar or ndarray
    Value of the function
yp : scalar or ndarray
    Value of the derivative vs x

See Also
--------
mathieu_modcem2mathieu_semmathieu_sem(m, q, x, out=None)

Odd Mathieu function and its derivative

Returns the odd Mathieu function, se_m(x, q), of order `m` and
parameter `q` evaluated at `x` (given in degrees).  Also returns the
derivative with respect to `x` of se_m(x, q).

Parameters
----------
m : array_like
    Order of the function
q : array_like
    Parameter of the function
x : array_like
    Argument of the function, *given in degrees, not radians*.
out : tuple of ndarray, optional
    Optional output arrays for the function results

Returns
-------
y : scalar or ndarray
    Value of the function
yp : scalar or ndarray
    Value of the derivative vs x

See Also
--------
mathieu_a, mathieu_b, mathieu_cemmodfresnelmmodfresnelm(x, out=None)

Modified Fresnel negative integrals

Parameters
----------
x : array_like
    Function argument
out : tuple of ndarray, optional
    Optional output arrays for the function results

Returns
-------
fm : scalar or ndarray
    Integral ``F_-(x)``: ``integral(exp(-1j*t*t), t=x..inf)``
km : scalar or ndarray
    Integral ``K_-(x)``: ``1/sqrt(pi)*exp(1j*(x*x+pi/4))*fp``

See Also
--------
modfresnelpmodfresnelpmodfresnelp(x, out=None)

Modified Fresnel positive integrals

Parameters
----------
x : array_like
    Function argument
out : tuple of ndarray, optional
    Optional output arrays for the function results

Returns
-------
fp : scalar or ndarray
    Integral ``F_+(x)``: ``integral(exp(1j*t*t), t=x..inf)``
kp : scalar or ndarray
    Integral ``K_+(x)``: ``1/sqrt(pi)*exp(-1j*(x*x+pi/4))*fp``

See Also
--------
modfresnelmmodstruvemodstruve(v, x, out=None)

Modified Struve function.

Return the value of the modified Struve function of order `v` at `x`.  The
modified Struve function is defined as,

.. math::
    L_v(x) = -\imath \exp(-\pi\imath v/2) H_v(\imath x),

where :math:`H_v` is the Struve function.

Parameters
----------
v : array_like
    Order of the modified Struve function (float).
x : array_like
    Argument of the Struve function (float; must be positive unless `v` is
    an integer).
out : ndarray, optional
    Optional output array for the function results

Returns
-------
L : scalar or ndarray
    Value of the modified Struve function of order `v` at `x`.

See Also
--------
struve

Notes
-----
Three methods discussed in [1]_ are used to evaluate the function:

- power series
- expansion in Bessel functions (if :math:`|x| < |v| + 20`)
- asymptotic large-x expansion (if :math:`x \geq 0.7v + 12`)

Rounding errors are estimated based on the largest terms in the sums, and
the result associated with the smallest error is returned.

References
----------
.. [1] NIST Digital Library of Mathematical Functions
       https://dlmf.nist.gov/11

Examples
--------
Calculate the modified Struve function of order 1 at 2.

>>> import numpy as np
>>> from scipy.special import modstruve
>>> import matplotlib.pyplot as plt
>>> modstruve(1, 2.)
1.102759787367716

Calculate the modified Struve function at 2 for orders 1, 2 and 3 by
providing a list for the order parameter `v`.

>>> modstruve([1, 2, 3], 2.)
array([1.10275979, 0.41026079, 0.11247294])

Calculate the modified Struve function of order 1 for several points
by providing an array for `x`.

>>> points = np.array([2., 5., 8.])
>>> modstruve(1, points)
array([  1.10275979,  23.72821578, 399.24709139])

Compute the modified Struve function for several orders at several
points by providing arrays for `v` and `z`. The arrays have to be
broadcastable to the correct shapes.

>>> orders = np.array([[1], [2], [3]])
>>> points.shape, orders.shape
((3,), (3, 1))

>>> modstruve(orders, points)
array([[1.10275979e+00, 2.37282158e+01, 3.99247091e+02],
       [4.10260789e-01, 1.65535979e+01, 3.25973609e+02],
       [1.12472937e-01, 9.42430454e+00, 2.33544042e+02]])

Plot the modified Struve functions of order 0 to 3 from -5 to 5.

>>> fig, ax = plt.subplots()
>>> x = np.linspace(-5., 5., 1000)
>>> for i in range(4):
...     ax.plot(x, modstruve(i, x), label=f'$L_{i!r}$')
>>> ax.legend(ncol=2)
>>> ax.set_xlim(-5, 5)
>>> ax.set_title(r"Modified Struve functions $L_{\nu}$")
>>> plt.show()nbdtrnbdtr(k, n, p, out=None)

Negative binomial cumulative distribution function.

Returns the sum of the terms 0 through `k` of the negative binomial
distribution probability mass function,

.. math::

    F = \sum_{j=0}^k {{n + j - 1}\choose{j}} p^n (1 - p)^j.

In a sequence of Bernoulli trials with individual success probabilities
`p`, this is the probability that `k` or fewer failures precede the nth
success.

Parameters
----------
k : array_like
    The maximum number of allowed failures (nonnegative int).
n : array_like
    The target number of successes (positive int).
p : array_like
    Probability of success in a single event (float).
out : ndarray, optional
    Optional output array for the function results

Returns
-------
F : scalar or ndarray
    The probability of `k` or fewer failures before `n` successes in a
    sequence of events with individual success probability `p`.

See Also
--------
nbdtrc : Negative binomial survival function
nbdtrik : Negative binomial quantile function
scipy.stats.nbinom : Negative binomial distribution

Notes
-----
If floating point values are passed for `k` or `n`, they will be truncated
to integers.

The terms are not summed directly; instead the regularized incomplete beta
function is employed, according to the formula,

.. math::
    \mathrm{nbdtr}(k, n, p) = I_{p}(n, k + 1).

Wrapper for the Cephes [1]_ routine `nbdtr`.

The negative binomial distribution is also available as
`scipy.stats.nbinom`. Using `nbdtr` directly can improve performance
compared to the ``cdf`` method of `scipy.stats.nbinom` (see last example).

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/

Examples
--------
Compute the function for ``k=10`` and ``n=5`` at ``p=0.5``.

>>> import numpy as np
>>> from scipy.special import nbdtr
>>> nbdtr(10, 5, 0.5)
0.940765380859375

Compute the function for ``n=10`` and ``p=0.5`` at several points by
providing a NumPy array or list for `k`.

>>> nbdtr([5, 10, 15], 10, 0.5)
array([0.15087891, 0.58809853, 0.88523853])

Plot the function for four different parameter sets.

>>> import matplotlib.pyplot as plt
>>> k = np.arange(130)
>>> n_parameters = [20, 20, 20, 80]
>>> p_parameters = [0.2, 0.5, 0.8, 0.5]
>>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']
>>> parameters_list = list(zip(p_parameters, n_parameters,
...                            linestyles))
>>> fig, ax = plt.subplots(figsize=(8, 8))
>>> for parameter_set in parameters_list:
...     p, n, style = parameter_set
...     nbdtr_vals = nbdtr(k, n, p)
...     ax.plot(k, nbdtr_vals, label=rf"$n={n},\, p={p}$",
...             ls=style)
>>> ax.legend()
>>> ax.set_xlabel("$k$")
>>> ax.set_title("Negative binomial cumulative distribution function")
>>> plt.show()

The negative binomial distribution is also available as
`scipy.stats.nbinom`. Using `nbdtr` directly can be much faster than
calling the ``cdf`` method of `scipy.stats.nbinom`, especially for small
arrays or individual values. To get the same results one must use the
following parametrization: ``nbinom(n, p).cdf(k)=nbdtr(k, n, p)``.

>>> from scipy.stats import nbinom
>>> k, n, p = 5, 3, 0.5
>>> nbdtr_res = nbdtr(k, n, p)  # this will often be faster than below
>>> stats_res = nbinom(n, p).cdf(k)
>>> stats_res, nbdtr_res  # test that results are equal
(0.85546875, 0.85546875)

`nbdtr` can evaluate different parameter sets by providing arrays with
shapes compatible for broadcasting for `k`, `n` and `p`. Here we compute
the function for three different `k` at four locations `p`, resulting in
a 3x4 array.

>>> k = np.array([[5], [10], [15]])
>>> p = np.array([0.3, 0.5, 0.7, 0.9])
>>> k.shape, p.shape
((3, 1), (4,))

>>> nbdtr(k, 5, p)
array([[0.15026833, 0.62304687, 0.95265101, 0.9998531 ],
       [0.48450894, 0.94076538, 0.99932777, 0.99999999],
       [0.76249222, 0.99409103, 0.99999445, 1.        ]])nbdtrcnbdtrc(k, n, p, out=None)

Negative binomial survival function.

Returns the sum of the terms `k + 1` to infinity of the negative binomial
distribution probability mass function,

.. math::

    F = \sum_{j=k + 1}^\infty {{n + j - 1}\choose{j}} p^n (1 - p)^j.

In a sequence of Bernoulli trials with individual success probabilities
`p`, this is the probability that more than `k` failures precede the nth
success.

Parameters
----------
k : array_like
    The maximum number of allowed failures (nonnegative int).
n : array_like
    The target number of successes (positive int).
p : array_like
    Probability of success in a single event (float).
out : ndarray, optional
    Optional output array for the function results

Returns
-------
F : scalar or ndarray
    The probability of `k + 1` or more failures before `n` successes in a
    sequence of events with individual success probability `p`.

See Also
--------
nbdtr : Negative binomial cumulative distribution function
nbdtrik : Negative binomial percentile function
scipy.stats.nbinom : Negative binomial distribution

Notes
-----
If floating point values are passed for `k` or `n`, they will be truncated
to integers.

The terms are not summed directly; instead the regularized incomplete beta
function is employed, according to the formula,

.. math::
    \mathrm{nbdtrc}(k, n, p) = I_{1 - p}(k + 1, n).

Wrapper for the Cephes [1]_ routine `nbdtrc`.

The negative binomial distribution is also available as
`scipy.stats.nbinom`. Using `nbdtrc` directly can improve performance
compared to the ``sf`` method of `scipy.stats.nbinom` (see last example).

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/

Examples
--------
Compute the function for ``k=10`` and ``n=5`` at ``p=0.5``.

>>> import numpy as np
>>> from scipy.special import nbdtrc
>>> nbdtrc(10, 5, 0.5)
0.059234619140624986

Compute the function for ``n=10`` and ``p=0.5`` at several points by
providing a NumPy array or list for `k`.

>>> nbdtrc([5, 10, 15], 10, 0.5)
array([0.84912109, 0.41190147, 0.11476147])

Plot the function for four different parameter sets.

>>> import matplotlib.pyplot as plt
>>> k = np.arange(130)
>>> n_parameters = [20, 20, 20, 80]
>>> p_parameters = [0.2, 0.5, 0.8, 0.5]
>>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']
>>> parameters_list = list(zip(p_parameters, n_parameters,
...                            linestyles))
>>> fig, ax = plt.subplots(figsize=(8, 8))
>>> for parameter_set in parameters_list:
...     p, n, style = parameter_set
...     nbdtrc_vals = nbdtrc(k, n, p)
...     ax.plot(k, nbdtrc_vals, label=rf"$n={n},\, p={p}$",
...             ls=style)
>>> ax.legend()
>>> ax.set_xlabel("$k$")
>>> ax.set_title("Negative binomial distribution survival function")
>>> plt.show()

The negative binomial distribution is also available as
`scipy.stats.nbinom`. Using `nbdtrc` directly can be much faster than
calling the ``sf`` method of `scipy.stats.nbinom`, especially for small
arrays or individual values. To get the same results one must use the
following parametrization: ``nbinom(n, p).sf(k)=nbdtrc(k, n, p)``.

>>> from scipy.stats import nbinom
>>> k, n, p = 3, 5, 0.5
>>> nbdtr_res = nbdtrc(k, n, p)  # this will often be faster than below
>>> stats_res = nbinom(n, p).sf(k)
>>> stats_res, nbdtr_res  # test that results are equal
(0.6367187499999999, 0.6367187499999999)

`nbdtrc` can evaluate different parameter sets by providing arrays with
shapes compatible for broadcasting for `k`, `n` and `p`. Here we compute
the function for three different `k` at four locations `p`, resulting in
a 3x4 array.

>>> k = np.array([[5], [10], [15]])
>>> p = np.array([0.3, 0.5, 0.7, 0.9])
>>> k.shape, p.shape
((3, 1), (4,))

>>> nbdtrc(k, 5, p)
array([[8.49731667e-01, 3.76953125e-01, 4.73489874e-02, 1.46902600e-04],
       [5.15491059e-01, 5.92346191e-02, 6.72234070e-04, 9.29610100e-09],
       [2.37507779e-01, 5.90896606e-03, 5.55025308e-06, 3.26346760e-13]])nbdtrinbdtri(k, n, y, out=None)

Returns the inverse with respect to the parameter `p` of
`y = nbdtr(k, n, p)`, the negative binomial cumulative distribution
function.

Parameters
----------
k : array_like
    The maximum number of allowed failures (nonnegative int).
n : array_like
    The target number of successes (positive int).
y : array_like
    The probability of `k` or fewer failures before `n` successes (float).
out : ndarray, optional
    Optional output array for the function results

Returns
-------
p : scalar or ndarray
    Probability of success in a single event (float) such that
    `nbdtr(k, n, p) = y`.

See Also
--------
nbdtr : Cumulative distribution function of the negative binomial.
nbdtrc : Negative binomial survival function.
scipy.stats.nbinom : negative binomial distribution.
nbdtrik : Inverse with respect to `k` of `nbdtr(k, n, p)`.
nbdtrin : Inverse with respect to `n` of `nbdtr(k, n, p)`.
scipy.stats.nbinom : Negative binomial distribution

Notes
-----
Wrapper for the Cephes [1]_ routine `nbdtri`.

The negative binomial distribution is also available as
`scipy.stats.nbinom`. Using `nbdtri` directly can improve performance
compared to the ``ppf`` method of `scipy.stats.nbinom`.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/

Examples
--------
`nbdtri` is the inverse of `nbdtr` with respect to `p`.
Up to floating point errors the following holds:
``nbdtri(k, n, nbdtr(k, n, p))=p``.

>>> import numpy as np
>>> from scipy.special import nbdtri, nbdtr
>>> k, n, y = 5, 10, 0.2
>>> cdf_val = nbdtr(k, n, y)
>>> nbdtri(k, n, cdf_val)
0.20000000000000004

Compute the function for ``k=10`` and ``n=5`` at several points by
providing a NumPy array or list for `y`.

>>> y = np.array([0.1, 0.4, 0.8])
>>> nbdtri(3, 5, y)
array([0.34462319, 0.51653095, 0.69677416])

Plot the function for three different parameter sets.

>>> import matplotlib.pyplot as plt
>>> n_parameters = [5, 20, 30, 30]
>>> k_parameters = [20, 20, 60, 80]
>>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']
>>> parameters_list = list(zip(n_parameters, k_parameters, linestyles))
>>> cdf_vals = np.linspace(0, 1, 1000)
>>> fig, ax = plt.subplots(figsize=(8, 8))
>>> for parameter_set in parameters_list:
...     n, k, style = parameter_set
...     nbdtri_vals = nbdtri(k, n, cdf_vals)
...     ax.plot(cdf_vals, nbdtri_vals, label=rf"$k={k},\ n={n}$",
...             ls=style)
>>> ax.legend()
>>> ax.set_ylabel("$p$")
>>> ax.set_xlabel("$CDF$")
>>> title = "nbdtri: inverse of negative binomial CDF with respect to $p$"
>>> ax.set_title(title)
>>> plt.show()

`nbdtri` can evaluate different parameter sets by providing arrays with
shapes compatible for broadcasting for `k`, `n` and `p`. Here we compute
the function for three different `k` at four locations `p`, resulting in
a 3x4 array.

>>> k = np.array([[5], [10], [15]])
>>> y = np.array([0.3, 0.5, 0.7, 0.9])
>>> k.shape, y.shape
((3, 1), (4,))

>>> nbdtri(k, 5, y)
array([[0.37258157, 0.45169416, 0.53249956, 0.64578407],
       [0.24588501, 0.30451981, 0.36778453, 0.46397088],
       [0.18362101, 0.22966758, 0.28054743, 0.36066188]])nbdtrik(y, n, p, out=None)

Negative binomial percentile function.

Returns the inverse with respect to the parameter `k` of
`y = nbdtr(k, n, p)`, the negative binomial cumulative distribution
function.

Parameters
----------
y : array_like
    The probability of `k` or fewer failures before `n` successes (float).
n : array_like
    The target number of successes (positive int).
p : array_like
    Probability of success in a single event (float).
out : ndarray, optional
    Optional output array for the function results

Returns
-------
k : scalar or ndarray
    The maximum number of allowed failures such that `nbdtr(k, n, p) = y`.

