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�@UUUUUUտ��+��D@@��\AL�,@G}g���)@��\AL�(@��������;f����;f���_�#B����;�d~�QJ�חA�
�:�<polynomial only defined for nonnegative nscipy.special._boxcox.boxcox1pscipy.special._hyp0f1._hyp0f1_realscipy.special._hyp0f1._hyp0f1_asyscipy.special._hyp0f1._hyp0f1_cmplxInterpreter change detected - this module can only be loaded into one interpreter per process.Module '_ufuncs' has already been imported. Re-initialisation is not supported.compiletime version %s of module '%.100s' does not match runtime version %sShared Cython type %.200s is not a type objectShared Cython type %.200s has the wrong size, try recompiling_export_faddeeva_dawsn_complex_export_faddeeva_erfcx_complex_export_faddeeva_voigt_profile%.200s does not export expected C function %.200svoid (sf_error_t, sf_action_t)C function %.200s.%.200s has wrong signature (expected %.500s, got %.500s)_ARRAY_API is not PyCapsule objectmodule compiled against ABI version 0x%x but this version of numpy is 0x%xmodule compiled against API version 0x%x but this version of numpy is 0x%xFATAL: module compiled as unknown endianFATAL: module compiled as little endian, but detected different endianness at runtimenumpy.core._multiarray_umath failed to import_UFUNC_API is not PyCapsule object_cosine_cdf(x)

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
-------
float
    The cosine distribution CDF evaluated at `x`._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
-------
float
    The inverse of the cosine distribution CDF evaluated at `p`.Internal function, do not use.Internal function, use `ellip_harm` instead.Internal function, use `lambertw` instead.Internal function, use `zeta` instead._smirnovc(n, d)
 Internal function, do not use._smirnovp(n, p)
 Internal function, do not use.Internal function, use `spherical_in` instead.Internal function, use `spherical_jn` instead.Internal function, use `spherical_kn` instead.Internal function, use `spherical_yn` instead._struve_asymp_large_z(v, z, is_h)

Internal function for testing `struve` & `modstruve`

Evaluates using asymptotic expansion

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

Internal function for testing `struve` & `modstruve`

Evaluates using Bessel function series

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

Internal function for testing `struve` & `modstruve`

Evaluates using power series

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

Internal function, Hurwitz zeta.agm(a, b)

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.

Returns
-------
float
    The arithmetic-geometric mean of `a` and `b`.

Examples
--------
>>> 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]])airy(z)

Airy functions and their derivatives.

Parameters
----------
z : array_like
    Real or complex argument.

Returns
-------
Ai, Aip, Bi, Bip : ndarrays
    Airy functions Ai and Bi, and their derivatives Aip and Bip.

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)

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

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].

>>> 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()airye(z)

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.

Returns
-------
eAi, eAip, eBi, eBip : array_like
    Exponentially scaled Airy functions eAi and eBi, and their derivatives
    eAip and eBip

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

See also
--------
airy

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:

>>> 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)bdtr(k, n, p)

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).

Returns
-------
y : 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/bdtrc(k, n, p)

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.

Returns
-------
y : 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/bdtri(k, n, y)

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).

Returns
-------
p : 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/bdtrik(y, n, p)

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).

Returns
-------
k : 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.bdtrin(k, y, p)

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).

Returns
-------
n : 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.bei(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 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])beip(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#PT5ber(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 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])berp(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#PT5besselpoly(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.beta(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 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)
0.25betainc(a, b, x, out=None)

Incomplete beta function.

Computes the 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`.

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
-------
array-like
    Value of the incomplete beta function

See Also
--------
beta : beta function
betaincinv : inverse of the incomplete beta function

Notes
-----
The incomplete beta function is also sometimes defined
without the `gamma` terms, in which case the above
definition is the so-called regularized incomplete beta
function. Under this definition, you can get the incomplete
beta function by multiplying the result of the SciPy
function by `beta`.

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)
0.49339638807619446betaincinv(a, b, y, out=None)

Inverse of the 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
-------
array-like
    Value of the inverse of the incomplete beta function

See Also
--------
betainc : 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)
0.5betaln(a, b)

Natural logarithm of absolute value of beta function.

Computes ``ln(abs(beta(a, b)))``.binom(n, k)

Binomial coefficient

See Also
--------
comb : The number of combinations of N things taken k at a time.boxcox(x, lmbda)

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.

Returns
-------
y : array
    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       ])boxcox1p(x, lmbda)

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.

Returns
-------
y : array
    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])btdtr(a, b, x)

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.

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].

Returns
-------
I : 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/btdtri(a, b, p)

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

Parameters
----------
a : array_like
    Shape parameter (`a` > 0).
b : array_like
    Shape parameter (`b` > 0).
p : array_like
    Cumulative probability, in [0, 1].

Returns
-------
x : 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)

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].

Returns
-------
a : 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)

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].

Returns
-------
b : 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.cbrt(x)

Element-wise cube root of `x`.

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

Returns
-------
float
    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       ])chdtr(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 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])chdtrc(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 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])chdtri(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)

Non-central chi square cumulative distribution functionchndtridf(x, p, nc)

Inverse to `chndtr` vs `df`chndtrinc(x, df, p)

Inverse to `chndtr` vs `nc`chndtrix(p, df, nc)

Inverse to `chndtr` vs `x`cosdg(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 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])cosm1(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 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-61cotdg(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 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])dawsn(x)

Dawson's integral.

Computes::

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

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

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

Examples
--------
>>> 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()ellipe(m)

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.

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

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) .

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.

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`.

>>> 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
21.991148575128552ellipeinc(phi, m)

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.

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

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) .

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.

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#iellipj(u, m)

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.

Returns
-------
sn, cn, dn, ph : ndarrays
    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)`.

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`.