See Also
--------
nbdtr : Cumulative distribution function of the negative binomial.
nbdtrc : Survival function of the negative binomial.
nbdtri : Inverse with respect to `p` of `nbdtr(k, n, p)`.
nbdtrin : Inverse with respect to `n` of `nbdtr(k, n, p)`.
scipy.stats.nbinom : Negative binomial distribution

Notes
-----
Wrapper for the CDFLIB [1]_ Fortran routine `cdfnbn`.

Formula 26.5.26 of [2]_,

.. math::
    \sum_{j=k + 1}^\infty {{n + j - 1}
    \choose{j}} p^n (1 - p)^j = I_{1 - p}(k + 1, n),

is used to reduce calculation of the cumulative distribution function to
that of a regularized incomplete beta :math:`I`.

Computation of `k` involves a search for a value that produces the desired
value of `y`.  The search relies on the monotonicity of `y` with `k`.

References
----------
.. [1] Barry Brown, James Lovato, and Kathy Russell,
       CDFLIB: Library of Fortran Routines for Cumulative Distribution
       Functions, Inverses, and Other Parameters.
.. [2] Milton Abramowitz and Irene A. Stegun, eds.
       Handbook of Mathematical Functions with Formulas,
       Graphs, and Mathematical Tables. New York: Dover, 1972.

Examples
--------
Compute the negative binomial cumulative distribution function for an
exemplary parameter set.

>>> import numpy as np
>>> from scipy.special import nbdtr, nbdtrik
>>> k, n, p = 5, 2, 0.5
>>> cdf_value = nbdtr(k, n, p)
>>> cdf_value
0.9375

Verify that `nbdtrik` recovers the original value for `k`.

>>> nbdtrik(cdf_value, n, p)
5.0

Plot the function for different parameter sets.

>>> import matplotlib.pyplot as plt
>>> p_parameters = [0.2, 0.5, 0.7, 0.5]
>>> n_parameters = [30, 30, 30, 80]
>>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']
>>> parameters_list = list(zip(p_parameters, n_parameters, linestyles))
>>> cdf_vals = np.linspace(0, 1, 1000)
>>> fig, ax = plt.subplots(figsize=(8, 8))
>>> for parameter_set in parameters_list:
...     p, n, style = parameter_set
...     nbdtrik_vals = nbdtrik(cdf_vals, n, p)
...     ax.plot(cdf_vals, nbdtrik_vals, label=rf"$n={n},\ p={p}$",
...             ls=style)
>>> ax.legend()
>>> ax.set_ylabel("$k$")
>>> ax.set_xlabel("$CDF$")
>>> ax.set_title("Negative binomial percentile function")
>>> plt.show()

The negative binomial distribution is also available as
`scipy.stats.nbinom`. The percentile function  method ``ppf``
returns the result of `nbdtrik` rounded up to integers:

>>> from scipy.stats import nbinom
>>> q, n, p = 0.6, 5, 0.5
>>> nbinom.ppf(q, n, p), nbdtrik(q, n, p)
(5.0, 4.800428460273882)nbdtrin(k, y, p, out=None)

Inverse of `nbdtr` vs `n`.

Returns the inverse with respect to the parameter `n` of
`y = nbdtr(k, n, p)`, the negative binomial cumulative distribution
function.

Parameters
----------
k : array_like
    The maximum number of allowed failures (nonnegative int).
y : array_like
    The probability of `k` or fewer failures before `n` successes (float).
p : array_like
    Probability of success in a single event (float).
out : ndarray, optional
    Optional output array for the function results

Returns
-------
n : scalar or ndarray
    The number of successes `n` such that `nbdtr(k, n, p) = y`.

See Also
--------
nbdtr : Cumulative distribution function of the negative binomial.
nbdtri : Inverse with respect to `p` of `nbdtr(k, n, p)`.
nbdtrik : Inverse with respect to `k` of `nbdtr(k, n, p)`.

Notes
-----
Wrapper for the CDFLIB [1]_ Fortran routine `cdfnbn`.

Formula 26.5.26 of [2]_,

.. math::
    \sum_{j=k + 1}^\infty {{n + j - 1}
    \choose{j}} p^n (1 - p)^j = I_{1 - p}(k + 1, n),

is used to reduce calculation of the cumulative distribution function to
that of a regularized incomplete beta :math:`I`.

Computation of `n` involves a search for a value that produces the desired
value of `y`.  The search relies on the monotonicity of `y` with `n`.

References
----------
.. [1] Barry Brown, James Lovato, and Kathy Russell,
       CDFLIB: Library of Fortran Routines for Cumulative Distribution
       Functions, Inverses, and Other Parameters.
.. [2] Milton Abramowitz and Irene A. Stegun, eds.
       Handbook of Mathematical Functions with Formulas,
       Graphs, and Mathematical Tables. New York: Dover, 1972.

Examples
--------
Compute the negative binomial cumulative distribution function for an
exemplary parameter set.

>>> from scipy.special import nbdtr, nbdtrin
>>> k, n, p = 5, 2, 0.5
>>> cdf_value = nbdtr(k, n, p)
>>> cdf_value
0.9375

Verify that `nbdtrin` recovers the original value for `n` up to floating
point accuracy.

>>> nbdtrin(k, cdf_value, p)
1.999999999998137ncfdtr(dfn, dfd, nc, f, out=None)

Cumulative distribution function of the non-central F distribution.

The non-central F describes the distribution of,

.. math::
    Z = \frac{X/d_n}{Y/d_d}

where :math:`X` and :math:`Y` are independently distributed, with
:math:`X` distributed non-central :math:`\chi^2` with noncentrality
parameter `nc` and :math:`d_n` degrees of freedom, and :math:`Y`
distributed :math:`\chi^2` with :math:`d_d` degrees of freedom.

Parameters
----------
dfn : array_like
    Degrees of freedom of the numerator sum of squares.  Range (0, inf).
dfd : array_like
    Degrees of freedom of the denominator sum of squares.  Range (0, inf).
nc : array_like
    Noncentrality parameter.  Should be in range (0, 1e4).
f : array_like
    Quantiles, i.e. the upper limit of integration.
out : ndarray, optional
    Optional output array for the function results

Returns
-------
cdf : scalar or ndarray
    The calculated CDF.  If all inputs are scalar, the return will be a
    float.  Otherwise it will be an array.

See Also
--------
ncfdtri : Quantile function; inverse of `ncfdtr` with respect to `f`.
ncfdtridfd : Inverse of `ncfdtr` with respect to `dfd`.
ncfdtridfn : Inverse of `ncfdtr` with respect to `dfn`.
ncfdtrinc : Inverse of `ncfdtr` with respect to `nc`.

Notes
-----
Wrapper for the CDFLIB [1]_ Fortran routine `cdffnc`.

The cumulative distribution function is computed using Formula 26.6.20 of
[2]_:

.. math::
    F(d_n, d_d, n_c, f) = \sum_{j=0}^\infty e^{-n_c/2}
    \frac{(n_c/2)^j}{j!} I_{x}(\frac{d_n}{2} + j, \frac{d_d}{2}),

where :math:`I` is the regularized incomplete beta function, and
:math:`x = f d_n/(f d_n + d_d)`.

The computation time required for this routine is proportional to the
noncentrality parameter `nc`.  Very large values of this parameter can
consume immense computer resources.  This is why the search range is
bounded by 10,000.

References
----------
.. [1] Barry Brown, James Lovato, and Kathy Russell,
       CDFLIB: Library of Fortran Routines for Cumulative Distribution
       Functions, Inverses, and Other Parameters.
.. [2] Milton Abramowitz and Irene A. Stegun, eds.
       Handbook of Mathematical Functions with Formulas,
       Graphs, and Mathematical Tables. New York: Dover, 1972.

Examples
--------
>>> import numpy as np
>>> from scipy import special
>>> from scipy import stats
>>> import matplotlib.pyplot as plt

Plot the CDF of the non-central F distribution, for nc=0.  Compare with the
F-distribution from scipy.stats:

>>> x = np.linspace(-1, 8, num=500)
>>> dfn = 3
>>> dfd = 2
>>> ncf_stats = stats.f.cdf(x, dfn, dfd)
>>> ncf_special = special.ncfdtr(dfn, dfd, 0, x)

>>> fig = plt.figure()
>>> ax = fig.add_subplot(111)
>>> ax.plot(x, ncf_stats, 'b-', lw=3)
>>> ax.plot(x, ncf_special, 'r-')
>>> plt.show()ncfdtri(dfn, dfd, nc, p, out=None)

Inverse with respect to `f` of the CDF of the non-central F distribution.

See `ncfdtr` for more details.

Parameters
----------
dfn : array_like
    Degrees of freedom of the numerator sum of squares.  Range (0, inf).
dfd : array_like
    Degrees of freedom of the denominator sum of squares.  Range (0, inf).
nc : array_like
    Noncentrality parameter.  Should be in range (0, 1e4).
p : array_like
    Value of the cumulative distribution function.  Must be in the
    range [0, 1].
out : ndarray, optional
    Optional output array for the function results

Returns
-------
f : scalar or ndarray
    Quantiles, i.e., the upper limit of integration.

See Also
--------
ncfdtr : CDF of the non-central F distribution.
ncfdtridfd : Inverse of `ncfdtr` with respect to `dfd`.
ncfdtridfn : Inverse of `ncfdtr` with respect to `dfn`.
ncfdtrinc : Inverse of `ncfdtr` with respect to `nc`.

Examples
--------
>>> from scipy.special import ncfdtr, ncfdtri

Compute the CDF for several values of `f`:

>>> f = [0.5, 1, 1.5]
>>> p = ncfdtr(2, 3, 1.5, f)
>>> p
array([ 0.20782291,  0.36107392,  0.47345752])

Compute the inverse.  We recover the values of `f`, as expected:

>>> ncfdtri(2, 3, 1.5, p)
array([ 0.5,  1. ,  1.5])ncfdtridfd(dfn, p, nc, f, out=None)

Calculate degrees of freedom (denominator) for the noncentral F-distribution.

This is the inverse with respect to `dfd` of `ncfdtr`.
See `ncfdtr` for more details.

Parameters
----------
dfn : array_like
    Degrees of freedom of the numerator sum of squares.  Range (0, inf).
p : array_like
    Value of the cumulative distribution function.  Must be in the
    range [0, 1].
nc : array_like
    Noncentrality parameter.  Should be in range (0, 1e4).
f : array_like
    Quantiles, i.e., the upper limit of integration.
out : ndarray, optional
    Optional output array for the function results

Returns
-------
dfd : scalar or ndarray
    Degrees of freedom of the denominator sum of squares.

See Also
--------
ncfdtr : CDF of the non-central F distribution.
ncfdtri : Quantile function; inverse of `ncfdtr` with respect to `f`.
ncfdtridfn : Inverse of `ncfdtr` with respect to `dfn`.
ncfdtrinc : Inverse of `ncfdtr` with respect to `nc`.

Notes
-----
The value of the cumulative noncentral F distribution is not necessarily
monotone in either degrees of freedom. There thus may be two values that
provide a given CDF value. This routine assumes monotonicity and will
find an arbitrary one of the two values.

Examples
--------
>>> from scipy.special import ncfdtr, ncfdtridfd

Compute the CDF for several values of `dfd`:

>>> dfd = [1, 2, 3]
>>> p = ncfdtr(2, dfd, 0.25, 15)
>>> p
array([ 0.8097138 ,  0.93020416,  0.96787852])

Compute the inverse.  We recover the values of `dfd`, as expected:

>>> ncfdtridfd(2, p, 0.25, 15)
array([ 1.,  2.,  3.])ncfdtridfn(p, dfd, nc, f, out=None)

Calculate degrees of freedom (numerator) for the noncentral F-distribution.

This is the inverse with respect to `dfn` of `ncfdtr`.
See `ncfdtr` for more details.

Parameters
----------
p : array_like
    Value of the cumulative distribution function. Must be in the
    range [0, 1].
dfd : array_like
    Degrees of freedom of the denominator sum of squares. Range (0, inf).
nc : array_like
    Noncentrality parameter.  Should be in range (0, 1e4).
f : float
    Quantiles, i.e., the upper limit of integration.
out : ndarray, optional
    Optional output array for the function results

Returns
-------
dfn : scalar or ndarray
    Degrees of freedom of the numerator sum of squares.

See Also
--------
ncfdtr : CDF of the non-central F distribution.
ncfdtri : Quantile function; inverse of `ncfdtr` with respect to `f`.
ncfdtridfd : Inverse of `ncfdtr` with respect to `dfd`.
ncfdtrinc : Inverse of `ncfdtr` with respect to `nc`.

Notes
-----
The value of the cumulative noncentral F distribution is not necessarily
monotone in either degrees of freedom. There thus may be two values that
provide a given CDF value. This routine assumes monotonicity and will
find an arbitrary one of the two values.

Examples
--------
>>> from scipy.special import ncfdtr, ncfdtridfn

Compute the CDF for several values of `dfn`:

>>> dfn = [1, 2, 3]
>>> p = ncfdtr(dfn, 2, 0.25, 15)
>>> p
array([ 0.92562363,  0.93020416,  0.93188394])

Compute the inverse. We recover the values of `dfn`, as expected:

>>> ncfdtridfn(p, 2, 0.25, 15)
array([ 1.,  2.,  3.])ncfdtrinc(dfn, dfd, p, f, out=None)

Calculate non-centrality parameter for non-central F distribution.

This is the inverse with respect to `nc` of `ncfdtr`.
See `ncfdtr` for more details.

Parameters
----------
dfn : array_like
    Degrees of freedom of the numerator sum of squares. Range (0, inf).
dfd : array_like
    Degrees of freedom of the denominator sum of squares. Range (0, inf).
p : array_like
    Value of the cumulative distribution function. Must be in the
    range [0, 1].
f : array_like
    Quantiles, i.e., the upper limit of integration.
out : ndarray, optional
    Optional output array for the function results

Returns
-------
nc : scalar or ndarray
    Noncentrality parameter.

See Also
--------
ncfdtr : CDF of the non-central F distribution.
ncfdtri : Quantile function; inverse of `ncfdtr` with respect to `f`.
ncfdtridfd : Inverse of `ncfdtr` with respect to `dfd`.
ncfdtridfn : Inverse of `ncfdtr` with respect to `dfn`.

Examples
--------
>>> from scipy.special import ncfdtr, ncfdtrinc

Compute the CDF for several values of `nc`:

>>> nc = [0.5, 1.5, 2.0]
>>> p = ncfdtr(2, 3, nc, 15)
>>> p
array([ 0.96309246,  0.94327955,  0.93304098])

Compute the inverse. We recover the values of `nc`, as expected:

>>> ncfdtrinc(2, 3, p, 15)
array([ 0.5,  1.5,  2. ])nctdtr(df, nc, t, out=None)

Cumulative distribution function of the non-central `t` distribution.

Parameters
----------
df : array_like
    Degrees of freedom of the distribution. Should be in range (0, inf).
nc : array_like
    Noncentrality parameter. Should be in range (-1e6, 1e6).
t : array_like
    Quantiles, i.e., the upper limit of integration.
out : ndarray, optional
    Optional output array for the function results

Returns
-------
cdf : scalar or ndarray
    The calculated CDF. If all inputs are scalar, the return will be a
    float. Otherwise, it will be an array.

See Also
--------
nctdtrit : Inverse CDF (iCDF) of the non-central t distribution.
nctdtridf : Calculate degrees of freedom, given CDF and iCDF values.
nctdtrinc : Calculate non-centrality parameter, given CDF iCDF values.

Examples
--------
>>> import numpy as np
>>> from scipy import special
>>> from scipy import stats
>>> import matplotlib.pyplot as plt

Plot the CDF of the non-central t distribution, for nc=0. Compare with the
t-distribution from scipy.stats:

>>> x = np.linspace(-5, 5, num=500)
>>> df = 3
>>> nct_stats = stats.t.cdf(x, df)
>>> nct_special = special.nctdtr(df, 0, x)

>>> fig = plt.figure()
>>> ax = fig.add_subplot(111)
>>> ax.plot(x, nct_stats, 'b-', lw=3)
>>> ax.plot(x, nct_special, 'r-')
>>> plt.show()nctdtridf(p, nc, t, out=None)

Calculate degrees of freedom for non-central t distribution.

See `nctdtr` for more details.

Parameters
----------
p : array_like
    CDF values, in range (0, 1].
nc : array_like
    Noncentrality parameter. Should be in range (-1e6, 1e6).
t : array_like
    Quantiles, i.e., the upper limit of integration.
out : ndarray, optional
    Optional output array for the function results

Returns
-------
df : scalar or ndarray
    The degrees of freedom. If all inputs are scalar, the return will be a
    float. Otherwise, it will be an array.

See Also
--------
nctdtr :  CDF of the non-central `t` distribution.
nctdtrit : Inverse CDF (iCDF) of the non-central t distribution.
nctdtrinc : Calculate non-centrality parameter, given CDF iCDF values.

Examples
--------
>>> from scipy.special import nctdtr, nctdtridf

Compute the CDF for several values of `df`:

>>> df = [1, 2, 3]
>>> p = nctdtr(df, 0.25, 1)
>>> p
array([0.67491974, 0.716464  , 0.73349456])

Compute the inverse. We recover the values of `df`, as expected:

>>> nctdtridf(p, 0.25, 1)
array([1., 2., 3.])nctdtrinc(df, p, t, out=None)

Calculate non-centrality parameter for non-central t distribution.

See `nctdtr` for more details.

Parameters
----------
df : array_like
    Degrees of freedom of the distribution. Should be in range (0, inf).
p : array_like
    CDF values, in range (0, 1].
t : array_like
    Quantiles, i.e., the upper limit of integration.
out : ndarray, optional
    Optional output array for the function results

Returns
-------
nc : scalar or ndarray
    Noncentrality parameter

See Also
--------
nctdtr :  CDF of the non-central `t` distribution.
nctdtrit : Inverse CDF (iCDF) of the non-central t distribution.
nctdtridf : Calculate degrees of freedom, given CDF and iCDF values.

Examples
--------
>>> from scipy.special import nctdtr, nctdtrinc

Compute the CDF for several values of `nc`:

>>> nc = [0.5, 1.5, 2.5]
>>> p = nctdtr(3, nc, 1.5)
>>> p
array([0.77569497, 0.45524533, 0.1668691 ])

Compute the inverse. We recover the values of `nc`, as expected:

>>> nctdtrinc(3, p, 1.5)
array([0.5, 1.5, 2.5])nctdtrit(df, nc, p, out=None)

Inverse cumulative distribution function of the non-central t distribution.

See `nctdtr` for more details.

Parameters
----------
df : array_like
    Degrees of freedom of the distribution. Should be in range (0, inf).
nc : array_like
    Noncentrality parameter. Should be in range (-1e6, 1e6).
p : array_like
    CDF values, in range (0, 1].
out : ndarray, optional
    Optional output array for the function results

Returns
-------
t : scalar or ndarray
    Quantiles

See Also
--------
nctdtr :  CDF of the non-central `t` distribution.
nctdtridf : Calculate degrees of freedom, given CDF and iCDF values.
nctdtrinc : Calculate non-centrality parameter, given CDF iCDF values.

Examples
--------
>>> from scipy.special import nctdtr, nctdtrit

Compute the CDF for several values of `t`:

>>> t = [0.5, 1, 1.5]
>>> p = nctdtr(3, 1, t)
>>> p
array([0.29811049, 0.46922687, 0.6257559 ])

Compute the inverse. We recover the values of `t`, as expected:

>>> nctdtrit(3, 1, p)
array([0.5, 1. , 1.5])ndtrndtr(x, out=None)

Cumulative distribution of the standard normal distribution.

Returns the area under the standard Gaussian probability
density function, integrated from minus infinity to `x`

.. math::

   \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x \exp(-t^2/2) dt

Parameters
----------
x : array_like, real or complex
    Argument
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    The value of the normal CDF evaluated at `x`

See Also
--------
log_ndtr : Logarithm of ndtr
ndtri : Inverse of ndtr, standard normal percentile function
erf : Error function
erfc : 1 - erf
scipy.stats.norm : Normal distribution

Examples
--------
Evaluate `ndtr` at one point.

>>> import numpy as np
>>> from scipy.special import ndtr
>>> ndtr(0.5)
0.6914624612740131

Evaluate the function at several points by providing a NumPy array
or list for `x`.

>>> ndtr([0, 0.5, 2])
array([0.5       , 0.69146246, 0.97724987])

Plot the function.

>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-5, 5, 100)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, ndtr(x))
>>> ax.set_title(r"Standard normal cumulative distribution function $\Phi$")
>>> plt.show()ndtrindtri(y, out=None)

Inverse of `ndtr` vs x

Returns the argument x for which the area under the standard normal
probability density function (integrated from minus infinity to `x`)
is equal to y.

Parameters
----------
p : array_like
    Probability
out : ndarray, optional
    Optional output array for the function results

Returns
-------
x : scalar or ndarray
    Value of x such that ``ndtr(x) == p``.

See Also
--------
ndtr : Standard normal cumulative probability distribution
ndtri_exp : Inverse of log_ndtr

Examples
--------
`ndtri` is the percentile function of the standard normal distribution.
This means it returns the inverse of the cumulative density `ndtr`. First,
let us compute a cumulative density value.

>>> import numpy as np
>>> from scipy.special import ndtri, ndtr
>>> cdf_val = ndtr(2)
>>> cdf_val
0.9772498680518208

Verify that `ndtri` yields the original value for `x` up to floating point
errors.

>>> ndtri(cdf_val)
2.0000000000000004

Plot the function. For that purpose, we provide a NumPy array as argument.

>>> import matplotlib.pyplot as plt
>>> x = np.linspace(0.01, 1, 200)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, ndtri(x))
>>> ax.set_title("Standard normal percentile function")
>>> plt.show()ndtri_expndtri_exp(y, out=None)

Inverse of `log_ndtr` vs x. Allows for greater precision than
`ndtri` composed with `numpy.exp` for very small values of y and for
y close to 0.

Parameters
----------
y : array_like of float
    Function argument
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    Inverse of the log CDF of the standard normal distribution, evaluated
    at y.

See Also
--------
log_ndtr : log of the standard normal cumulative distribution function
ndtr : standard normal cumulative distribution function
ndtri : standard normal percentile function

Examples
--------
>>> import numpy as np
>>> import scipy.special as sc

`ndtri_exp` agrees with the naive implementation when the latter does
not suffer from underflow.

>>> sc.ndtri_exp(-1)
-0.33747496376420244
>>> sc.ndtri(np.exp(-1))
-0.33747496376420244

For extreme values of y, the naive approach fails

>>> sc.ndtri(np.exp(-800))
-inf
>>> sc.ndtri(np.exp(-1e-20))
inf

whereas `ndtri_exp` is still able to compute the result to high precision.

>>> sc.ndtri_exp(-800)
-39.88469483825668
>>> sc.ndtri_exp(-1e-20)
9.262340089798409nrdtrimn(p, std, x, out=None)

Calculate mean of normal distribution given other params.

Parameters
----------
p : array_like
    CDF values, in range (0, 1].
std : array_like
    Standard deviation.
x : array_like
    Quantiles, i.e. the upper limit of integration.
out : ndarray, optional
    Optional output array for the function results

Returns
-------
mn : scalar or ndarray
    The mean of the normal distribution.

See Also
--------
scipy.stats.norm : Normal distribution
ndtr : Standard normal cumulative probability distribution
ndtri : Inverse of standard normal CDF with respect to quantile
nrdtrisd : Inverse of normal distribution CDF with respect to
           standard deviation

Examples
--------
`nrdtrimn` can be used to recover the mean of a normal distribution
if we know the CDF value `p` for a given quantile `x` and the
standard deviation `std`. First, we calculate
the normal distribution CDF for an exemplary parameter set.

>>> from scipy.stats import norm
>>> mean = 3.
>>> std = 2.
>>> x = 6.
>>> p = norm.cdf(x, loc=mean, scale=std)
>>> p
0.9331927987311419

Verify that `nrdtrimn` returns the original value for `mean`.

>>> from scipy.special import nrdtrimn
>>> nrdtrimn(p, std, x)
3.0000000000000004nrdtrisd(mn, p, x, out=None)

Calculate standard deviation of normal distribution given other params.

Parameters
----------
mn : scalar or ndarray
    The mean of the normal distribution.
p : array_like
    CDF values, in range (0, 1].
x : array_like
    Quantiles, i.e. the upper limit of integration.

out : ndarray, optional
    Optional output array for the function results

Returns
-------
std : scalar or ndarray
    Standard deviation.

See Also
--------
scipy.stats.norm : Normal distribution
ndtr : Standard normal cumulative probability distribution
ndtri : Inverse of standard normal CDF with respect to quantile
nrdtrimn : Inverse of normal distribution CDF with respect to
           mean

Examples
--------
`nrdtrisd` can be used to recover the standard deviation of a normal
distribution if we know the CDF value `p` for a given quantile `x` and
the mean `mn`. First, we calculate the normal distribution CDF for an
exemplary parameter set.

>>> from scipy.stats import norm
>>> mean = 3.
>>> std = 2.
>>> x = 6.
>>> p = norm.cdf(x, loc=mean, scale=std)
>>> p
0.9331927987311419

Verify that `nrdtrisd` returns the original value for `std`.

>>> from scipy.special import nrdtrisd
>>> nrdtrisd(mean, p, x)
2.0000000000000004obl_ang1obl_ang1(m, n, c, x, out=None)

Oblate spheroidal angular function of the first kind and its derivative

Computes the oblate spheroidal angular function of the first kind
and its derivative (with respect to `x`) for mode parameters m>=0
and n>=m, spheroidal parameter `c` and ``|x| < 1.0``.

Parameters
----------
m : array_like
    Mode parameter m (nonnegative)
n : array_like
    Mode parameter n (>= m)
c : array_like
    Spheroidal parameter
x : array_like
    Parameter x (``|x| < 1.0``)
out : ndarray, optional
    Optional output array for the function results

Returns
-------
s : scalar or ndarray
    Value of the function
sp : scalar or ndarray
    Value of the derivative vs x

See Also
--------
obl_ang1_cvobl_ang1_cvobl_ang1_cv(m, n, c, cv, x, out=None)

Oblate spheroidal angular function obl_ang1 for precomputed characteristic value

Computes the oblate spheroidal angular function of the first kind
and its derivative (with respect to `x`) for mode parameters m>=0
and n>=m, spheroidal parameter `c` and ``|x| < 1.0``. Requires
pre-computed characteristic value.

Parameters
----------
m : array_like
    Mode parameter m (nonnegative)
n : array_like
    Mode parameter n (>= m)
c : array_like
    Spheroidal parameter
cv : array_like
    Characteristic value
x : array_like
    Parameter x (``|x| < 1.0``)
out : ndarray, optional
    Optional output array for the function results

Returns
-------
s : scalar or ndarray
    Value of the function
sp : scalar or ndarray
    Value of the derivative vs x

See Also
--------
obl_ang1obl_cvobl_cv(m, n, c, out=None)

Characteristic value of oblate spheroidal function

Computes the characteristic value of oblate spheroidal wave
functions of order `m`, `n` (n>=m) and spheroidal parameter `c`.

Parameters
----------
m : array_like
    Mode parameter m (nonnegative)
n : array_like
    Mode parameter n (>= m)
c : array_like
    Spheroidal parameter
out : ndarray, optional
    Optional output array for the function results

Returns
-------
cv : scalar or ndarray
    Characteristic valueobl_rad1obl_rad1(m, n, c, x, out=None)

Oblate spheroidal radial function of the first kind and its derivative

Computes the oblate spheroidal radial function of the first kind
and its derivative (with respect to `x`) for mode parameters m>=0
and n>=m, spheroidal parameter `c` and ``|x| < 1.0``.

Parameters
----------
m : array_like
    Mode parameter m (nonnegative)
n : array_like
    Mode parameter n (>= m)
c : array_like
    Spheroidal parameter
x : array_like
    Parameter x (``|x| < 1.0``)
out : ndarray, optional
    Optional output array for the function results

Returns
-------
s : scalar or ndarray
    Value of the function
sp : scalar or ndarray
    Value of the derivative vs x

See Also
--------
obl_rad1_cvobl_rad1_cvobl_rad1_cv(m, n, c, cv, x, out=None)

Oblate spheroidal radial function obl_rad1 for precomputed characteristic value

Computes the oblate spheroidal radial function of the first kind
and its derivative (with respect to `x`) for mode parameters m>=0
and n>=m, spheroidal parameter `c` and ``|x| < 1.0``. Requires
pre-computed characteristic value.

Parameters
----------
m : array_like
    Mode parameter m (nonnegative)
n : array_like
    Mode parameter n (>= m)
c : array_like
    Spheroidal parameter
cv : array_like
    Characteristic value
x : array_like
    Parameter x (``|x| < 1.0``)
out : ndarray, optional
    Optional output array for the function results

Returns
-------
s : scalar or ndarray
    Value of the function
sp : scalar or ndarray
    Value of the derivative vs x

See Also
--------
obl_rad1obl_rad2obl_rad2(m, n, c, x, out=None)

Oblate spheroidal radial function of the second kind and its derivative.

Computes the oblate spheroidal radial function of the second kind
and its derivative (with respect to `x`) for mode parameters m>=0
and n>=m, spheroidal parameter `c` and ``|x| < 1.0``.

Parameters
----------
m : array_like
    Mode parameter m (nonnegative)
n : array_like
    Mode parameter n (>= m)
c : array_like
    Spheroidal parameter
x : array_like
    Parameter x (``|x| < 1.0``)
out : ndarray, optional
    Optional output array for the function results

Returns
-------
s : scalar or ndarray
    Value of the function
sp : scalar or ndarray
    Value of the derivative vs x

See Also
--------
obl_rad2_cvobl_rad2_cvobl_rad2_cv(m, n, c, cv, x, out=None)

Oblate spheroidal radial function obl_rad2 for precomputed characteristic value

Computes the oblate spheroidal radial function of the second kind
and its derivative (with respect to `x`) for mode parameters m>=0
and n>=m, spheroidal parameter `c` and ``|x| < 1.0``. Requires
pre-computed characteristic value.

Parameters
----------
m : array_like
    Mode parameter m (nonnegative)
n : array_like
    Mode parameter n (>= m)
c : array_like
    Spheroidal parameter
cv : array_like
    Characteristic value
x : array_like
    Parameter x (``|x| < 1.0``)
out : ndarray, optional
    Optional output array for the function results

Returns
-------
s : scalar or ndarray
    Value of the function
sp : scalar or ndarray
    Value of the derivative vs x

See Also
--------
obl_rad2owens_towens_t(h, a, out=None)

Owen's T Function.

The function T(h, a) gives the probability of the event
(X > h and 0 < Y < a * X) where X and Y are independent
standard normal random variables.

Parameters
----------
h: array_like
    Input value.
a: array_like
    Input value.
out : ndarray, optional
    Optional output array for the function results

Returns
-------
t: scalar or ndarray
    Probability of the event (X > h and 0 < Y < a * X),
    where X and Y are independent standard normal random variables.

References
----------
.. [1] M. Patefield and D. Tandy, "Fast and accurate calculation of
       Owen's T Function", Statistical Software vol. 5, pp. 1-25, 2000.

Examples
--------
>>> from scipy import special
>>> a = 3.5
>>> h = 0.78
>>> special.owens_t(h, a)
0.10877216734852274pbdvpbdv(v, x, out=None)

Parabolic cylinder function D

Returns (d, dp) the parabolic cylinder function Dv(x) in d and the
derivative, Dv'(x) in dp.

Parameters
----------
v : array_like
    Real parameter
x : array_like
    Real argument
out : ndarray, optional
    Optional output array for the function results

Returns
-------
d : scalar or ndarray
    Value of the function
dp : scalar or ndarray
    Value of the derivative vs xpbvvpbvv(v, x, out=None)

Parabolic cylinder function V

Returns the parabolic cylinder function Vv(x) in v and the
derivative, Vv'(x) in vp.

Parameters
----------
v : array_like
    Real parameter
x : array_like
    Real argument
out : ndarray, optional
    Optional output array for the function results

Returns
-------
v : scalar or ndarray
    Value of the function
vp : scalar or ndarray
    Value of the derivative vs xpbwapbwa(a, x, out=None)

Parabolic cylinder function W.

The function is a particular solution to the differential equation

.. math::

    y'' + \left(\frac{1}{4}x^2 - a\right)y = 0,

for a full definition see section 12.14 in [1]_.

Parameters
----------
a : array_like
    Real parameter
x : array_like
    Real argument
out : ndarray, optional
    Optional output array for the function results

Returns
-------
w : scalar or ndarray
    Value of the function
wp : scalar or ndarray
    Value of the derivative in x

Notes
-----
The function is a wrapper for a Fortran routine by Zhang and Jin
[2]_. The implementation is accurate only for ``|a|, |x| < 5`` and
returns NaN outside that range.

References
----------
.. [1] Digital Library of Mathematical Functions, 14.30.
       https://dlmf.nist.gov/14.30
.. [2] Zhang, Shanjie and Jin, Jianming. "Computation of Special
       Functions", John Wiley and Sons, 1996.
       https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.htmlpdtrpdtr(k, m, out=None)

Poisson cumulative distribution function.

Defined as the probability that a Poisson-distributed random
variable with event rate :math:`m` is less than or equal to
:math:`k`. More concretely, this works out to be [1]_

.. math::

   \exp(-m) \sum_{j = 0}^{\lfloor{k}\rfloor} \frac{m^j}{j!}.