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

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

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.

Returns
-------
K : array_like
    Value of the elliptic integral.

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) .

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.

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#iellipkinc(phi, m)

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

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

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) .

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.

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#iellipkm1(p)

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`.

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

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.

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.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/elliprc(x, y)

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.

Returns
-------
R : 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.

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

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.

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/BF02198293elliprd(x, y, z)

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.

Returns
-------
R : 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.

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

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.

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/BF02198293elliprf(x, y, z)

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.

Returns
-------
R : 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.

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

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.

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/BF02198293elliprg(x, y, z)

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.

Returns
-------
R : 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.

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

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.

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
--------
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).

>>> from scipy.special import elliprg
>>> 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))
108.62688289491807elliprj(x, y, z, p)

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.

Returns
-------
R : 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]_

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
constaints 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

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.

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/S0036141092228477entr(x)

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.

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

See Also
--------
kl_div, rel_entr

Notes
-----
This function is concave.

.. versionadded:: 0.15.0erf(z)

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.

Returns
-------
res : 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
--------
>>> 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()erfc(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
--------
>>> 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()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 have multi-value.
    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]

    Returns
    -------
    erfcinv : 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
    --------
    1) evaluating a float number

    >>> from scipy import special
    >>> special.erfcinv(0.5)
    0.4769362762044698

    2) evaluating an ndarray

    >>> from scipy import special
    >>> y = np.linspace(0.0, 2.0, num=11)
    >>> special.erfcinv(y)
    array([        inf,  0.9061938 ,  0.59511608,  0.37080716,  0.17914345,
           -0.        , -0.17914345, -0.37080716, -0.59511608, -0.9061938 ,
                  -inf])erfcx(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
--------
>>> 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()erfi(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
--------
>>> 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()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 have multi-value.
    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]

    Returns
    -------
    erfinv : 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
    --------
    1) evaluating a float number

    >>> from scipy import special
    >>> special.erfinv(0.5)
    0.4769362762044698

    2) evaluating an ndarray

    >>> from scipy import special
    >>> y = np.linspace(-1.0, 1.0, num=10)
    >>> special.erfinv(y)
    array([       -inf, -0.86312307, -0.5407314 , -0.30457019, -0.0987901 ,
            0.0987901 ,  0.30457019,  0.5407314 ,  0.86312307,         inf])eval_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

Returns
-------
C : 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 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_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

Returns
-------
S : 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 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_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

Returns
-------
T : 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_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

Returns
-------
U : 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_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

Returns
-------
C : 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

Returns
-------
L : 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

Returns
-------
H : 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

Returns
-------
He : 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_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

Returns
-------
P : 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_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

Returns
-------
L : 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_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

Returns
-------
P : 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
--------
>>> 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_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

Returns
-------
T : 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_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

Returns
-------
U : 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_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

Returns
-------
G : 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_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

Returns
-------
P : 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.exp1(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 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])exp10(x)

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

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

Returns
-------
float
    ``10**x``, computed element-wise.

Examples
--------
>>> 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 ]])exp2(x)

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

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

Returns
-------
float
    ``2**x``, computed element-wise.

Examples
--------
>>> 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]])expi(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 principle
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

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

.. math::

    E_1(x) = -Ei(-x)

for :math:`x > 0`.

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

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

Examples
--------
>>> 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.

>>> import scipy.special as sc
>>> 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)expit(x)

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.

Returns
-------
out : 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
--------
>>> 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()expm1(x)

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.

Returns
-------
float
    ``exp(x) - 1`` computed element-wise.

Examples
--------
>>> 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-13expn(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 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])exprel(x)

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.

Returns
-------
float
    ``(exp(x) - 1)/x``, computed element-wise.

See Also
--------
expm1

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

Examples
--------
>>> 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
0.99999999392252903fdtr(dfn, dfd, x)

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).

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

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`.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/fdtrc(dfn, dfd, x)

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).

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

See also
--------
fdtr

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`.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/fdtri(dfn, dfd, p)

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].

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

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`.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/fdtridfd(dfn, p, x)

Inverse to `fdtr` vs dfd

Finds the F density argument dfd such that ``fdtr(dfn, dfd, x) == p``.fresnel(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 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])gamma(z)

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

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
--------
>>> 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()gammainc(a, x)

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

Returns
-------
scalar or ndarray
    Values of the lower 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]_.

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

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)
0.6289066304773024gammaincc(a, x)

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

Returns
-------
scalar or ndarray
    Values of the 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]_.

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

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)
0.37109336952269756gammainccinv(a, y)

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

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.])gammaincinv(a, y)

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

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])gammaln(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 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])gammasgn(x)

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

Returns
-------
scalar or ndarray
    Sign of the gamma function

Notes
-----
The gamma function can be computed as ``gammasgn(x) *
np.exp(gammaln(x))``.

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

References
----------
.. [dlmf] NIST Digital Library of Mathematical Functions
          https://dlmf.nist.gov/5.2#E1

Examples
--------
>>> 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 ])gdtr(a, b, x)

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).

See also
--------
gdtrc : 1 - CDF of the gamma distribution.

Returns
-------
F : ndarray
    The CDF of the gamma distribution with parameters `a` and `b`
    evaluated at `x`.

Notes
-----
The evaluation is carried out using the relation to the incomplete gamma
integral (regularized gamma function).

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

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/gdtrc(a, b, x)

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).

Returns
-------
F : ndarray
    The survival function of the gamma distribution with parameters `a`
    and `b` evaluated at `x`.

See Also
--------
gdtr, 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`.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/gdtria(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 : 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 : 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 : 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)
5.5999999999999996hankel1(v, z)

Hankel function of the first kind

Parameters
----------
v : array_like
    Order (float).
z : array_like
    Argument (float or complex).

Returns
-------
out : Values of the Hankel function of the first kind.

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.

See also
--------
hankel1e : this function with leading exponential behavior stripped off.

References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
       of a Complex Argument and Nonnegative Order",
       http://netlib.org/amos/hankel1e(v, z)

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).

Returns
-------
out : 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/hankel2(v, z)

Hankel function of the second kind

Parameters
----------
v : array_like
    Order (float).
z : array_like
    Argument (float or complex).

Returns
-------
out : Values of the 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(\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.

See also
--------
hankel2e : this function with leading exponential behavior stripped off.

References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
       of a Complex Argument and Nonnegative Order",
       http://netlib.org/amos/hankel2e(v, z)

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).

Returns
-------
out : 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/huber(delta, r)

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.

Returns
-------
res : ndarray
    The computed Huber loss function values.

Notes
-----
This function is convex in r.

.. versionadded:: 0.15.0hyp0f1(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 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])hyp1f1(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 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)

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.

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 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
0.9272952180016123hyperu(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 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 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)
0.1926947246463881i0(x)

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)

Returns
-------
I : ndarray
    Value of the modified Bessel function of order 0 at `x`.

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`.

See also
--------
iv
i0e

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/i0e(x)

Exponentially scaled modified Bessel function of order 0.

Defined as::

    i0e(x) = exp(-abs(x)) * i0(x).

Parameters
----------
x : array_like
    Argument (float)

Returns
-------
I : ndarray
    Value of the exponentially scaled modified Bessel function of order 0
    at `x`.

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`.

See also
--------
iv
i0

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/i1(x)

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)

Returns
-------
I : ndarray
    Value of the modified Bessel function of order 1 at `x`.

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`.

See also
--------
iv
i1e

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/i1e(x)

Exponentially scaled modified Bessel function of order 1.

Defined as::

    i1e(x) = exp(-abs(x)) * i1(x)

Parameters
----------
x : array_like
    Argument (float)

Returns
-------
I : ndarray
    Value of the exponentially scaled modified Bessel function of order 1
    at `x`.

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`.

See also
--------
iv
i1

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/inv_boxcox(y, lmbda)

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.

Returns
-------
x : array
    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_boxcox1p(y, lmbda)

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.

Returns
-------
x : array
    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.])it2i0k0(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`it2j0y0(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`it2struve0(x)

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.

Returns
-------
I : 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.htmlitairy(x)

Integrals of Airy functions

Calculates the integrals of Airy functions from 0 to `x`.

Parameters
----------

x: array_like
    Upper limit of integration (float).

Returns
-------
Apt
    Integral of Ai(t) from 0 to x.
Bpt
    Integral of Bi(t) from 0 to x.
Ant
    Integral of Ai(-t) from 0 to x.
Bnt
    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.htmliti0k0(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`itj0y0(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`itmodstruve0(x)

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).

Returns
-------
I : ndarray
    The integral of :math:`L_0` 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.htmlitstruve0(x)

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).

Returns
-------
I : ndarray
    The integral of :math:`H_0` from 0 to `x`.

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.htmliv(v, z)

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.

Returns
-------
out : ndarray
    Values of the modified Bessel function.

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`.

See also
--------
kve : This function with leading exponential behavior stripped off.

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/ive(v, z)

Exponentially scaled modified Bessel function of the first kind

Defined as::

    ive(v, z) = iv(v, z) * exp(-abs(z.real))

Parameters
----------
v : array_like of float
    Order.
z : array_like of float or complex
    Argument.

Returns
-------
out : ndarray
    Values of the exponentially scaled modified Bessel function.

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`.

References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
       of a Complex Argument and Nonnegative Order",
       http://netlib.org/amos/j0(x)

Bessel function of the first kind of order 0.

Parameters
----------
x : array_like
    Argument (float).

Returns
-------
J : ndarray
    Value of the Bessel function of the first kind of order 0 at `x`.

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`).

See also
--------
jv : Bessel function of real order and complex argument.
spherical_jn : spherical Bessel functions.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/j1(x)

Bessel function of the first kind of order 1.

Parameters
----------
x : array_like
    Argument (float).

Returns
-------
J : ndarray
    Value of the Bessel function of the first kind of order 1 at `x`.

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`).

See also
--------
jv
spherical_jn : spherical Bessel functions.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/jv(v, z)

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).

Returns
-------
J : ndarray
    Value of the Bessel function, :math:`J_v(z)`.

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`).

See also
--------
jve : :math:`J_v` with leading exponential behavior stripped off.
spherical_jn : spherical Bessel functions.

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

Exponentially scaled Bessel function 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).

Returns
-------
J : ndarray
    Value of the exponentially scaled Bessel function.

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)`.

References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
       of a Complex Argument and Nonnegative Order",
       http://netlib.org/amos/k0(x)

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).

Returns
-------
K : ndarray
    Value of the modified Bessel function :math:`K_0` at `x`.

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`.

See also
--------
kv
k0e

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/k0e(x)

Exponentially scaled modified Bessel function K of order 0

Defined as::

    k0e(x) = exp(x) * k0(x).

Parameters
----------
x : array_like
    Argument (float)

Returns
-------
K : ndarray
    Value of the exponentially scaled modified Bessel function K of order
    0 at `x`.

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`.

See also
--------
kv
k0

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/k1(x)

Modified Bessel function of the second kind of order 1, :math:`K_1(x)`.

Parameters
----------
x : array_like
    Argument (float)

Returns
-------
K : ndarray
    Value of the modified Bessel function K of order 1 at `x`.

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`.

See also
--------
kv
k1e

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/k1e(x)

Exponentially scaled modified Bessel function K of order 1

Defined as::

    k1e(x) = exp(x) * k1(x)

Parameters
----------
x : array_like
    Argument (float)

Returns
-------
K : ndarray
    Value of the exponentially scaled modified Bessel function K of order
    1 at `x`.