Parameters
----------
k : array_like
    Number of occurrences (nonnegative, real)
m : array_like
    Shape parameter (nonnegative, real)
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    Values of the Poisson cumulative distribution function

See Also
--------
pdtrc : Poisson survival function
pdtrik : inverse of `pdtr` with respect to `k`
pdtri : inverse of `pdtr` with respect to `m`

References
----------
.. [1] https://en.wikipedia.org/wiki/Poisson_distribution

Examples
--------
>>> import numpy as np
>>> import scipy.special as sc

It is a cumulative distribution function, so it converges to 1
monotonically as `k` goes to infinity.

>>> sc.pdtr([1, 10, 100, np.inf], 1)
array([0.73575888, 0.99999999, 1.        , 1.        ])

It is discontinuous at integers and constant between integers.

>>> sc.pdtr([1, 1.5, 1.9, 2], 1)
array([0.73575888, 0.73575888, 0.73575888, 0.9196986 ])pdtrcpdtrc(k, m, out=None)

Poisson survival function

Returns the sum of the terms from k+1 to infinity of the Poisson
distribution: sum(exp(-m) * m**j / j!, j=k+1..inf) = gammainc(
k+1, m). Arguments must both be non-negative doubles.

Parameters
----------
k : array_like
    Number of occurrences (nonnegative, real)
m : array_like
    Shape parameter (nonnegative, real)
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    Values of the Poisson survival function

See Also
--------
pdtr : Poisson cumulative distribution function
pdtrik : inverse of `pdtr` with respect to `k`
pdtri : inverse of `pdtr` with respect to `m`pdtripdtri(k, y, out=None)

Inverse to `pdtr` vs m

Returns the Poisson variable `m` such that the sum from 0 to `k` of
the Poisson density is equal to the given probability `y`:
calculated by ``gammaincinv(k + 1, y)``. `k` must be a nonnegative
integer and `y` between 0 and 1.

Parameters
----------
k : array_like
    Number of occurrences (nonnegative, real)
y : array_like
    Probability
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    Values of the shape parameter `m` such that ``pdtr(k, m) = p``

See Also
--------
pdtr : Poisson cumulative distribution function
pdtrc : Poisson survival function
pdtrik : inverse of `pdtr` with respect to `k`pdtrik(p, m, out=None)

Inverse to `pdtr` vs `m`.

Parameters
----------
m : array_like
    Shape parameter (nonnegative, real)
p : array_like
    Probability
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    The number of occurrences `k` such that ``pdtr(k, m) = p``

See Also
--------
pdtr : Poisson cumulative distribution function
pdtrc : Poisson survival function
pdtri : inverse of `pdtr` with respect to `m`pochpoch(z, m, out=None)

Pochhammer symbol.

The Pochhammer symbol (rising factorial) is defined as

.. math::

    (z)_m = \frac{\Gamma(z + m)}{\Gamma(z)}

For positive integer `m` it reads

.. math::

    (z)_m = z (z + 1) ... (z + m - 1)

See [dlmf]_ for more details.

Parameters
----------
z, m : array_like
    Real-valued arguments.
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    The value of the function.

References
----------
.. [dlmf] Nist, Digital Library of Mathematical Functions
    https://dlmf.nist.gov/5.2#iii

Examples
--------
>>> import scipy.special as sc

It is 1 when m is 0.

>>> sc.poch([1, 2, 3, 4], 0)
array([1., 1., 1., 1.])

For z equal to 1 it reduces to the factorial function.

>>> sc.poch(1, 5)
120.0
>>> 1 * 2 * 3 * 4 * 5
120

It can be expressed in terms of the gamma function.

>>> z, m = 3.7, 2.1
>>> sc.poch(z, m)
20.529581933776953
>>> sc.gamma(z + m) / sc.gamma(z)
20.52958193377696powm1powm1(x, y, out=None)

Computes ``x**y - 1``.

This function is useful when `y` is near 0, or when `x` is near 1.

The function is implemented for real types only (unlike ``numpy.power``,
which accepts complex inputs).

Parameters
----------
x : array_like
    The base. Must be a real type (i.e. integer or float, not complex).
y : array_like
    The exponent. Must be a real type (i.e. integer or float, not complex).

Returns
-------
array_like
    Result of the calculation

Notes
-----
.. versionadded:: 1.10.0

The underlying code is implemented for single precision and double
precision floats only.  Unlike `numpy.power`, integer inputs to
`powm1` are converted to floating point, and complex inputs are
not accepted.

Note the following edge cases:

* ``powm1(x, 0)`` returns 0 for any ``x``, including 0, ``inf``
  and ``nan``.
* ``powm1(1, y)`` returns 0 for any ``y``, including ``nan``
  and ``inf``.

Examples
--------
>>> import numpy as np
>>> from scipy.special import powm1

>>> x = np.array([1.2, 10.0, 0.9999999975])
>>> y = np.array([1e-9, 1e-11, 0.1875])
>>> powm1(x, y)
array([ 1.82321557e-10,  2.30258509e-11, -4.68749998e-10])

It can be verified that the relative errors in those results
are less than 2.5e-16.

Compare that to the result of ``x**y - 1``, where the
relative errors are all larger than 8e-8:

>>> x**y - 1
array([ 1.82321491e-10,  2.30258035e-11, -4.68750039e-10])pro_ang1pro_ang1(m, n, c, x, out=None)

Prolate spheroidal angular function of the first kind and its derivative

Computes the prolate spheroidal angular function of the first kind
and its derivative (with respect to `x`) for mode parameters m>=0
and n>=m, spheroidal parameter `c` and ``|x| < 1.0``.

Parameters
----------
m : array_like
    Nonnegative mode parameter m
n : array_like
    Mode parameter n (>= m)
c : array_like
    Spheroidal parameter
x : array_like
    Real parameter (``|x| < 1.0``)
out : ndarray, optional
    Optional output array for the function results

Returns
-------
s : scalar or ndarray
    Value of the function
sp : scalar or ndarray
    Value of the derivative vs xpro_ang1_cvpro_ang1_cv(m, n, c, cv, x, out=None)

Prolate spheroidal angular function pro_ang1 for precomputed characteristic value

Computes the prolate spheroidal angular function of the first kind
and its derivative (with respect to `x`) for mode parameters m>=0
and n>=m, spheroidal parameter `c` and ``|x| < 1.0``. Requires
pre-computed characteristic value.

Parameters
----------
m : array_like
    Nonnegative mode parameter m
n : array_like
    Mode parameter n (>= m)
c : array_like
    Spheroidal parameter
cv : array_like
    Characteristic value
x : array_like
    Real parameter (``|x| < 1.0``)
out : ndarray, optional
    Optional output array for the function results

Returns
-------
s : scalar or ndarray
    Value of the function
sp : scalar or ndarray
    Value of the derivative vs xpro_cvpro_cv(m, n, c, out=None)

Characteristic value of prolate spheroidal function

Computes the characteristic value of prolate spheroidal wave
functions of order `m`, `n` (n>=m) and spheroidal parameter `c`.

Parameters
----------
m : array_like
    Nonnegative mode parameter m
n : array_like
    Mode parameter n (>= m)
c : array_like
    Spheroidal parameter
out : ndarray, optional
    Optional output array for the function results

Returns
-------
cv : scalar or ndarray
    Characteristic valuepro_rad1pro_rad1(m, n, c, x, out=None)

Prolate spheroidal radial function of the first kind and its derivative

Computes the prolate spheroidal radial function of the first kind
and its derivative (with respect to `x`) for mode parameters m>=0
and n>=m, spheroidal parameter `c` and ``|x| < 1.0``.

Parameters
----------
m : array_like
    Nonnegative mode parameter m
n : array_like
    Mode parameter n (>= m)
c : array_like
    Spheroidal parameter
x : array_like
    Real parameter (``|x| < 1.0``)
out : ndarray, optional
    Optional output array for the function results

Returns
-------
s : scalar or ndarray
    Value of the function
sp : scalar or ndarray
    Value of the derivative vs xpro_rad1_cvpro_rad1_cv(m, n, c, cv, x, out=None)

Prolate spheroidal radial function pro_rad1 for precomputed characteristic value

Computes the prolate spheroidal radial function of the first kind
and its derivative (with respect to `x`) for mode parameters m>=0
and n>=m, spheroidal parameter `c` and ``|x| < 1.0``. Requires
pre-computed characteristic value.

Parameters
----------
m : array_like
    Nonnegative mode parameter m
n : array_like
    Mode parameter n (>= m)
c : array_like
    Spheroidal parameter
cv : array_like
    Characteristic value
x : array_like
    Real parameter (``|x| < 1.0``)
out : ndarray, optional
    Optional output array for the function results

Returns
-------
s : scalar or ndarray
    Value of the function
sp : scalar or ndarray
    Value of the derivative vs xpro_rad2pro_rad2(m, n, c, x, out=None)

Prolate spheroidal radial function of the second kind and its derivative

Computes the prolate spheroidal radial function of the second kind
and its derivative (with respect to `x`) for mode parameters m>=0
and n>=m, spheroidal parameter `c` and ``|x| < 1.0``.

Parameters
----------
m : array_like
    Nonnegative mode parameter m
n : array_like
    Mode parameter n (>= m)
c : array_like
    Spheroidal parameter
cv : array_like
    Characteristic value
x : array_like
    Real parameter (``|x| < 1.0``)
out : ndarray, optional
    Optional output array for the function results

Returns
-------
s : scalar or ndarray
    Value of the function
sp : scalar or ndarray
    Value of the derivative vs xpro_rad2_cvpro_rad2_cv(m, n, c, cv, x, out=None)

Prolate spheroidal radial function pro_rad2 for precomputed characteristic value

Computes the prolate spheroidal radial function of the second kind
and its derivative (with respect to `x`) for mode parameters m>=0
and n>=m, spheroidal parameter `c` and ``|x| < 1.0``. Requires
pre-computed characteristic value.

Parameters
----------
m : array_like
    Nonnegative mode parameter m
n : array_like
    Mode parameter n (>= m)
c : array_like
    Spheroidal parameter
cv : array_like
    Characteristic value
x : array_like
    Real parameter (``|x| < 1.0``)
out : ndarray, optional
    Optional output array for the function results

Returns
-------
s : scalar or ndarray
    Value of the function
sp : scalar or ndarray
    Value of the derivative vs xpseudo_huberpseudo_huber(delta, r, out=None)

Pseudo-Huber loss function.

.. math:: \mathrm{pseudo\_huber}(\delta, r) =
          \delta^2 \left( \sqrt{ 1 + \left( \frac{r}{\delta} \right)^2 } - 1 \right)

Parameters
----------
delta : array_like
    Input array, indicating the soft quadratic vs. linear loss changepoint.
r : array_like
    Input array, possibly representing residuals.
out : ndarray, optional
    Optional output array for the function results

Returns
-------
res : scalar or ndarray
    The computed Pseudo-Huber loss function values.

See Also
--------
huber: Similar function which this function approximates

Notes
-----
Like `huber`, `pseudo_huber` often serves as a robust loss function
in statistics or machine learning to reduce the influence of outliers.
Unlike `huber`, `pseudo_huber` is smooth.

Typically, `r` represents residuals, the difference
between a model prediction and data. Then, for :math:`|r|\leq\delta`,
`pseudo_huber` resembles the squared error and for :math:`|r|>\delta` the
absolute error. This way, the Pseudo-Huber loss often achieves
a fast convergence in model fitting for small residuals like the squared
error loss function and still reduces the influence of outliers
(:math:`|r|>\delta`) like the absolute error loss. As :math:`\delta` is
the cutoff between squared and absolute error regimes, it has
to be tuned carefully for each problem. `pseudo_huber` is also
convex, making it suitable for gradient based optimization. [1]_ [2]_

.. versionadded:: 0.15.0

References
----------
.. [1] Hartley, Zisserman, "Multiple View Geometry in Computer Vision".
       2003. Cambridge University Press. p. 619
.. [2] Charbonnier et al. "Deterministic edge-preserving regularization
       in computed imaging". 1997. IEEE Trans. Image Processing.
       6 (2): 298 - 311.

Examples
--------
Import all necessary modules.

>>> import numpy as np
>>> from scipy.special import pseudo_huber, huber
>>> import matplotlib.pyplot as plt

Calculate the function for ``delta=1`` at ``r=2``.

>>> pseudo_huber(1., 2.)
1.2360679774997898

Calculate the function at ``r=2`` for different `delta` by providing
a list or NumPy array for `delta`.

>>> pseudo_huber([1., 2., 4.], 3.)
array([2.16227766, 3.21110255, 4.        ])

Calculate the function for ``delta=1`` at several points by providing
a list or NumPy array for `r`.

>>> pseudo_huber(2., np.array([1., 1.5, 3., 4.]))
array([0.47213595, 1.        , 3.21110255, 4.94427191])

The function can be calculated for different `delta` and `r` by
providing arrays for both with compatible shapes for broadcasting.

>>> r = np.array([1., 2.5, 8., 10.])
>>> deltas = np.array([[1.], [5.], [9.]])
>>> print(r.shape, deltas.shape)
(4,) (3, 1)

>>> pseudo_huber(deltas, r)
array([[ 0.41421356,  1.6925824 ,  7.06225775,  9.04987562],
       [ 0.49509757,  2.95084972, 22.16990566, 30.90169944],
       [ 0.49846624,  3.06693762, 27.37435121, 40.08261642]])

Plot the function for different `delta`.

>>> x = np.linspace(-4, 4, 500)
>>> deltas = [1, 2, 3]
>>> linestyles = ["dashed", "dotted", "dashdot"]
>>> fig, ax = plt.subplots()
>>> combined_plot_parameters = list(zip(deltas, linestyles))
>>> for delta, style in combined_plot_parameters:
...     ax.plot(x, pseudo_huber(delta, x), label=rf"$\delta={delta}$",
...             ls=style)
>>> ax.legend(loc="upper center")
>>> ax.set_xlabel("$x$")
>>> ax.set_title(r"Pseudo-Huber loss function $h_{\delta}(x)$")
>>> ax.set_xlim(-4, 4)
>>> ax.set_ylim(0, 8)
>>> plt.show()

Finally, illustrate the difference between `huber` and `pseudo_huber` by
plotting them and their gradients with respect to `r`. The plot shows
that `pseudo_huber` is continuously differentiable while `huber` is not
at the points :math:`\pm\delta`.

>>> def huber_grad(delta, x):
...     grad = np.copy(x)
...     linear_area = np.argwhere(np.abs(x) > delta)
...     grad[linear_area]=delta*np.sign(x[linear_area])
...     return grad
>>> def pseudo_huber_grad(delta, x):
...     return x* (1+(x/delta)**2)**(-0.5)
>>> x=np.linspace(-3, 3, 500)
>>> delta = 1.
>>> fig, ax = plt.subplots(figsize=(7, 7))
>>> ax.plot(x, huber(delta, x), label="Huber", ls="dashed")
>>> ax.plot(x, huber_grad(delta, x), label="Huber Gradient", ls="dashdot")
>>> ax.plot(x, pseudo_huber(delta, x), label="Pseudo-Huber", ls="dotted")
>>> ax.plot(x, pseudo_huber_grad(delta, x), label="Pseudo-Huber Gradient",
...         ls="solid")
>>> ax.legend(loc="upper center")
>>> plt.show()psipsi(z, out=None)

The digamma function.

The logarithmic derivative of the gamma function evaluated at ``z``.

Parameters
----------
z : array_like
    Real or complex argument.
out : ndarray, optional
    Array for the computed values of ``psi``.

Returns
-------
digamma : scalar or ndarray
    Computed values of ``psi``.

Notes
-----
For large values not close to the negative real axis, ``psi`` is
computed using the asymptotic series (5.11.2) from [1]_. For small
arguments not close to the negative real axis, the recurrence
relation (5.5.2) from [1]_ is used until the argument is large
enough to use the asymptotic series. For values close to the
negative real axis, the reflection formula (5.5.4) from [1]_ is
used first. Note that ``psi`` has a family of zeros on the
negative real axis which occur between the poles at nonpositive
integers. Around the zeros the reflection formula suffers from
cancellation and the implementation loses precision. The sole
positive zero and the first negative zero, however, are handled
separately by precomputing series expansions using [2]_, so the
function should maintain full accuracy around the origin.

References
----------
.. [1] NIST Digital Library of Mathematical Functions
       https://dlmf.nist.gov/5
.. [2] Fredrik Johansson and others.
       "mpmath: a Python library for arbitrary-precision floating-point arithmetic"
       (Version 0.19) http://mpmath.org/

Examples
--------
>>> from scipy.special import psi
>>> z = 3 + 4j
>>> psi(z)
(1.55035981733341+1.0105022091860445j)

Verify psi(z) = psi(z + 1) - 1/z:

>>> psi(z + 1) - 1/z
(1.55035981733341+1.0105022091860445j)radianradian(d, m, s, out=None)

Convert from degrees to radians.

Returns the angle given in (d)egrees, (m)inutes, and (s)econds in
radians.