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`.

See also
--------
kv
k1

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/kei(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 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 ])keip(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#PT5kelvin(x)

Kelvin functions as complex numbers

Returns
-------
Be, Ke, Bep, Kep
    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.ker(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.

See Also
--------
kei : the corresponding imaginary part
kerp : the derivative of ker
kv : modified Bessel function of the second kind

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

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 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])kerp(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_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

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 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] Grant, Boyd, and Ye, "CVX: Matlab Software for Disciplined Convex
    Programming", http://cvxr.com/cvx/kn(n, x)

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)
z : array_like of float
    Argument at which to evaluate the Bessel functions

Returns
-------
out : ndarray
    The results

Notes
-----
Wrapper for AMOS [1]_ routine `zbesk`.  For a discussion of the
algorithm used, see [2]_ and the references therein.

See Also
--------
kv : Same function, but accepts real order and complex argument
kvp : Derivative of this function

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:

>>> 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])kolmogi(p)

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

Returns
-------
float
    The value(s) of kolmogi(p)

Notes
-----
`kolmogorov` is used by `stats.kstest` in the application of the
Kolmogorov-Smirnov Goodness of Fit test. For historial 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.

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

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.        ])kolmogorov(y)

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).

Returns
-------
float
    The value(s) of kolmogorov(y)

Notes
-----
`kolmogorov` is used by `stats.kstest` in the application of the
Kolmogorov-Smirnov Goodness of Fit test. For historial 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.

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

Examples
--------
Show the probability of a gap at least as big as 0, 0.5 and 1.0.

>>> 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()kv(v, z)

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

Returns
-------
out : 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)``.

Notes
-----
Wrapper for AMOS [1]_ routine `zbesk`.  For a discussion of the
algorithm used, see [2]_ and the references therein.

See Also
--------
kve : This function with leading exponential behavior stripped off.
kvp : Derivative of this function

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:

>>> 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])kve(v, z)

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

Returns
-------
out : ndarray
    The exponentially scaled modified Bessel function of the second kind.

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. 265log1p(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 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_expit(x)

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.

Returns
-------
out : 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
--------
>>> 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_ndtr(x)

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

Returns
-------
ndarray
    The value of the log of the normal CDF evaluated at `x`

See Also
--------
erf
erfc
scipy.stats.norm
ndtrloggamma(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 plain at which to compute ``loggamma``
out : ndarray, optional
    Output array for computed values of ``loggamma``

Returns
-------
loggamma : ndarray
    Values of ``loggamma`` at z.

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]_.

See also
--------
gammaln : logarithm of the absolute value of the gamma function
gammasgn : sign of the gamma function

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.logit(x)

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.

Returns
-------
out : 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
--------
>>> 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()lpmv(m, v, x)

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``.

Returns
-------
pmv : 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_a(m, q)

Characteristic value of even Mathieu functions

Returns the characteristic value for the even solution,
``ce_m(z, q)``, of Mathieu's equation.mathieu_b(m, q)

Characteristic value of odd Mathieu functions

Returns the characteristic value for the odd solution,
``se_m(z, q)``, of Mathieu's equation.mathieu_cem(m, q, x)

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
    Order of the function
q
    Parameter of the function
x
    Argument of the function, *given in degrees, not radians*

Returns
-------
y
    Value of the function
yp
    Value of the derivative vs xmathieu_modcem1(m, q, x)

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`.

Returns
-------
y
    Value of the function
yp
    Value of the derivative vs xmathieu_modcem2(m, q, x)

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`.

Returns
-------
y
    Value of the function
yp
    Value of the derivative vs xmathieu_modsem1(m, q, x)

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`.

Returns
-------
y
    Value of the function
yp
    Value of the derivative vs xmathieu_modsem2(m, q, x)

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.

Returns
-------
y
    Value of the function
yp
    Value of the derivative vs xmathieu_sem(m, q, x)

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
    Order of the function
q
    Parameter of the function
x
    Argument of the function, *given in degrees, not radians*.

Returns
-------
y
    Value of the function
yp
    Value of the derivative vs xmodfresnelm(x)

Modified Fresnel negative integrals

Returns
-------
fm
    Integral ``F_-(x)``: ``integral(exp(-1j*t*t), t=x..inf)``
km
    Integral ``K_-(x)``: ``1/sqrt(pi)*exp(1j*(x*x+pi/4))*fp``modfresnelp(x)

Modified Fresnel positive integrals

Returns
-------
fp
    Integral ``F_+(x)``: ``integral(exp(1j*t*t), t=x..inf)``
kp
    Integral ``K_+(x)``: ``1/sqrt(pi)*exp(-1j*(x*x+pi/4))*fp``modstruve(v, x)

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).

Returns
-------
L : ndarray
    Value of the modified Struve function of order `v` at `x`.

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.

See also
--------
struve

References
----------
.. [1] NIST Digital Library of Mathematical Functions
       https://dlmf.nist.gov/11nbdtr(k, n, p)

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).

Returns
-------
F : ndarray
    The probability of `k` or fewer failures before `n` successes in a
    sequence of events with individual success probability `p`.

See also
--------
nbdtrc

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`.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/nbdtrc(k, n, p)

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).

Returns
-------
F : ndarray
    The probability of `k + 1` or more failures before `n` successes in a
    sequence of events with individual success probability `p`.

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`.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/nbdtri(k, n, y)

Inverse of `nbdtr` vs `p`.

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).

Returns
-------
p : 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.
nbdtrik : Inverse with respect to `k` of `nbdtr(k, n, p)`.
nbdtrin : Inverse with respect to `n` of `nbdtr(k, n, p)`.

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

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/nbdtrik(y, n, p)

Inverse of `nbdtr` vs `k`.

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).

Returns
-------
k : ndarray
    The maximum number of allowed failures 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)`.
nbdtrin : Inverse with respect to `n` 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 `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.nbdtrin(k, y, p)

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).

Returns
-------
n : 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.ncfdtr(dfn, dfd, nc, f)

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.

Returns
-------
cdf : float 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
--------
>>> 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)

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].

Returns
-------
f : float
    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)

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.

Returns
-------
dfd : float
    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)

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.

Returns
-------
dfn : float
    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)

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.

Returns
-------
nc : float
    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)

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.

Returns
-------
cdf : float 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
--------
>>> 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)

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.nctdtrinc(df, p, t)

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.nctdtrit(df, nc, p)

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].ndtr(x)

Gaussian cumulative distribution function.

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

Returns
-------
ndarray
    The value of the normal CDF evaluated at `x`

See Also
--------
erf
erfc
scipy.stats.norm
log_ndtrndtri(y)

Inverse of `ndtr` vs x

Returns the argument x for which the area under the Gaussian
probability density function (integrated from minus infinity to `x`)
is equal to y.ndtri_exp(y)

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

Returns
-------
scalar or ndarray
    Inverse of the log CDF of the standard normal distribution, evaluated
    at y.

Examples
--------
>>> 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)
9.262340089798409

See Also
--------
log_ndtr, ndtri, ndtrnrdtrimn(p, x, std)

Calculate mean of normal distribution given other params.

Parameters
----------
p : array_like
    CDF values, in range (0, 1].
x : array_like
    Quantiles, i.e. the upper limit of integration.
std : array_like
    Standard deviation.

Returns
-------
mn : float or ndarray
    The mean of the normal distribution.

See Also
--------
nrdtrimn, ndtrnrdtrisd(p, x, mn)

Calculate standard deviation of normal distribution given other params.

Parameters
----------
p : array_like
    CDF values, in range (0, 1].
x : array_like
    Quantiles, i.e. the upper limit of integration.
mn : float or ndarray
    The mean of the normal distribution.

Returns
-------
std : array_like
    Standard deviation.

See Also
--------
ndtrobl_ang1(m, n, c, x)

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``.

Returns
-------
s
    Value of the function
sp
    Value of the derivative vs xobl_ang1_cv(m, n, c, cv, x)

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.

Returns
-------
s
    Value of the function
sp
    Value of the derivative vs xobl_cv(m, n, c)

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`.obl_rad1(m, n, c, x)

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``.

Returns
-------
s
    Value of the function
sp
    Value of the derivative vs xobl_rad1_cv(m, n, c, cv, x)

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.

Returns
-------
s
    Value of the function
sp
    Value of the derivative vs xobl_rad2(m, n, c, x)

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``.

Returns
-------
s
    Value of the function
sp
    Value of the derivative vs xobl_rad2_cv(m, n, c, cv, x)

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.

Returns
-------
s
    Value of the function
sp
    Value of the derivative vs xowens_t(h, a)

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.

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.

Examples
--------
>>> from scipy import special
>>> a = 3.5
>>> h = 0.78
>>> special.owens_t(h, a)
0.10877216734852274

References
----------
.. [1] M. Patefield and D. Tandy, "Fast and accurate calculation of
       Owen's T Function", Statistical Software vol. 5, pp. 1-25, 2000.pbdv(v, x)

Parabolic cylinder function D

Returns (d, dp) the parabolic cylinder function Dv(x) in d and the
derivative, Dv'(x) in dp.

Returns
-------
d
    Value of the function
dp
    Value of the derivative vs xpbvv(v, x)

Parabolic cylinder function V

Returns the parabolic cylinder function Vv(x) in v and the
derivative, Vv'(x) in vp.

Returns
-------
v
    Value of the function
vp
    Value of the derivative vs xpbwa(a, x)

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

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.htmlpdtr(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
    Nonnegative real argument
m : array_like
    Nonnegative real shape parameter
out : ndarray
    Optional output array for the function results

See Also
--------
pdtrc : Poisson survival function
pdtrik : inverse of `pdtr` with respect to `k`
pdtri : inverse of `pdtr` with respect to `m`

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

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

Examples
--------
>>> 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 ])pdtrc(k, m)

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.pdtri(k, y)

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.pdtrik(p, m)

Inverse to `pdtr` vs k

Returns the quantile k such that ``pdtr(k, m) = p``poch(z, m)

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.

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)
20.52958193377696pro_ang1(m, n, c, x)

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``.

Returns
-------
s
    Value of the function
sp
    Value of the derivative vs xpro_ang1_cv(m, n, c, cv, x)

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.

Returns
-------
s
    Value of the function
sp
    Value of the derivative vs xpro_cv(m, n, c)

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`.pro_rad1(m, n, c, x)

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``.

Returns
-------
s
    Value of the function
sp
    Value of the derivative vs xpro_rad1_cv(m, n, c, cv, x)

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.

Returns
-------
s
    Value of the function
sp
    Value of the derivative vs xpro_rad2(m, n, c, x)

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``.

Returns
-------
s
    Value of the function
sp
    Value of the derivative vs xpro_rad2_cv(m, n, c, cv, x)

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.

Returns
-------
s
    Value of the function
sp
    Value of the derivative vs xpseudo_huber(delta, r)

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 : ndarray
    Input array, indicating the soft quadratic vs. linear loss changepoint.
r : ndarray
    Input array, possibly representing residuals.

Returns
-------
res : ndarray
    The computed Pseudo-Huber loss function values.

Notes
-----
This function is convex in :math:`r`.

.. versionadded:: 0.15.0psi(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 : 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)radian(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_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

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`, to get the relative
entropy of statistics compute the sum

.. math::

    \sum_{i = 1}^n \mathrm{rel\_entr}(p_i, q_i).