Parameters
----------
d : array_like
    Degrees, can be real-valued.
m : array_like
    Minutes, can be real-valued.
s : array_like
    Seconds, can be real-valued.
out : ndarray, optional
    Optional output array for the function results.

Returns
-------
scalar or ndarray
    Values of the inputs in radians.

Examples
--------
>>> import scipy.special as sc

There are many ways to specify an angle.

>>> sc.radian(90, 0, 0)
1.5707963267948966
>>> sc.radian(0, 60 * 90, 0)
1.5707963267948966
>>> sc.radian(0, 0, 60**2 * 90)
1.5707963267948966

The inputs can be real-valued.

>>> sc.radian(1.5, 0, 0)
0.02617993877991494
>>> sc.radian(1, 30, 0)
0.02617993877991494rel_entrrel_entr(x, y, out=None)

Elementwise function for computing relative entropy.

.. math::

    \mathrm{rel\_entr}(x, y) =
        \begin{cases}
            x \log(x / y) & x > 0, y > 0 \\
            0 & x = 0, y \ge 0 \\
            \infty & \text{otherwise}
        \end{cases}

Parameters
----------
x, y : array_like
    Input arrays
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    Relative entropy of the inputs

See Also
--------
entr, kl_div, scipy.stats.entropy

Notes
-----
.. versionadded:: 0.15.0

This function is jointly convex in x and y.

The origin of this function is in convex programming; see
[1]_. Given two discrete probability distributions :math:`p_1,
\ldots, p_n` and :math:`q_1, \ldots, q_n`, the definition of relative
entropy in the context of *information theory* is

.. math::

    \sum_{i = 1}^n \mathrm{rel\_entr}(p_i, q_i).

To compute the latter quantity, use `scipy.stats.entropy`.

See [2]_ for details.

References
----------
.. [1] Boyd, Stephen and Lieven Vandenberghe. *Convex optimization*.
       Cambridge University Press, 2004.
       :doi:`https://doi.org/10.1017/CBO9780511804441`
.. [2] Kullback-Leibler divergence,
       https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergencergammargamma(z, out=None)

Reciprocal of the gamma function.

Defined as :math:`1 / \Gamma(z)`, where :math:`\Gamma` is the
gamma function. For more on the gamma function see `gamma`.

Parameters
----------
z : array_like
    Real or complex valued input
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    Function results

See Also
--------
gamma, gammaln, loggamma

Notes
-----
The gamma function has no zeros and has simple poles at
nonpositive integers, so `rgamma` is an entire function with zeros
at the nonpositive integers. See the discussion in [dlmf]_ for
more details.

References
----------
.. [dlmf] Nist, Digital Library of Mathematical functions,
    https://dlmf.nist.gov/5.2#i

Examples
--------
>>> import scipy.special as sc

It is the reciprocal of the gamma function.

>>> sc.rgamma([1, 2, 3, 4])
array([1.        , 1.        , 0.5       , 0.16666667])
>>> 1 / sc.gamma([1, 2, 3, 4])
array([1.        , 1.        , 0.5       , 0.16666667])

It is zero at nonpositive integers.

>>> sc.rgamma([0, -1, -2, -3])
array([0., 0., 0., 0.])

It rapidly underflows to zero along the positive real axis.

>>> sc.rgamma([10, 100, 179])
array([2.75573192e-006, 1.07151029e-156, 0.00000000e+000])roundround(x, out=None)

Round to the nearest integer.

Returns the nearest integer to `x`.  If `x` ends in 0.5 exactly,
the nearest even integer is chosen.

Parameters
----------
x : array_like
    Real valued input.
out : ndarray, optional
    Optional output array for the function results.

Returns
-------
scalar or ndarray
    The nearest integers to the elements of `x`. The result is of
    floating type, not integer type.

Examples
--------
>>> import scipy.special as sc

It rounds to even.

>>> sc.round([0.5, 1.5])
array([0., 2.])shichi(x, out=None)

Hyperbolic sine and cosine integrals.

The hyperbolic sine integral is

.. math::

  \int_0^x \frac{\sinh{t}}{t}dt

and the hyperbolic cosine integral is

.. math::

  \gamma + \log(x) + \int_0^x \frac{\cosh{t} - 1}{t} dt

where :math:`\gamma` is Euler's constant and :math:`\log` is the
principal branch of the logarithm [1]_.

Parameters
----------
x : array_like
    Real or complex points at which to compute the hyperbolic sine
    and cosine integrals.
out : tuple of ndarray, optional
    Optional output arrays for the function results

Returns
-------
si : scalar or ndarray
    Hyperbolic sine integral at ``x``
ci : scalar or ndarray
    Hyperbolic cosine integral at ``x``

See Also
--------
sici : Sine and cosine integrals.
exp1 : Exponential integral E1.
expi : Exponential integral Ei.

Notes
-----
For real arguments with ``x < 0``, ``chi`` is the real part of the
hyperbolic cosine integral. For such points ``chi(x)`` and ``chi(x
+ 0j)`` differ by a factor of ``1j*pi``.

For real arguments the function is computed by calling Cephes'
[2]_ *shichi* routine. For complex arguments the algorithm is based
on Mpmath's [3]_ *shi* and *chi* routines.

References
----------
.. [1] Milton Abramowitz and Irene A. Stegun, eds.
       Handbook of Mathematical Functions with Formulas,
       Graphs, and Mathematical Tables. New York: Dover, 1972.
       (See Section 5.2.)
.. [2] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/
.. [3] Fredrik Johansson and others.
       "mpmath: a Python library for arbitrary-precision floating-point
       arithmetic" (Version 0.19) http://mpmath.org/

Examples
--------
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.special import shichi, sici

`shichi` accepts real or complex input:

>>> shichi(0.5)
(0.5069967498196671, -0.05277684495649357)
>>> shichi(0.5 + 2.5j)
((0.11772029666668238+1.831091777729851j),
 (0.29912435887648825+1.7395351121166562j))

The hyperbolic sine and cosine integrals Shi(z) and Chi(z) are
related to the sine and cosine integrals Si(z) and Ci(z) by

* Shi(z) = -i*Si(i*z)
* Chi(z) = Ci(-i*z) + i*pi/2

>>> z = 0.25 + 5j
>>> shi, chi = shichi(z)
>>> shi, -1j*sici(1j*z)[0]            # Should be the same.
((-0.04834719325101729+1.5469354086921228j),
 (-0.04834719325101729+1.5469354086921228j))
>>> chi, sici(-1j*z)[1] + 1j*np.pi/2  # Should be the same.
((-0.19568708973868087+1.556276312103824j),
 (-0.19568708973868087+1.556276312103824j))

Plot the functions evaluated on the real axis:

>>> xp = np.geomspace(1e-8, 4.0, 250)
>>> x = np.concatenate((-xp[::-1], xp))
>>> shi, chi = shichi(x)

>>> fig, ax = plt.subplots()
>>> ax.plot(x, shi, label='Shi(x)')
>>> ax.plot(x, chi, '--', label='Chi(x)')
>>> ax.set_xlabel('x')
>>> ax.set_title('Hyperbolic Sine and Cosine Integrals')
>>> ax.legend(shadow=True, framealpha=1, loc='lower right')
>>> ax.grid(True)
>>> plt.show()sici(x, out=None)

Sine and cosine integrals.

The sine integral is

.. math::

  \int_0^x \frac{\sin{t}}{t}dt

and the cosine integral is

.. math::

  \gamma + \log(x) + \int_0^x \frac{\cos{t} - 1}{t}dt

where :math:`\gamma` is Euler's constant and :math:`\log` is the
principal branch of the logarithm [1]_.

Parameters
----------
x : array_like
    Real or complex points at which to compute the sine and cosine
    integrals.
out : tuple of ndarray, optional
    Optional output arrays for the function results

Returns
-------
si : scalar or ndarray
    Sine integral at ``x``
ci : scalar or ndarray
    Cosine integral at ``x``

See Also
--------
shichi : Hyperbolic sine and cosine integrals.
exp1 : Exponential integral E1.
expi : Exponential integral Ei.

Notes
-----
For real arguments with ``x < 0``, ``ci`` is the real part of the
cosine integral. For such points ``ci(x)`` and ``ci(x + 0j)``
differ by a factor of ``1j*pi``.

For real arguments the function is computed by calling Cephes'
[2]_ *sici* routine. For complex arguments the algorithm is based
on Mpmath's [3]_ *si* and *ci* routines.

References
----------
.. [1] Milton Abramowitz and Irene A. Stegun, eds.
       Handbook of Mathematical Functions with Formulas,
       Graphs, and Mathematical Tables. New York: Dover, 1972.
       (See Section 5.2.)
.. [2] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/
.. [3] Fredrik Johansson and others.
       "mpmath: a Python library for arbitrary-precision floating-point
       arithmetic" (Version 0.19) http://mpmath.org/

Examples
--------
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.special import sici, exp1

`sici` accepts real or complex input:

>>> sici(2.5)
(1.7785201734438267, 0.2858711963653835)
>>> sici(2.5 + 3j)
((4.505735874563953+0.06863305018999577j),
(0.0793644206906966-2.935510262937543j))

For z in the right half plane, the sine and cosine integrals are
related to the exponential integral E1 (implemented in SciPy as
`scipy.special.exp1`) by

* Si(z) = (E1(i*z) - E1(-i*z))/2i + pi/2
* Ci(z) = -(E1(i*z) + E1(-i*z))/2

See [1]_ (equations 5.2.21 and 5.2.23).

We can verify these relations:

>>> z = 2 - 3j
>>> sici(z)
((4.54751388956229-1.3991965806460565j),
(1.408292501520851+2.9836177420296055j))

>>> (exp1(1j*z) - exp1(-1j*z))/2j + np.pi/2  # Same as sine integral
(4.54751388956229-1.3991965806460565j)

>>> -(exp1(1j*z) + exp1(-1j*z))/2            # Same as cosine integral
(1.408292501520851+2.9836177420296055j)

Plot the functions evaluated on the real axis; the dotted horizontal
lines are at pi/2 and -pi/2:

>>> x = np.linspace(-16, 16, 150)
>>> si, ci = sici(x)

>>> fig, ax = plt.subplots()
>>> ax.plot(x, si, label='Si(x)')
>>> ax.plot(x, ci, '--', label='Ci(x)')
>>> ax.legend(shadow=True, framealpha=1, loc='upper left')
>>> ax.set_xlabel('x')
>>> ax.set_title('Sine and Cosine Integrals')
>>> ax.axhline(np.pi/2, linestyle=':', alpha=0.5, color='k')
>>> ax.axhline(-np.pi/2, linestyle=':', alpha=0.5, color='k')
>>> ax.grid(True)
>>> plt.show()sindgsindg(x, out=None)

Sine of the angle `x` given in degrees.

Parameters
----------
x : array_like
    Angle, given in degrees.
out : ndarray, optional
    Optional output array for the function results.

Returns
-------
scalar or ndarray
    Sine at the input.

See Also
--------
cosdg, tandg, cotdg

Examples
--------
>>> import numpy as np
>>> import scipy.special as sc

It is more accurate than using sine directly.

>>> x = 180 * np.arange(3)
>>> sc.sindg(x)
array([ 0., -0.,  0.])
>>> np.sin(x * np.pi / 180)
array([ 0.0000000e+00,  1.2246468e-16, -2.4492936e-16])smirnovsmirnov(n, d, out=None)

Kolmogorov-Smirnov complementary cumulative distribution function

Returns the exact Kolmogorov-Smirnov complementary cumulative
distribution function,(aka the Survival Function) of Dn+ (or Dn-)
for a one-sided test of equality between an empirical and a
theoretical distribution. It is equal to the probability that the
maximum difference between a theoretical distribution and an empirical
one based on `n` samples is greater than d.

Parameters
----------
n : int
  Number of samples
d : float array_like
  Deviation between the Empirical CDF (ECDF) and the target CDF.
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    The value(s) of smirnov(n, d), Prob(Dn+ >= d) (Also Prob(Dn- >= d))

See Also
--------
smirnovi : The Inverse Survival Function for the distribution
scipy.stats.ksone : Provides the functionality as a continuous distribution
kolmogorov, kolmogi : Functions for the two-sided distribution

Notes
-----
`smirnov` is used by `stats.kstest` in the application of the
Kolmogorov-Smirnov Goodness of Fit test. For historical reasons this
function is exposed in `scpy.special`, but the recommended way to achieve
the most accurate CDF/SF/PDF/PPF/ISF computations is to use the
`stats.ksone` distribution.

Examples
--------
>>> import numpy as np
>>> from scipy.special import smirnov
>>> from scipy.stats import norm

Show the probability of a gap at least as big as 0, 0.5 and 1.0 for a
sample of size 5.

>>> smirnov(5, [0, 0.5, 1.0])
array([ 1.   ,  0.056,  0.   ])

Compare a sample of size 5 against N(0, 1), the standard normal
distribution with mean 0 and standard deviation 1.

`x` is the sample.

>>> x = np.array([-1.392, -0.135, 0.114, 0.190, 1.82])

>>> target = norm(0, 1)
>>> cdfs = target.cdf(x)
>>> cdfs
array([0.0819612 , 0.44630594, 0.5453811 , 0.57534543, 0.9656205 ])

Construct the empirical CDF and the K-S statistics (Dn+, Dn-, Dn).

>>> n = len(x)
>>> ecdfs = np.arange(n+1, dtype=float)/n
>>> cols = np.column_stack([x, ecdfs[1:], cdfs, cdfs - ecdfs[:n],
...                        ecdfs[1:] - cdfs])
>>> with np.printoptions(precision=3):
...    print(cols)
[[-1.392  0.2    0.082  0.082  0.118]
 [-0.135  0.4    0.446  0.246 -0.046]
 [ 0.114  0.6    0.545  0.145  0.055]
 [ 0.19   0.8    0.575 -0.025  0.225]
 [ 1.82   1.     0.966  0.166  0.034]]
>>> gaps = cols[:, -2:]
>>> Dnpm = np.max(gaps, axis=0)
>>> print(f'Dn-={Dnpm[0]:f}, Dn+={Dnpm[1]:f}')
Dn-=0.246306, Dn+=0.224655
>>> probs = smirnov(n, Dnpm)
>>> print(f'For a sample of size {n} drawn from N(0, 1):',
...       f' Smirnov n={n}: Prob(Dn- >= {Dnpm[0]:f}) = {probs[0]:.4f}',
...       f' Smirnov n={n}: Prob(Dn+ >= {Dnpm[1]:f}) = {probs[1]:.4f}',
...       sep='\n')
For a sample of size 5 drawn from N(0, 1):
 Smirnov n=5: Prob(Dn- >= 0.246306) = 0.4711
 Smirnov n=5: Prob(Dn+ >= 0.224655) = 0.5245

Plot the empirical CDF and the standard normal CDF.

>>> import matplotlib.pyplot as plt
>>> plt.step(np.concatenate(([-2.5], x, [2.5])),
...          np.concatenate((ecdfs, [1])),
...          where='post', label='Empirical CDF')
>>> xx = np.linspace(-2.5, 2.5, 100)
>>> plt.plot(xx, target.cdf(xx), '--', label='CDF for N(0, 1)')

Add vertical lines marking Dn+ and Dn-.

>>> iminus, iplus = np.argmax(gaps, axis=0)
>>> plt.vlines([x[iminus]], ecdfs[iminus], cdfs[iminus], color='r',
...            alpha=0.5, lw=4)
>>> plt.vlines([x[iplus]], cdfs[iplus], ecdfs[iplus+1], color='m',
...            alpha=0.5, lw=4)

>>> plt.grid(True)
>>> plt.legend(framealpha=1, shadow=True)
>>> plt.show()smirnovismirnovi(n, p, out=None)

Inverse to `smirnov`

Returns `d` such that ``smirnov(n, d) == p``, the critical value
corresponding to `p`.

Parameters
----------
n : int
  Number of samples
p : float array_like
    Probability
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    The value(s) of smirnovi(n, p), the critical values.

See Also
--------
smirnov : The Survival Function (SF) for the distribution
scipy.stats.ksone : Provides the functionality as a continuous distribution
kolmogorov, kolmogi : Functions for the two-sided distribution
scipy.stats.kstwobign : Two-sided Kolmogorov-Smirnov distribution, large n

Notes
-----
`smirnov` is used by `stats.kstest` in the application of the
Kolmogorov-Smirnov Goodness of Fit test. For historical reasons this
function is exposed in `scpy.special`, but the recommended way to achieve
the most accurate CDF/SF/PDF/PPF/ISF computations is to use the
`stats.ksone` distribution.

Examples
--------
>>> from scipy.special import smirnovi, smirnov

>>> n = 24
>>> deviations = [0.1, 0.2, 0.3]

Use `smirnov` to compute the complementary CDF of the Smirnov
distribution for the given number of samples and deviations.

>>> p = smirnov(n, deviations)
>>> p
array([0.58105083, 0.12826832, 0.01032231])

The inverse function ``smirnovi(n, p)`` returns ``deviations``.