See [2]_ for details.

References
----------
.. [1] Grant, Boyd, and Ye, "CVX: Matlab Software for Disciplined Convex
    Programming", http://cvxr.com/cvx/
.. [2] Kullback-Leibler divergence,
    https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergencergamma(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

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.

See Also
--------
gamma, gammaln, loggamma

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])round(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
principle branch of the logarithm.

Parameters
----------
x : array_like
    Real or complex points at which to compute the hyperbolic sine
    and cosine integrals.

Returns
-------
si : ndarray
    Hyperbolic sine integral at ``x``
ci : ndarray
    Hyperbolic cosine integral at ``x``

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'
[1]_ *shichi* routine. For complex arguments the algorithm is based
on Mpmath's [2]_ *shi* and *chi* routines.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/
.. [2] Fredrik Johansson and others.
       "mpmath: a Python library for arbitrary-precision floating-point arithmetic"
       (Version 0.19) http://mpmath.org/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
principle branch of the logarithm.

Parameters
----------
x : array_like
    Real or complex points at which to compute the sine and cosine
    integrals.

Returns
-------
si : ndarray
    Sine integral at ``x``
ci : ndarray
    Cosine integral at ``x``

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'
[1]_ *sici* routine. For complex arguments the algorithm is based
on Mpmath's [2]_ *si* and *ci* routines.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/
.. [2] Fredrik Johansson and others.
       "mpmath: a Python library for arbitrary-precision floating-point arithmetic"
       (Version 0.19) http://mpmath.org/sindg(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 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])smirnov(n, d)

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.

Returns
-------
float
    The value(s) of smirnov(n, d), Prob(Dn+ >= d) (Also Prob(Dn- >= d))

Notes
-----
`smirnov` is used by `stats.kstest` in the application of the
Kolmogorov-Smirnov Goodness of Fit test. For historial 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.

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

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

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 drawn from a source N(0.5, 1) distribution against
a target N(0, 1) CDF.

>>> from scipy.stats import norm
>>> rng = np.random.default_rng()
>>> n = 5
>>> gendist = norm(0.5, 1)       # Normal distribution, mean 0.5, stddev 1
>>> x = np.sort(gendist.rvs(size=n, random_state=rng))
>>> x
array([-1.3922078 , -0.13526532,  0.1371477 ,  0.18981686,  1.81948167])
>>> target = norm(0, 1)
>>> cdfs = target.cdf(x)
>>> cdfs
array([0.08192974, 0.44620105, 0.55454297, 0.57527368, 0.96558101])
# Construct the Empirical CDF and the K-S statistics (Dn+, Dn-, Dn)
>>> ecdfs = np.arange(n+1, dtype=float)/n
>>> cols = np.column_stack([x, ecdfs[1:], cdfs, cdfs - ecdfs[:n], ecdfs[1:] - cdfs])
>>> np.set_printoptions(precision=3)
>>> cols
array([[-1.392,  0.2  ,  0.082,  0.082,  0.118],
       [-0.135,  0.4  ,  0.446,  0.246, -0.046],
       [ 0.137,  0.6  ,  0.555,  0.155,  0.045],
       [ 0.19 ,  0.8  ,  0.575, -0.025,  0.225],
       [ 1.819,  1.   ,  0.966,  0.166,  0.034]])
>>> gaps = cols[:, -2:]
>>> Dnpm = np.max(gaps, axis=0)
>>> print('Dn-=%f, Dn+=%f' % (Dnpm[0], Dnpm[1]))
Dn-=0.246201, Dn+=0.224726
>>> probs = smirnov(n, Dnpm)
>>> print(chr(10).join(['For a sample of size %d drawn from a N(0, 1) distribution:' % n,
...      ' Smirnov n=%d: Prob(Dn- >= %f) = %.4f' % (n, Dnpm[0], probs[0]),
...      ' Smirnov n=%d: Prob(Dn+ >= %f) = %.4f' % (n, Dnpm[1], probs[1])]))
For a sample of size 5 drawn from a N(0, 1) distribution:
 Smirnov n=5: Prob(Dn- >= 0.246201) = 0.4713
 Smirnov n=5: Prob(Dn+ >= 0.224726) = 0.5243

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='m', linestyle='dashed', lw=4)
>>> plt.show()smirnovi(n, p)

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

Returns
-------
float
    The value(s) of smirnovi(n, p), the critical values.

Notes
-----
`smirnov` is used by `stats.kstest` in the application of the
Kolmogorov-Smirnov Goodness of Fit test. For historial 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.

See Also
--------
smirnov  : The Survival Function (SF) for the distribution
scipy.stats.ksone : Provides the functionality as a continuous distribution
kolmogorov, kolmogi, scipy.stats.kstwobign : Functions for the two-sided distributionspence(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

Returns
-------
s : 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)``.sph_harm(m, n, theta, phi)

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]``.

Returns
-------
y_mn : complex float
   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)

Student t distribution cumulative distribution function

Returns the integral from minus infinity to t of the Student t
distribution with df > 0 degrees of freedom::

   gamma((df+1)/2)/(sqrt(df*pi)*gamma(df/2)) *
   integral((1+x**2/df)**(-df/2-1/2), x=-inf..t)stdtridf(p, t)

Inverse of `stdtr` vs df

Returns the argument df such that stdtr(df, t) is equal to `p`.stdtrit(df, p)

Inverse of `stdtr` vs `t`

Returns the argument `t` such that stdtr(df, t) is equal to `p`.struve(v, x)

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).

Returns
-------
H : ndarray
    Value of the Struve function of order `v` at `x`.

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.

See also
--------
modstruve

References
----------
.. [1] NIST Digital Library of Mathematical Functions
       https://dlmf.nist.gov/11tandg(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 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])tklmbda(x, lmbda)

Tukey-Lambda cumulative distribution functionvoigt_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

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.

See Also
--------
wofz : Faddeeva function

References
----------
.. [1] https://en.wikipedia.org/wiki/Voigt_profilewofz(z)

Faddeeva function

Returns the value of the Faddeeva function for complex argument::

    exp(-z**2) * erfc(-i*z)

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

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

Examples
--------
>>> 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)

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

Notes
-----
Due to the compexity of the function with its three parameters, only
non-negative arguments are implemented.

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)
4.5314465939443025

References
----------
.. [1] Digital Library of Mathematical Functions, 10.46.
       https://dlmf.nist.gov/10.46.E1wrightomega(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

Returns
-------
omega : ndarray
    Values of the Wright Omega 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]_.

See Also
--------
lambertw : The Lambert W function

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`.xlog1py(x, y)

Compute ``x*log1p(y)`` so that the result is 0 if ``x = 0``.

Parameters
----------
x : array_like
    Multiplier
y : array_like
    Argument

Returns
-------
z : array_like
    Computed x*log1p(y)

Notes
-----

.. versionadded:: 0.13.0xlogy(x, y)

Compute ``x*log(y)`` so that the result is 0 if ``x = 0``.

Parameters
----------
x : array_like
    Multiplier
y : array_like
    Argument

Returns
-------
z : array_like
    Computed x*log(y)

Notes
-----

.. versionadded:: 0.13.0y0(x)

Bessel function of the second kind of order 0.

Parameters
----------
x : array_like
    Argument (float).

Returns
-------
Y : ndarray
    Value of the Bessel function of the second kind of order 0 at `x`.

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`.

See also
--------
j0
yv

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/y1(x)

Bessel function of the second kind of order 1.

Parameters
----------
x : array_like
    Argument (float).

Returns
-------
Y : ndarray
    Value of the Bessel function of the second kind of order 1 at `x`.

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`.

See also
--------
j1
yn
yv

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/yn(n, x)

Bessel function of the second kind of integer order and real argument.

Parameters
----------
n : array_like
    Order (integer).
z : array_like
    Argument (float).

Returns
-------
Y : ndarray
    Value of the Bessel function, :math:`Y_n(x)`.

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.

See also
--------
yv : For real order and real or complex argument.

References
----------
.. [1] Cephes Mathematical Functions Library,
       http://www.netlib.org/cephes/yv(v, z)

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).

Returns
-------
Y : ndarray
    Value of the Bessel function of the second kind, :math:`Y_v(x)`.

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)`.

See also
--------
yve : :math:`Y_v` with leading exponential behavior stripped off.

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

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).

Returns
-------
Y : ndarray
    Value of the exponentially scaled Bessel function.

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/zetac(x)

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).

Returns
-------
out : array_like
    Values of zeta(x) - 1.

See Also
--------
zeta

Examples
--------
>>> 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)