>>> smirnovi(n, p)
array([0.1, 0.2, 0.3])spencespence(z, out=None)

Spence's function, also known as the dilogarithm.

It is defined to be

.. math::
  \int_1^z \frac{\log(t)}{1 - t}dt

for complex :math:`z`, where the contour of integration is taken
to avoid the branch cut of the logarithm. Spence's function is
analytic everywhere except the negative real axis where it has a
branch cut.

Parameters
----------
z : array_like
    Points at which to evaluate Spence's function
out : ndarray, optional
    Optional output array for the function results

Returns
-------
s : scalar or ndarray
    Computed values of Spence's function

Notes
-----
There is a different convention which defines Spence's function by
the integral

.. math::
  -\int_0^z \frac{\log(1 - t)}{t}dt;

this is our ``spence(1 - z)``.

Examples
--------
>>> import numpy as np
>>> from scipy.special import spence
>>> import matplotlib.pyplot as plt

The function is defined for complex inputs:

>>> spence([1-1j, 1.5+2j, 3j, -10-5j])
array([-0.20561676+0.91596559j, -0.86766909-1.39560134j,
       -0.59422064-2.49129918j, -1.14044398+6.80075924j])

For complex inputs on the branch cut, which is the negative real axis,
the function returns the limit for ``z`` with positive imaginary part.
For example, in the following, note the sign change of the imaginary
part of the output for ``z = -2`` and ``z = -2 - 1e-8j``:

>>> spence([-2 + 1e-8j, -2, -2 - 1e-8j])
array([2.32018041-3.45139229j, 2.32018042-3.4513923j ,
       2.32018041+3.45139229j])

The function returns ``nan`` for real inputs on the branch cut:

>>> spence(-1.5)
nan

Verify some particular values: ``spence(0) = pi**2/6``,
``spence(1) = 0`` and ``spence(2) = -pi**2/12``.

>>> spence([0, 1, 2])
array([ 1.64493407,  0.        , -0.82246703])
>>> np.pi**2/6, -np.pi**2/12
(1.6449340668482264, -0.8224670334241132)

Verify the identity::

    spence(z) + spence(1 - z) = pi**2/6 - log(z)*log(1 - z)

>>> z = 3 + 4j
>>> spence(z) + spence(1 - z)
(-2.6523186143876067+1.8853470951513935j)
>>> np.pi**2/6 - np.log(z)*np.log(1 - z)
(-2.652318614387606+1.885347095151394j)

Plot the function for positive real input.

>>> fig, ax = plt.subplots()
>>> x = np.linspace(0, 6, 400)
>>> ax.plot(x, spence(x))
>>> ax.grid()
>>> ax.set_xlabel('x')
>>> ax.set_title('spence(x)')
>>> plt.show()sph_harm(m, n, theta, phi, out=None)

Compute spherical harmonics.

The spherical harmonics are defined as

.. math::

    Y^m_n(\theta,\phi) = \sqrt{\frac{2n+1}{4\pi} \frac{(n-m)!}{(n+m)!}}
      e^{i m \theta} P^m_n(\cos(\phi))

where :math:`P_n^m` are the associated Legendre functions; see `lpmv`.

Parameters
----------
m : array_like
    Order of the harmonic (int); must have ``|m| <= n``.
n : array_like
   Degree of the harmonic (int); must have ``n >= 0``. This is
   often denoted by ``l`` (lower case L) in descriptions of
   spherical harmonics.
theta : array_like
   Azimuthal (longitudinal) coordinate; must be in ``[0, 2*pi]``.
phi : array_like
   Polar (colatitudinal) coordinate; must be in ``[0, pi]``.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
y_mn : complex scalar or ndarray
   The harmonic :math:`Y^m_n` sampled at ``theta`` and ``phi``.

Notes
-----
There are different conventions for the meanings of the input
arguments ``theta`` and ``phi``. In SciPy ``theta`` is the
azimuthal angle and ``phi`` is the polar angle. It is common to
see the opposite convention, that is, ``theta`` as the polar angle
and ``phi`` as the azimuthal angle.

Note that SciPy's spherical harmonics include the Condon-Shortley
phase [2]_ because it is part of `lpmv`.

With SciPy's conventions, the first several spherical harmonics
are

.. math::

    Y_0^0(\theta, \phi) &= \frac{1}{2} \sqrt{\frac{1}{\pi}} \\
    Y_1^{-1}(\theta, \phi) &= \frac{1}{2} \sqrt{\frac{3}{2\pi}}
                                e^{-i\theta} \sin(\phi) \\
    Y_1^0(\theta, \phi) &= \frac{1}{2} \sqrt{\frac{3}{\pi}}
                             \cos(\phi) \\
    Y_1^1(\theta, \phi) &= -\frac{1}{2} \sqrt{\frac{3}{2\pi}}
                             e^{i\theta} \sin(\phi).

References
----------
.. [1] Digital Library of Mathematical Functions, 14.30.
       https://dlmf.nist.gov/14.30
.. [2] https://en.wikipedia.org/wiki/Spherical_harmonics#Condon.E2.80.93Shortley_phasestdtr(df, t, out=None)

Student t distribution cumulative distribution function

Returns the integral:

.. math::
    \frac{\Gamma((df+1)/2)}{\sqrt{\pi df} \Gamma(df/2)}
    \int_{-\infty}^t (1+x^2/df)^{-(df+1)/2}\, dx

Parameters
----------
df : array_like
    Degrees of freedom
t : array_like
    Upper bound of the integral
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    Value of the Student t CDF at t

See Also
--------
stdtridf : inverse of stdtr with respect to `df`
stdtrit : inverse of stdtr with respect to `t`
scipy.stats.t : student t distribution

Notes
-----
The student t distribution is also available as `scipy.stats.t`.
Calling `stdtr` directly can improve performance compared to the
``cdf`` method of `scipy.stats.t` (see last example below).

Examples
--------
Calculate the function for ``df=3`` at ``t=1``.

>>> import numpy as np
>>> from scipy.special import stdtr
>>> import matplotlib.pyplot as plt
>>> stdtr(3, 1)
0.8044988905221148

Plot the function for three different degrees of freedom.

>>> x = np.linspace(-10, 10, 1000)
>>> fig, ax = plt.subplots()
>>> parameters = [(1, "solid"), (3, "dashed"), (10, "dotted")]
>>> for (df, linestyle) in parameters:
...     ax.plot(x, stdtr(df, x), ls=linestyle, label=f"$df={df}$")
>>> ax.legend()
>>> ax.set_title("Student t distribution cumulative distribution function")
>>> plt.show()

The function can be computed for several degrees of freedom at the same
time by providing a NumPy array or list for `df`:

>>> stdtr([1, 2, 3], 1)
array([0.75      , 0.78867513, 0.80449889])

It is possible to calculate the function at several points for several
different degrees of freedom simultaneously by providing arrays for `df`
and `t` with shapes compatible for broadcasting. Compute `stdtr` at
4 points for 3 degrees of freedom resulting in an array of shape 3x4.

>>> dfs = np.array([[1], [2], [3]])
>>> t = np.array([2, 4, 6, 8])
>>> dfs.shape, t.shape
((3, 1), (4,))

>>> stdtr(dfs, t)
array([[0.85241638, 0.92202087, 0.94743154, 0.96041658],
       [0.90824829, 0.97140452, 0.98666426, 0.99236596],
       [0.93033702, 0.98599577, 0.99536364, 0.99796171]])

The t distribution is also available as `scipy.stats.t`. Calling `stdtr`
directly can be much faster than calling the ``cdf`` method of
`scipy.stats.t`. To get the same results, one must use the following
parametrization: ``scipy.stats.t(df).cdf(x) = stdtr(df, x)``.

>>> from scipy.stats import t
>>> df, x = 3, 1
>>> stdtr_result = stdtr(df, x)  # this can be faster than below
>>> stats_result = t(df).cdf(x)
>>> stats_result == stdtr_result  # test that results are equal
Truestdtridf(p, t, out=None)

Inverse of `stdtr` vs df

Returns the argument df such that stdtr(df, t) is equal to `p`.

Parameters
----------
p : array_like
    Probability
t : array_like
    Upper bound of the integral
out : ndarray, optional
    Optional output array for the function results

Returns
-------
df : scalar or ndarray
    Value of `df` such that ``stdtr(df, t) == p``

See Also
--------
stdtr : Student t CDF
stdtrit : inverse of stdtr with respect to `t`
scipy.stats.t : Student t distribution

Examples
--------
Compute the student t cumulative distribution function for one
parameter set.

>>> from scipy.special import stdtr, stdtridf
>>> df, x = 5, 2
>>> cdf_value = stdtr(df, x)
>>> cdf_value
0.9490302605850709

Verify that `stdtridf` recovers the original value for `df` given
the CDF value and `x`.

>>> stdtridf(cdf_value, x)
5.0stdtrit(df, p, out=None)

The `p`-th quantile of the student t distribution.

This function is the inverse of the student t distribution cumulative
distribution function (CDF), returning `t` such that `stdtr(df, t) = p`.

Returns the argument `t` such that stdtr(df, t) is equal to `p`.

Parameters
----------
df : array_like
    Degrees of freedom
p : array_like
    Probability
out : ndarray, optional
    Optional output array for the function results

Returns
-------
t : scalar or ndarray
    Value of `t` such that ``stdtr(df, t) == p``

See Also
--------
stdtr : Student t CDF
stdtridf : inverse of stdtr with respect to `df`
scipy.stats.t : Student t distribution

Notes
-----
The student t distribution is also available as `scipy.stats.t`. Calling
`stdtrit` directly can improve performance compared to the ``ppf``
method of `scipy.stats.t` (see last example below).

Examples
--------
`stdtrit` represents the inverse of the student t distribution CDF which
is available as `stdtr`. Here, we calculate the CDF for ``df`` at
``x=1``. `stdtrit` then returns ``1`` up to floating point errors
given the same value for `df` and the computed CDF value.

>>> import numpy as np
>>> from scipy.special import stdtr, stdtrit
>>> import matplotlib.pyplot as plt
>>> df = 3
>>> x = 1
>>> cdf_value = stdtr(df, x)
>>> stdtrit(df, cdf_value)
0.9999999994418539

Plot the function for three different degrees of freedom.

>>> x = np.linspace(0, 1, 1000)
>>> parameters = [(1, "solid"), (2, "dashed"), (5, "dotted")]
>>> fig, ax = plt.subplots()
>>> for (df, linestyle) in parameters:
...     ax.plot(x, stdtrit(df, x), ls=linestyle, label=f"$df={df}$")
>>> ax.legend()
>>> ax.set_ylim(-10, 10)
>>> ax.set_title("Student t distribution quantile function")
>>> plt.show()

The function can be computed for several degrees of freedom at the same
time by providing a NumPy array or list for `df`:

>>> stdtrit([1, 2, 3], 0.7)
array([0.72654253, 0.6172134 , 0.58438973])

It is possible to calculate the function at several points for several
different degrees of freedom simultaneously by providing arrays for `df`
and `p` with shapes compatible for broadcasting. Compute `stdtrit` at
4 points for 3 degrees of freedom resulting in an array of shape 3x4.

>>> dfs = np.array([[1], [2], [3]])
>>> p = np.array([0.2, 0.4, 0.7, 0.8])
>>> dfs.shape, p.shape
((3, 1), (4,))

>>> stdtrit(dfs, p)
array([[-1.37638192, -0.3249197 ,  0.72654253,  1.37638192],
       [-1.06066017, -0.28867513,  0.6172134 ,  1.06066017],
       [-0.97847231, -0.27667066,  0.58438973,  0.97847231]])

The t distribution is also available as `scipy.stats.t`. Calling `stdtrit`
directly can be much faster than calling the ``ppf`` method of
`scipy.stats.t`. To get the same results, one must use the following
parametrization: ``scipy.stats.t(df).ppf(x) = stdtrit(df, x)``.

>>> from scipy.stats import t
>>> df, x = 3, 0.5
>>> stdtrit_result = stdtrit(df, x)  # this can be faster than below
>>> stats_result = t(df).ppf(x)
>>> stats_result == stdtrit_result  # test that results are equal
Truestruvestruve(v, x, out=None)

Struve function.

Return the value of the Struve function of order `v` at `x`.  The Struve
function is defined as,

.. math::
    H_v(x) = (z/2)^{v + 1} \sum_{n=0}^\infty
    \frac{(-1)^n (z/2)^{2n}}{\Gamma(n + \frac{3}{2}) \Gamma(n + v + \frac{3}{2})},

where :math:`\Gamma` is the gamma function.

Parameters
----------
v : array_like
    Order of the Struve function (float).
x : array_like
    Argument of the Struve function (float; must be positive unless `v` is
    an integer).
out : ndarray, optional
    Optional output array for the function results

Returns
-------
H : scalar or ndarray
    Value of the Struve function of order `v` at `x`.

See Also
--------
modstruve: Modified Struve function

Notes
-----
Three methods discussed in [1]_ are used to evaluate the Struve function:

- power series
- expansion in Bessel functions (if :math:`|z| < |v| + 20`)
- asymptotic large-z expansion (if :math:`z \geq 0.7v + 12`)

Rounding errors are estimated based on the largest terms in the sums, and
the result associated with the smallest error is returned.

References
----------
.. [1] NIST Digital Library of Mathematical Functions
       https://dlmf.nist.gov/11

Examples
--------
Calculate the Struve function of order 1 at 2.

>>> import numpy as np
>>> from scipy.special import struve
>>> import matplotlib.pyplot as plt
>>> struve(1, 2.)
0.6467637282835622

Calculate the Struve function at 2 for orders 1, 2 and 3 by providing
a list for the order parameter `v`.

>>> struve([1, 2, 3], 2.)
array([0.64676373, 0.28031806, 0.08363767])

Calculate the Struve function of order 1 for several points by providing
an array for `x`.

>>> points = np.array([2., 5., 8.])
>>> struve(1, points)
array([0.64676373, 0.80781195, 0.48811605])

Compute the Struve function for several orders at several points by
providing arrays for `v` and `z`. The arrays have to be broadcastable
to the correct shapes.

>>> orders = np.array([[1], [2], [3]])
>>> points.shape, orders.shape
((3,), (3, 1))

>>> struve(orders, points)
array([[0.64676373, 0.80781195, 0.48811605],
       [0.28031806, 1.56937455, 1.51769363],
       [0.08363767, 1.50872065, 2.98697513]])

Plot the Struve functions of order 0 to 3 from -10 to 10.

>>> fig, ax = plt.subplots()
>>> x = np.linspace(-10., 10., 1000)
>>> for i in range(4):
...     ax.plot(x, struve(i, x), label=f'$H_{i!r}$')
>>> ax.legend(ncol=2)
>>> ax.set_xlim(-10, 10)
>>> ax.set_title(r"Struve functions $H_{\nu}$")
>>> plt.show()tandgtandg(x, out=None)

Tangent of angle `x` given in degrees.

Parameters
----------
x : array_like
    Angle, given in degrees.
out : ndarray, optional
    Optional output array for the function results.

Returns
-------
scalar or ndarray
    Tangent at the input.

See Also
--------
sindg, cosdg, cotdg

Examples
--------
>>> import numpy as np
>>> import scipy.special as sc

It is more accurate than using tangent directly.

>>> x = 180 * np.arange(3)
>>> sc.tandg(x)
array([0., 0., 0.])
>>> np.tan(x * np.pi / 180)
array([ 0.0000000e+00, -1.2246468e-16, -2.4492936e-16])tklmbdatklmbda(x, lmbda, out=None)

Cumulative distribution function of the Tukey lambda distribution.

Parameters
----------
x, lmbda : array_like
    Parameters
out : ndarray, optional
    Optional output array for the function results

Returns
-------
cdf : scalar or ndarray
    Value of the Tukey lambda CDF

See Also
--------
scipy.stats.tukeylambda : Tukey lambda distribution

Examples
--------
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.special import tklmbda, expit

Compute the cumulative distribution function (CDF) of the Tukey lambda
distribution at several ``x`` values for `lmbda` = -1.5.

>>> x = np.linspace(-2, 2, 9)
>>> x
array([-2. , -1.5, -1. , -0.5,  0. ,  0.5,  1. ,  1.5,  2. ])
>>> tklmbda(x, -1.5)
array([0.34688734, 0.3786554 , 0.41528805, 0.45629737, 0.5       ,
       0.54370263, 0.58471195, 0.6213446 , 0.65311266])

When `lmbda` is 0, the function is the logistic sigmoid function,
which is implemented in `scipy.special` as `expit`.

>>> tklmbda(x, 0)
array([0.11920292, 0.18242552, 0.26894142, 0.37754067, 0.5       ,
       0.62245933, 0.73105858, 0.81757448, 0.88079708])
>>> expit(x)
array([0.11920292, 0.18242552, 0.26894142, 0.37754067, 0.5       ,
       0.62245933, 0.73105858, 0.81757448, 0.88079708])

When `lmbda` is 1, the Tukey lambda distribution is uniform on the
interval [-1, 1], so the CDF increases linearly.

>>> t = np.linspace(-1, 1, 9)
>>> tklmbda(t, 1)
array([0.   , 0.125, 0.25 , 0.375, 0.5  , 0.625, 0.75 , 0.875, 1.   ])

In the following, we generate plots for several values of `lmbda`.

The first figure shows graphs for `lmbda` <= 0.

>>> styles = ['-', '-.', '--', ':']
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-12, 12, 500)
>>> for k, lmbda in enumerate([-1.0, -0.5, 0.0]):
...     y = tklmbda(x, lmbda)
...     ax.plot(x, y, styles[k], label=rf'$\lambda$ = {lmbda:-4.1f}')

>>> ax.set_title(r'tklmbda(x, $\lambda$)')
>>> ax.set_label('x')
>>> ax.legend(framealpha=1, shadow=True)
>>> ax.grid(True)

The second figure shows graphs for `lmbda` > 0.  The dots in the
graphs show the bounds of the support of the distribution.

>>> fig, ax = plt.subplots()
>>> x = np.linspace(-4.2, 4.2, 500)
>>> lmbdas = [0.25, 0.5, 1.0, 1.5]
>>> for k, lmbda in enumerate(lmbdas):
...     y = tklmbda(x, lmbda)
...     ax.plot(x, y, styles[k], label=fr'$\lambda$ = {lmbda}')

>>> ax.set_prop_cycle(None)
>>> for lmbda in lmbdas:
...     ax.plot([-1/lmbda, 1/lmbda], [0, 1], '.', ms=8)

>>> ax.set_title(r'tklmbda(x, $\lambda$)')
>>> ax.set_xlabel('x')
>>> ax.legend(framealpha=1, shadow=True)
>>> ax.grid(True)

>>> plt.tight_layout()
>>> plt.show()

The CDF of the Tukey lambda distribution is also implemented as the
``cdf`` method of `scipy.stats.tukeylambda`.  In the following,
``tukeylambda.cdf(x, -0.5)`` and ``tklmbda(x, -0.5)`` compute the
same values:

>>> from scipy.stats import tukeylambda
>>> x = np.linspace(-2, 2, 9)

>>> tukeylambda.cdf(x, -0.5)
array([0.21995157, 0.27093858, 0.33541677, 0.41328161, 0.5       ,
       0.58671839, 0.66458323, 0.72906142, 0.78004843])

>>> tklmbda(x, -0.5)
array([0.21995157, 0.27093858, 0.33541677, 0.41328161, 0.5       ,
       0.58671839, 0.66458323, 0.72906142, 0.78004843])

The implementation in ``tukeylambda`` also provides location and scale
parameters, and other methods such as ``pdf()`` (the probability
density function) and ``ppf()`` (the inverse of the CDF), so for
working with the Tukey lambda distribution, ``tukeylambda`` is more
generally useful.  The primary advantage of ``tklmbda`` is that it is
significantly faster than ``tukeylambda.cdf``.voigt_profilevoigt_profile(x, sigma, gamma, out=None)

Voigt profile.

The Voigt profile is a convolution of a 1-D Normal distribution with
standard deviation ``sigma`` and a 1-D Cauchy distribution with half-width at
half-maximum ``gamma``.

If ``sigma = 0``, PDF of Cauchy distribution is returned.
Conversely, if ``gamma = 0``, PDF of Normal distribution is returned.
If ``sigma = gamma = 0``, the return value is ``Inf`` for ``x = 0``,
and ``0`` for all other ``x``.

Parameters
----------
x : array_like
    Real argument
sigma : array_like
    The standard deviation of the Normal distribution part
gamma : array_like
    The half-width at half-maximum of the Cauchy distribution part
out : ndarray, optional
    Optional output array for the function values

Returns
-------
scalar or ndarray
    The Voigt profile at the given arguments

See Also
--------
wofz : Faddeeva function

Notes
-----
It can be expressed in terms of Faddeeva function

.. math:: V(x; \sigma, \gamma) = \frac{Re[w(z)]}{\sigma\sqrt{2\pi}},
.. math:: z = \frac{x + i\gamma}{\sqrt{2}\sigma}

where :math:`w(z)` is the Faddeeva function.

References
----------
.. [1] https://en.wikipedia.org/wiki/Voigt_profile

Examples
--------
Calculate the function at point 2 for ``sigma=1`` and ``gamma=1``.

>>> from scipy.special import voigt_profile
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> voigt_profile(2, 1., 1.)
0.09071519942627544

Calculate the function at several points by providing a NumPy array
for `x`.

>>> values = np.array([-2., 0., 5])
>>> voigt_profile(values, 1., 1.)
array([0.0907152 , 0.20870928, 0.01388492])

Plot the function for different parameter sets.

>>> fig, ax = plt.subplots(figsize=(8, 8))
>>> x = np.linspace(-10, 10, 500)
>>> parameters_list = [(1.5, 0., "solid"), (1.3, 0.5, "dashed"),
...                    (0., 1.8, "dotted"), (1., 1., "dashdot")]
>>> for params in parameters_list:
...     sigma, gamma, linestyle = params
...     voigt = voigt_profile(x, sigma, gamma)
...     ax.plot(x, voigt, label=rf"$\sigma={sigma},\, \gamma={gamma}$",
...             ls=linestyle)
>>> ax.legend()
>>> plt.show()

Verify visually that the Voigt profile indeed arises as the convolution
of a normal and a Cauchy distribution.

>>> from scipy.signal import convolve
>>> x, dx = np.linspace(-10, 10, 500, retstep=True)
>>> def gaussian(x, sigma):
...     return np.exp(-0.5 * x**2/sigma**2)/(sigma * np.sqrt(2*np.pi))
>>> def cauchy(x, gamma):
...     return gamma/(np.pi * (np.square(x)+gamma**2))
>>> sigma = 2
>>> gamma = 1
>>> gauss_profile = gaussian(x, sigma)
>>> cauchy_profile = cauchy(x, gamma)
>>> convolved = dx * convolve(cauchy_profile, gauss_profile, mode="same")
>>> voigt = voigt_profile(x, sigma, gamma)
>>> fig, ax = plt.subplots(figsize=(8, 8))
>>> ax.plot(x, gauss_profile, label="Gauss: $G$", c='b')
>>> ax.plot(x, cauchy_profile, label="Cauchy: $C$", c='y', ls="dashed")
>>> xx = 0.5*(x[1:] + x[:-1])  # midpoints
>>> ax.plot(xx, convolved[1:], label="Convolution: $G * C$", ls='dashdot',
...         c='k')
>>> ax.plot(x, voigt, label="Voigt", ls='dotted', c='r')
>>> ax.legend()
>>> plt.show()wofzwofz(z, out=None)

Faddeeva function

Returns the value of the Faddeeva function for complex argument::

    exp(-z**2) * erfc(-i*z)

Parameters
----------
z : array_like
    complex argument
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    Value of the Faddeeva function

See Also
--------
dawsn, erf, erfc, erfcx, erfi

References
----------
.. [1] Steven G. Johnson, Faddeeva W function implementation.
   http://ab-initio.mit.edu/Faddeeva

Examples
--------
>>> import numpy as np
>>> from scipy import special
>>> import matplotlib.pyplot as plt

>>> x = np.linspace(-3, 3)
>>> z = special.wofz(x)

>>> plt.plot(x, z.real, label='wofz(x).real')
>>> plt.plot(x, z.imag, label='wofz(x).imag')
>>> plt.xlabel('$x$')
>>> plt.legend(framealpha=1, shadow=True)
>>> plt.grid(alpha=0.25)
>>> plt.show()wright_bessel(a, b, x, out=None)

Wright's generalized Bessel function.

Wright's generalized Bessel function is an entire function and defined as

.. math:: \Phi(a, b; x) = \sum_{k=0}^\infty \frac{x^k}{k! \Gamma(a k + b)}

See Also [1].

Parameters
----------
a : array_like of float
    a >= 0
b : array_like of float
    b >= 0
x : array_like of float
    x >= 0
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    Value of the Wright's generalized Bessel function

Notes
-----
Due to the complexity of the function with its three parameters, only
non-negative arguments are implemented.

References
----------
.. [1] Digital Library of Mathematical Functions, 10.46.
       https://dlmf.nist.gov/10.46.E1

Examples
--------
>>> from scipy.special import wright_bessel
>>> a, b, x = 1.5, 1.1, 2.5
>>> wright_bessel(a, b-1, x)
4.5314465939443025

Now, let us verify the relation

.. math:: \Phi(a, b-1; x) = a x \Phi(a, b+a; x) + (b-1) \Phi(a, b; x)

>>> a * x * wright_bessel(a, b+a, x) + (b-1) * wright_bessel(a, b, x)
4.5314465939443025wrightomegawrightomega(z, out=None)

Wright Omega function.

Defined as the solution to

.. math::

    \omega + \log(\omega) = z

where :math:`\log` is the principal branch of the complex logarithm.

Parameters
----------
z : array_like
    Points at which to evaluate the Wright Omega function
out : ndarray, optional
    Optional output array for the function values

Returns
-------
omega : scalar or ndarray
    Values of the Wright Omega function

See Also
--------
lambertw : The Lambert W function

Notes
-----
.. versionadded:: 0.19.0

The function can also be defined as

.. math::

    \omega(z) = W_{K(z)}(e^z)

where :math:`K(z) = \lceil (\Im(z) - \pi)/(2\pi) \rceil` is the
unwinding number and :math:`W` is the Lambert W function.

The implementation here is taken from [1]_.

References
----------
.. [1] Lawrence, Corless, and Jeffrey, "Algorithm 917: Complex
       Double-Precision Evaluation of the Wright :math:`\omega`
       Function." ACM Transactions on Mathematical Software,
       2012. :doi:`10.1145/2168773.2168779`.

Examples
--------
>>> import numpy as np
>>> from scipy.special import wrightomega, lambertw

>>> wrightomega([-2, -1, 0, 1, 2])
array([0.12002824, 0.27846454, 0.56714329, 1.        , 1.5571456 ])

Complex input:

>>> wrightomega(3 + 5j)
(1.5804428632097158+3.8213626783287937j)

Verify that ``wrightomega(z)`` satisfies ``w + log(w) = z``:

>>> w = -5 + 4j
>>> wrightomega(w + np.log(w))
(-5+4j)

Verify the connection to ``lambertw``:

>>> z = 0.5 + 3j
>>> wrightomega(z)
(0.0966015889280649+1.4937828458191993j)
>>> lambertw(np.exp(z))
(0.09660158892806493+1.4937828458191993j)

>>> z = 0.5 + 4j
>>> wrightomega(z)
(-0.3362123489037213+2.282986001579032j)
>>> lambertw(np.exp(z), k=1)
(-0.33621234890372115+2.282986001579032j)xlog1pyxlog1py(x, y, out=None)

Compute ``x*log1p(y)`` so that the result is 0 if ``x = 0``.

Parameters
----------
x : array_like
    Multiplier
y : array_like
    Argument
out : ndarray, optional
    Optional output array for the function results

Returns
-------
z : scalar or ndarray
    Computed x*log1p(y)

Notes
-----

.. versionadded:: 0.13.0

Examples
--------
This example shows how the function can be used to calculate the log of
the probability mass function for a geometric discrete random variable.
The probability mass function of the geometric distribution is defined
as follows:

.. math:: f(k) = (1-p)^{k-1} p

where :math:`p` is the probability of a single success
and :math:`1-p` is the probability of a single failure
and :math:`k` is the number of trials to get the first success.

>>> import numpy as np
>>> from scipy.special import xlog1py
>>> p = 0.5
>>> k = 100
>>> _pmf = np.power(1 - p, k - 1) * p
>>> _pmf
7.888609052210118e-31

If we take k as a relatively large number the value of the probability
mass function can become very low. In such cases taking the log of the
pmf would be more suitable as the log function can change the values
to a scale that is more appropriate to work with.

>>> _log_pmf = xlog1py(k - 1, -p) + np.log(p)
>>> _log_pmf
-69.31471805599453

We can confirm that we get a value close to the original pmf value by
taking the exponential of the log pmf.

>>> _orig_pmf = np.exp(_log_pmf)
>>> np.isclose(_pmf, _orig_pmf)
Truexlogyxlogy(x, y, out=None)

Compute ``x*log(y)`` so that the result is 0 if ``x = 0``.

Parameters
----------
x : array_like
    Multiplier
y : array_like
    Argument
out : ndarray, optional
    Optional output array for the function results

Returns
-------
z : scalar or ndarray
    Computed x*log(y)

Notes
-----
The log function used in the computation is the natural log.

.. versionadded:: 0.13.0

Examples
--------
We can use this function to calculate the binary logistic loss also
known as the binary cross entropy. This loss function is used for
binary classification problems and is defined as:

.. math::
    L = 1/n * \sum_{i=0}^n -(y_i*log(y\_pred_i) + (1-y_i)*log(1-y\_pred_i))

We can define the parameters `x` and `y` as y and y_pred respectively.
y is the array of the actual labels which over here can be either 0 or 1.
y_pred is the array of the predicted probabilities with respect to
the positive class (1).

>>> import numpy as np
>>> from scipy.special import xlogy
>>> y = np.array([0, 1, 0, 1, 1, 0])
>>> y_pred = np.array([0.3, 0.8, 0.4, 0.7, 0.9, 0.2])
>>> n = len(y)
>>> loss = -(xlogy(y, y_pred) + xlogy(1 - y, 1 - y_pred)).sum()
>>> loss /= n
>>> loss
0.29597052165495025

A lower loss is usually better as it indicates that the predictions are
similar to the actual labels. In this example since our predicted
probabilities are close to the actual labels, we get an overall loss
that is reasonably low and appropriate.y0y0(x, out=None)

Bessel function of the second kind of order 0.

Parameters
----------
x : array_like
    Argument (float).
out : ndarray, optional
    Optional output array for the function results

Returns
-------
Y : scalar or ndarray
    Value of the Bessel function of the second kind of order 0 at `x`.

See Also
--------
j0: Bessel function of the first kind of order 0
yv: Bessel function of the first kind

Notes
-----
The domain is divided into the intervals [0, 5] and (5, infinity). In the
first interval a rational approximation :math:`R(x)` is employed to
compute,

.. math::

    Y_0(x) = R(x) + \frac{2 \log(x) J_0(x)}{\pi},

where :math:`J_0` is the Bessel function of the first kind of order 0.

In the second interval, the Hankel asymptotic expansion is employed with
two rational functions of degree 6/6 and 7/7.

This function is a wrapper for the Cephes [1]_ routine `y0`.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/

Examples
--------
Calculate the function at one point:

>>> from scipy.special import y0
>>> y0(1.)
0.08825696421567697

Calculate at several points:

>>> import numpy as np
>>> y0(np.array([0.5, 2., 3.]))
array([-0.44451873,  0.51037567,  0.37685001])

Plot the function from 0 to 10.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 10., 1000)
>>> y = y0(x)
>>> ax.plot(x, y)
>>> plt.show()y1y1(x, out=None)

Bessel function of the second kind of order 1.

Parameters
----------
x : array_like
    Argument (float).
out : ndarray, optional
    Optional output array for the function results

Returns
-------
Y : scalar or ndarray
    Value of the Bessel function of the second kind of order 1 at `x`.

See Also
--------
j1: Bessel function of the first kind of order 1
yn: Bessel function of the second kind
yv: Bessel function of the second kind

Notes
-----
The domain is divided into the intervals [0, 8] and (8, infinity). In the
first interval a 25 term Chebyshev expansion is used, and computing
:math:`J_1` (the Bessel function of the first kind) is required. In the
second, the asymptotic trigonometric representation is employed using two
rational functions of degree 5/5.

This function is a wrapper for the Cephes [1]_ routine `y1`.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/

Examples
--------
Calculate the function at one point:

>>> from scipy.special import y1
>>> y1(1.)
-0.7812128213002888

Calculate at several points:

>>> import numpy as np
>>> y1(np.array([0.5, 2., 3.]))
array([-1.47147239, -0.10703243,  0.32467442])

Plot the function from 0 to 10.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 10., 1000)
>>> y = y1(x)
>>> ax.plot(x, y)
>>> plt.show()ynyn(n, x, out=None)

Bessel function of the second kind of integer order and real argument.

Parameters
----------
n : array_like
    Order (integer).
x : array_like
    Argument (float).
out : ndarray, optional
    Optional output array for the function results

Returns
-------
Y : scalar or ndarray
    Value of the Bessel function, :math:`Y_n(x)`.

See Also
--------
yv : For real order and real or complex argument.
y0: faster implementation of this function for order 0
y1: faster implementation of this function for order 1

Notes
-----
Wrapper for the Cephes [1]_ routine `yn`.

The function is evaluated by forward recurrence on `n`, starting with
values computed by the Cephes routines `y0` and `y1`. If `n = 0` or 1,
the routine for `y0` or `y1` is called directly.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/

Examples
--------
Evaluate the function of order 0 at one point.