References
----------
.. [dlmf] NIST Digital Library of Mathematical Functions
          https://dlmf.nist.gov/25eval_hermitenormeval_hermitefloat divisionscipy.special._boxcox.boxcoxscipy.special._exprel.exprelsicishichiscipy.special._cunity.clog1ploggammahyperuhypUdigammawright_bessel__loader__loader__file__origin__package__parent__path__submodule_search_locations%d.%d_cython_0_29_28builtinscython_runtime__builtins__numpydtypeflatiterbroadcastndarraygenericnumberunsignedintegerinexactcomplexfloatingflexiblecharacterscipy.special._ufuncs_cxx_export_faddeeva_dawsn_export_fellint_RC_export_cellint_RC_export_fellint_RD_export_cellint_RD_export_fellint_RF_export_cellint_RF_export_fellint_RG_export_cellint_RG_export_fellint_RJ_export_cellint_RJ_export_faddeeva_erf_export_faddeeva_erfc_export_faddeeva_erfcx_export_faddeeva_erfi_export_faddeeva_erfi_complex_export_expit_export_expitf_export_expitl_export_log_expit_export_log_expitf_export_log_expitl_export_faddeeva_log_ndtr_export_logit_export_logitf_export_logitl_export_faddeeva_ndtr_export_faddeeva_w_export_wrightomega_export_wrightomega_real__pyx_capi___set_actionnumpy.core._multiarray_umath_ARRAY_API_ARRAY_API not found_ARRAY_API is NULL pointernumpy.import_array_UFUNC_API_UFUNC_API not found_UFUNC_API is NULL pointernumpy.import_ufunc_cosine_cdf_cosine_invcdf_cospi_ellip_harm_factorial_igam_fac_kolmogc_kolmogci_kolmogp_lambertw_lanczos_sum_expg_scaled_lgam1p_log1pmx_riemann_zeta_sf_error_test_functionPrivate function; do not use._sinpi_smirnovc_smirnovci_smirnovp_spherical_in_spherical_in_d_spherical_jn_spherical_jn_d_spherical_kn_spherical_kn_d_spherical_yn_spherical_yn_d_struve_asymp_large_z_struve_bessel_series_struve_power_seriesagmairyebeibeipberpbesselpolybetaincbetaincinvbetalnbinombtdtrbtdtribtdtriabtdtribcbrtchdtrchdtrcchdtrichdtrivchndtrchndtridfchndtrincchndtrixcosdgcosm1cotdgellipeellipeincellipjellipkellipkincellipkm1elliprcelliprdelliprfelliprgelliprjerfcinverfinveval_chebyceval_chebyseval_chebyteval_chebyueval_gegenbauereval_jacobieval_laguerreeval_legendreeval_sh_chebyteval_sh_chebyueval_sh_jacobieval_sh_legendreexp10exp2expm1expnfdtrcfresnelgammaincgammainccgammainccinvgammaincinvgammalngammasgngdtrgdtrcgdtriagdtribgdtrixhankel1hankel1ehankel2hankel2ehyp0f1i0i0ei1i1einv_boxcoxinv_boxcox1pit2i0k0it2j0y0it2struve0itairyiti0k0itj0y0itmodstruve0itstruve0j0j1jvjvek0ek1k1ekeikeipkelvinkerkerpkl_divkolmogikolmogorovkvkvelpmvmathieu_amathieu_bmathieu_cemmathieu_modcem1mathieu_modcem2mathieu_modsem1mathieu_modsem2mathieu_semmodfresnelmmodfresnelpmodstruvenbdtrnbdtrcnbdtrinbdtriknbdtrinncfdtrncfdtrincfdtridfdncfdtridfnncfdtrincnctdtrnctdtridfnctdtrincnctdtritndtrindtri_expnrdtrimnnrdtrisdobl_ang1obl_ang1_cvobl_cvobl_rad1obl_rad1_cvobl_rad2obl_rad2_cvowens_tpbdvpbvvpbwapdtrpdtrcpdtripdtrikpochpro_ang1pro_ang1_cvpro_cvpro_rad1pro_rad1_cvpro_rad2pro_rad2_cvpseudo_huberpsiradianrel_entrrgammaroundsindgsmirnovsmirnovispencestdtrstdtridfstdtrittandgtklmbdavoigt_profilewofzxlog1pyxlogyy1yvyvezetacinit scipy.special._ufuncsfunc_doc__doc__func_name__name____qualname____self__func_dict__dict__func_globals__globals__func_closure__closure__func_code__code__func_defaults__defaults____kwdefaults____annotations____module____reduce__cython_function_or_method__enter__zetac_zetayveyvyny1y0xlogyxlog1pywrightomegawright_besselwofzwarnvoigt_profilevalueunderflow_ufuncs_extra_code.pxitracebacktklmbda__test__tandg_struve_power_series_struve_bessel_series_struve_asymp_large_zstruvestdtritstdtridfstdtr_spherical_yn_d_spherical_yn_spherical_kn_d_spherical_kn_spherical_jn_d_spherical_jn_spherical_in_d_spherical_insph_harmspence_smirnovpsmirnovi_smirnovci_smirnovcsmirnovslow_sinpisingularsindgsicishichi_sf_error_test_function_sf_error_code_map_sf_error_action_mapseterr (line 77)seterrselfscipy.special._ufuncsround_riemann_zetargammarel_entrrangeraiseradian__qualname__psipseudo_huberpro_rad2_cvpro_rad2pro_rad1_cvpro_rad1pro_cvpro_ang1_cvpro_ang1__prepare__poppochpdtrikpdtripdtrcpdtrpbwapbvvpbdvowens_toverflowotheroldstateolderrobl_rad2_cvobl_rad2obl_rad1_cvobl_rad1obl_cvobl_ang1_cvobl_ang1numpy.core.umath failed to importnumpy.core.multiarray failed to importnumpynrdtrisdnrdtrimnnpno_resultnewkwargsndtri_expndtrindtrnctdtritnctdtrincnctdtridfnctdtrncfdtrincncfdtridfnncfdtridfdncfdtrincfdtrnbdtrinnbdtriknbdtrinbdtrcnbdtr__name____module__modstruvemodfresnelpmodfresnelm__metaclass__mathieu_semmathieu_modsem2mathieu_modsem1mathieu_modcem2mathieu_modcem1mathieu_cemmathieu_bmathieu_a__main__lpmvlosslogitloggammalog_ndtrlog_expit_log1pmxlog1p_lgam1p_lanczos_sum_expg_scaled_lambertwkwargskvekv_kolmogpkolmogorovkolmogi_kolmogci_kolmogcknkl_divkeyskeykerpkerkelvinkeipkeik1ek1k0ek0jvejvjnj1j0iveivitstruve0itmodstruve0itj0y0iti0k0itemsitairyit2struve0it2j0y0it2i0k0inv_boxcox1pinv_boxcox__init____import__ignore_igam_faci1ei1i0ei0hyperuhyp2f1hyp1f1hyp0f1huberhankel2ehankel2hankel1ehankel1geterr (line 28)geterrgdtrixgdtribgdtriagdtrcgdtrgammasgngammalngammaincinvgammainccinvgammainccgammaincgammafresnelfdtridfdfdtrifdtrcfdtr_factorialexprelexpnexpm1expitexpiexp2exp10exp1__exit__exc_valueexc_typeeval_sh_legendreeval_sh_jacobieval_sh_chebyueval_sh_chebyteval_legendreeval_laguerreeval_jacobieval_hermitenormeval_hermiteeval_genlaguerreeval_gegenbauereval_chebyueval_chebyteval_chebyseval_chebycerrstate.__init__errstate.__exit__errstate.__enter__errstateerrorerrerfinverfierfcxerfcinverfcerfentr__enter__elliprjelliprgelliprfelliprdelliprcellipkm1ellipkincellipkellipjellipeincellipe_ellip_harmdomain__doc__dawsncotdg_cospicosm1_cosine_invcdf_cosine_cdfcosdgcodecline_in_tracebackchndtrixchndtrincchndtridfchndtrchdtrivchdtrichdtrcchdtrcbrtbtdtribbtdtriabtdtribtdtrboxcox1pboxcoxbinombetalnbetaincinvbetaincbetabesselpolyberpberbeipbeibdtrinbdtrikbdtribdtrcbdtrarg__all__allairyeairyagmactionSet 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)

    RuntimeWarningImportErrorGet 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

    DeprecationWarningContext 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 treatement 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

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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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Youez - 2016 - github.com/yon3zu
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