>>> from scipy.special import yn
>>> yn(0, 1.)
0.08825696421567697

Evaluate the function at one point for different orders.

>>> yn(0, 1.), yn(1, 1.), yn(2, 1.)
(0.08825696421567697, -0.7812128213002888, -1.6506826068162546)

The evaluation for different orders can be carried out in one call by
providing a list or NumPy array as argument for the `v` parameter:

>>> yn([0, 1, 2], 1.)
array([ 0.08825696, -0.78121282, -1.65068261])

Evaluate the function at several points for order 0 by providing an
array for `z`.

>>> import numpy as np
>>> points = np.array([0.5, 3., 8.])
>>> yn(0, points)
array([-0.44451873,  0.37685001,  0.22352149])

If `z` is an array, the order parameter `v` must be broadcastable to
the correct shape if different orders shall be computed in one call.
To calculate the orders 0 and 1 for an 1D array:

>>> orders = np.array([[0], [1]])
>>> orders.shape
(2, 1)

>>> yn(orders, points)
array([[-0.44451873,  0.37685001,  0.22352149],
       [-1.47147239,  0.32467442, -0.15806046]])

Plot the functions of order 0 to 3 from 0 to 10.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 10., 1000)
>>> for i in range(4):
...     ax.plot(x, yn(i, x), label=f'$Y_{i!r}$')
>>> ax.set_ylim(-3, 1)
>>> ax.legend()
>>> plt.show()yvyv(v, z, out=None)

Bessel function of the second kind of real order and complex argument.

Parameters
----------
v : array_like
    Order (float).
z : array_like
    Argument (float or complex).
out : ndarray, optional
    Optional output array for the function results

Returns
-------
Y : scalar or ndarray
    Value of the Bessel function of the second kind, :math:`Y_v(x)`.

See Also
--------
yve : :math:`Y_v` with leading exponential behavior stripped off.
y0: faster implementation of this function for order 0
y1: faster implementation of this function for order 1

Notes
-----
For positive `v` values, the computation is carried out using the
AMOS [1]_ `zbesy` routine, which exploits the connection to the Hankel
Bessel functions :math:`H_v^{(1)}` and :math:`H_v^{(2)}`,

.. math:: Y_v(z) = \frac{1}{2\imath} (H_v^{(1)} - H_v^{(2)}).

For negative `v` values the formula,

.. math:: Y_{-v}(z) = Y_v(z) \cos(\pi v) + J_v(z) \sin(\pi v)

is used, where :math:`J_v(z)` is the Bessel function of the first kind,
computed using the AMOS routine `zbesj`.  Note that the second term is
exactly zero for integer `v`; to improve accuracy the second term is
explicitly omitted for `v` values such that `v = floor(v)`.

References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
       of a Complex Argument and Nonnegative Order",
       http://netlib.org/amos/

Examples
--------
Evaluate the function of order 0 at one point.

>>> from scipy.special import yv
>>> yv(0, 1.)
0.088256964215677

Evaluate the function at one point for different orders.

>>> yv(0, 1.), yv(1, 1.), yv(1.5, 1.)
(0.088256964215677, -0.7812128213002889, -1.102495575160179)

The evaluation for different orders can be carried out in one call by
providing a list or NumPy array as argument for the `v` parameter:

>>> yv([0, 1, 1.5], 1.)
array([ 0.08825696, -0.78121282, -1.10249558])

Evaluate the function at several points for order 0 by providing an
array for `z`.

>>> import numpy as np
>>> points = np.array([0.5, 3., 8.])
>>> yv(0, points)
array([-0.44451873,  0.37685001,  0.22352149])

If `z` is an array, the order parameter `v` must be broadcastable to
the correct shape if different orders shall be computed in one call.
To calculate the orders 0 and 1 for an 1D array:

>>> orders = np.array([[0], [1]])
>>> orders.shape
(2, 1)

>>> yv(orders, points)
array([[-0.44451873,  0.37685001,  0.22352149],
       [-1.47147239,  0.32467442, -0.15806046]])

Plot the functions of order 0 to 3 from 0 to 10.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 10., 1000)
>>> for i in range(4):
...     ax.plot(x, yv(i, x), label=f'$Y_{i!r}$')
>>> ax.set_ylim(-3, 1)
>>> ax.legend()
>>> plt.show()yveyve(v, z, out=None)

Exponentially scaled Bessel function of the second kind of real order.

Returns the exponentially scaled Bessel function of the second
kind of real order `v` at complex `z`::

    yve(v, z) = yv(v, z) * exp(-abs(z.imag))

Parameters
----------
v : array_like
    Order (float).
z : array_like
    Argument (float or complex).
out : ndarray, optional
    Optional output array for the function results

Returns
-------
Y : scalar or ndarray
    Value of the exponentially scaled Bessel function.

See Also
--------
yv: Unscaled Bessel function of the second kind of real order.

Notes
-----
For positive `v` values, the computation is carried out using the
AMOS [1]_ `zbesy` routine, which exploits the connection to the Hankel
Bessel functions :math:`H_v^{(1)}` and :math:`H_v^{(2)}`,

.. math:: Y_v(z) = \frac{1}{2\imath} (H_v^{(1)} - H_v^{(2)}).

For negative `v` values the formula,

.. math:: Y_{-v}(z) = Y_v(z) \cos(\pi v) + J_v(z) \sin(\pi v)

is used, where :math:`J_v(z)` is the Bessel function of the first kind,
computed using the AMOS routine `zbesj`.  Note that the second term is
exactly zero for integer `v`; to improve accuracy the second term is
explicitly omitted for `v` values such that `v = floor(v)`.

Exponentially scaled Bessel functions are useful for large `z`:
for these, the unscaled Bessel functions can easily under-or overflow.

References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
       of a Complex Argument and Nonnegative Order",
       http://netlib.org/amos/

Examples
--------
Compare the output of `yv` and `yve` for large complex arguments for `z`
by computing their values for order ``v=1`` at ``z=1000j``. We see that
`yv` returns nan but `yve` returns a finite number:

>>> import numpy as np
>>> from scipy.special import yv, yve
>>> v = 1
>>> z = 1000j
>>> yv(v, z), yve(v, z)
((nan+nanj), (-0.012610930256928629+7.721967686709076e-19j))

For real arguments for `z`, `yve` returns the same as `yv` up to
floating point errors.

>>> v, z = 1, 1000
>>> yv(v, z), yve(v, z)
(-0.02478433129235178, -0.02478433129235179)

The function can be evaluated for several orders at the same time by
providing a list or NumPy array for `v`:

>>> yve([1, 2, 3], 1j)
array([-0.20791042+0.14096627j,  0.38053618-0.04993878j,
       0.00815531-1.66311097j])

In the same way, the function can be evaluated at several points in one
call by providing a list or NumPy array for `z`:

>>> yve(1, np.array([1j, 2j, 3j]))
array([-0.20791042+0.14096627j, -0.21526929+0.01205044j,
       -0.19682671+0.00127278j])

It is also possible to evaluate several orders at several points
at the same time by providing arrays for `v` and `z` with
broadcasting compatible shapes. Compute `yve` for two different orders
`v` and three points `z` resulting in a 2x3 array.

>>> v = np.array([[1], [2]])
>>> z = np.array([3j, 4j, 5j])
>>> v.shape, z.shape
((2, 1), (3,))

>>> yve(v, z)
array([[-1.96826713e-01+1.27277544e-03j, -1.78750840e-01+1.45558819e-04j,
        -1.63972267e-01+1.73494110e-05j],
       [1.94960056e-03-1.11782545e-01j,  2.02902325e-04-1.17626501e-01j,
        2.27727687e-05-1.17951906e-01j]])zetaczetac(x, out=None)

Riemann zeta function minus 1.

This function is defined as

.. math:: \zeta(x) = \sum_{k=2}^{\infty} 1 / k^x,

where ``x > 1``.  For ``x < 1`` the analytic continuation is
computed. For more information on the Riemann zeta function, see
[dlmf]_.

Parameters
----------
x : array_like of float
    Values at which to compute zeta(x) - 1 (must be real).
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    Values of zeta(x) - 1.

See Also
--------
zeta

References
----------
.. [dlmf] NIST Digital Library of Mathematical Functions
          https://dlmf.nist.gov/25

Examples
--------
>>> import numpy as np
>>> from scipy.special import zetac, zeta

Some special values:

>>> zetac(2), np.pi**2/6 - 1
(0.64493406684822641, 0.6449340668482264)

>>> zetac(-1), -1.0/12 - 1
(-1.0833333333333333, -1.0833333333333333)

Compare ``zetac(x)`` to ``zeta(x) - 1`` for large `x`:

>>> zetac(60), zeta(60) - 1
(8.673617380119933e-19, 0.0)init scipy.special._ufuncs_cython_3_0_10.cython_function_or_method__reduce____module__func_doc__doc__func_name__name____qualname__func_dict__dict__func_globals__globals__func_closure__closure__func_code__code__func_defaults__defaults____kwdefaults____annotations___is_coroutineCythonUnboundCMethod_ufuncsSet how special-function errors are handled.

    Parameters
    ----------
    all : {'ignore', 'warn' 'raise'}, optional
        Set treatment for all type of special-function errors at
        once. The options are:

        - 'ignore' Take no action when the error occurs
        - 'warn' Print a `SpecialFunctionWarning` when the error
          occurs (via the Python `warnings` module)
        - 'raise' Raise a `SpecialFunctionError` when the error
          occurs.

        The default is to not change the current behavior. If
        behaviors for additional categories of special-function errors
        are specified, then ``all`` is applied first, followed by the
        additional categories.
    singular : {'ignore', 'warn', 'raise'}, optional
        Treatment for singularities.
    underflow : {'ignore', 'warn', 'raise'}, optional
        Treatment for underflow.
    overflow : {'ignore', 'warn', 'raise'}, optional
        Treatment for overflow.
    slow : {'ignore', 'warn', 'raise'}, optional
        Treatment for slow convergence.
    loss : {'ignore', 'warn', 'raise'}, optional
        Treatment for loss of accuracy.
    no_result : {'ignore', 'warn', 'raise'}, optional
        Treatment for failing to find a result.
    domain : {'ignore', 'warn', 'raise'}, optional
        Treatment for an invalid argument to a function.
    arg : {'ignore', 'warn', 'raise'}, optional
        Treatment for an invalid parameter to a function.
    other : {'ignore', 'warn', 'raise'}, optional
        Treatment for an unknown error.

    Returns
    -------
    olderr : dict
        Dictionary containing the old settings.

    See Also
    --------
    geterr : get the current way of handling special-function errors
    errstate : context manager for special-function error handling
    numpy.seterr : similar numpy function for floating-point errors

    Examples
    --------
    >>> import scipy.special as sc
    >>> from pytest import raises
    >>> sc.gammaln(0)
    inf
    >>> olderr = sc.seterr(singular='raise')
    >>> with raises(sc.SpecialFunctionError):
    ...     sc.gammaln(0)
    ...
    >>> _ = sc.seterr(**olderr)

    We can also raise for every category except one.

    >>> olderr = sc.seterr(all='raise', singular='ignore')
    >>> sc.gammaln(0)
    inf
    >>> with raises(sc.SpecialFunctionError):
    ...     sc.spence(-1)
    ...
    >>> _ = sc.seterr(**olderr)

    geterrGet the current way of handling special-function errors.

    Returns
    -------
    err : dict
        A dictionary with keys "singular", "underflow", "overflow",
        "slow", "loss", "no_result", "domain", "arg", and "other",
        whose values are from the strings "ignore", "warn", and
        "raise". The keys represent possible special-function errors,
        and the values define how these errors are handled.

    See Also
    --------
    seterr : set how special-function errors are handled
    errstate : context manager for special-function error handling
    numpy.geterr : similar numpy function for floating-point errors

    Notes
    -----
    For complete documentation of the types of special-function errors
    and treatment options, see `seterr`.

    Examples
    --------
    By default all errors are ignored.

    >>> import scipy.special as sc
    >>> for key, value in sorted(sc.geterr().items()):
    ...     print("{}: {}".format(key, value))
    ...
    arg: ignore
    domain: ignore
    loss: ignore
    no_result: ignore
    other: ignore
    overflow: ignore
    singular: ignore
    slow: ignore
    underflow: ignore

    scipy\special\_ufuncs_extra_code.pxinumpy._core.umath failed to importnumpy._core.multiarray failed to importContext manager for special-function error handling.

    Using an instance of `errstate` as a context manager allows
    statements in that context to execute with a known error handling
    behavior. Upon entering the context the error handling is set with
    `seterr`, and upon exiting it is restored to what it was before.

    Parameters
    ----------
    kwargs : {all, singular, underflow, overflow, slow, loss, no_result, domain, arg, other}
        Keyword arguments. The valid keywords are possible
        special-function errors. Each keyword should have a string
        value that defines the treatment for the particular type of
        error. Values must be 'ignore', 'warn', or 'other'. See
        `seterr` for details.

    See Also
    --------
    geterr : get the current way of handling special-function errors
    seterr : set how special-function errors are handled
    numpy.errstate : similar numpy function for floating-point errors

    Examples
    --------
    >>> import scipy.special as sc
    >>> from pytest import raises
    >>> sc.gammaln(0)
    inf
    >>> with sc.errstate(singular='raise'):
    ...     with raises(sc.SpecialFunctionError):
    ...         sc.gammaln(0)
    ...
    >>> sc.gammaln(0)
    inf

    We can also raise on every category except one.

    >>> with sc.errstate(all='raise', singular='ignore'):
    ...     sc.gammaln(0)
    ...     with raises(sc.SpecialFunctionError):
    ...         sc.spence(-1)
    ...
    inf

    Set how special-function errors are handled.

    Parameters
    ----------
    all : {'ignore', 'warn' 'raise'}, optional
        Set treatment for all type of special-function errors at
        once. The options are:

        - 'ignore' Take no action when the error occurs
        - 'warn' Print a `SpecialFunctionWarning` when the error
          occurs (via the Python `warnings` module)
        - 'raise' Raise a `SpecialFunctionError` when the error
          occurs.

        The default is to not change the current behavior. If
        behaviors for additional categories of special-function errors
        are specified, then ``all`` is applied first, followed by the
        additional categories.
    singular : {'ignore', 'warn', 'raise'}, optional
        Treatment for singularities.
    underflow : {'ignore', 'warn', 'raise'}, optional
        Treatment for underflow.
    overflow : {'ignore', 'warn', 'raise'}, optional
        Treatment for overflow.
    slow : {'ignore', 'warn', 'raise'}, optional
        Treatment for slow convergence.
    loss : {'ignore', 'warn', 'raise'}, optional
        Treatment for loss of accuracy.
    no_result : {'ignore', 'warn', 'raise'}, optional
        Treatment for failing to find a result.
    domain : {'ignore', 'warn', 'raise'}, optional
        Treatment for an invalid argument to a function.
    arg : {'ignore', 'warn', 'raise'}, optional
        Treatment for an invalid parameter to a function.
    other : {'ignore', 'warn', 'raise'}, optional
        Treatment for an unknown error.

    Returns
    -------
    olderr : dict
        Dictionary containing the old settings.

    See Also
    --------
    geterr : get the current way of handling special-function errors
    errstate : context manager for special-function error handling
    numpy.seterr : similar numpy function for floating-point errors

    Examples
    --------
    >>> import scipy.special as sc
    >>> from pytest import raises
    >>> sc.gammaln(0)
    inf
    >>> olderr = sc.seterr(singular='raise')
    >>> with raises(sc.SpecialFunctionError):
    ...     sc.gammaln(0)
    ...
    >>> _ = sc.seterr(**olderr)

    We can also raise for every category except one.

    >>> olderr = sc.seterr(all='raise', singular='ignore')
    >>> sc.gammaln(0)
    inf
    >>> with raises(sc.SpecialFunctionError):
    ...     sc.spence(-1)
    ...
    >>> _ = sc.seterr(**olderr)

    Get the current way of handling special-function errors.

    Returns
    -------
    err : dict
        A dictionary with keys "singular", "underflow", "overflow",
        "slow", "loss", "no_result", "domain", "arg", and "other",
        whose values are from the strings "ignore", "warn", and
        "raise". The keys represent possible special-function errors,
        and the values define how these errors are handled.

    See Also
    --------
    seterr : set how special-function errors are handled
    errstate : context manager for special-function error handling
    numpy.geterr : similar numpy function for floating-point errors

    Notes
    -----
    For complete documentation of the types of special-function errors
    and treatment options, see `seterr`.

    Examples
    --------
    By default all errors are ignored.

    >>> import scipy.special as sc
    >>> for key, value in sorted(sc.geterr().items()):
    ...     print("{}: {}".format(key, value))
    ...
    arg: ignore
    domain: ignore
    loss: ignore
    no_result: ignore
    other: ignore
    overflow: ignore
    singular: ignore
    slow: ignore
    underflow: ignore

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