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""" |
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======================================== |
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Special functions (:mod:`scipy.special`) |
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======================================== |
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.. currentmodule:: scipy.special |
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Almost all of the functions below accept NumPy arrays as input |
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arguments as well as single numbers. This means they follow |
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broadcasting and automatic array-looping rules. Technically, |
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they are `NumPy universal functions |
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<https://numpy.org/doc/stable/user/basics.ufuncs.html#ufuncs-basics>`_. |
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Functions which do not accept NumPy arrays are marked by a warning |
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in the section description. |
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.. seealso:: |
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`scipy.special.cython_special` -- Typed Cython versions of special functions |
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Error handling |
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============== |
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Errors are handled by returning NaNs or other appropriate values. |
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Some of the special function routines can emit warnings or raise |
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exceptions when an error occurs. By default this is disabled, except |
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for memory allocation errors, which result in an exception being raised. |
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To query and control the current error handling state the following |
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functions are provided. |
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.. autosummary:: |
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:toctree: generated/ |
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geterr -- Get the current way of handling special-function errors. |
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seterr -- Set how special-function errors are handled. |
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errstate -- Context manager for special-function error handling. |
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SpecialFunctionWarning -- Warning that can be emitted by special functions. |
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SpecialFunctionError -- Exception that can be raised by special functions. |
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Available functions |
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=================== |
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Airy functions |
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-------------- |
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.. autosummary:: |
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:toctree: generated/ |
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airy -- Airy functions and their derivatives. |
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airye -- Exponentially scaled Airy functions and their derivatives. |
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ai_zeros -- Compute `nt` zeros and values of the Airy function Ai and its derivative. |
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bi_zeros -- Compute `nt` zeros and values of the Airy function Bi and its derivative. |
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itairy -- Integrals of Airy functions |
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Elliptic functions and integrals |
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-------------------------------- |
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.. autosummary:: |
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:toctree: generated/ |
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ellipj -- Jacobian elliptic functions. |
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ellipk -- Complete elliptic integral of the first kind. |
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ellipkm1 -- Complete elliptic integral of the first kind around `m` = 1. |
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ellipkinc -- Incomplete elliptic integral of the first kind. |
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ellipe -- Complete elliptic integral of the second kind. |
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ellipeinc -- Incomplete elliptic integral of the second kind. |
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elliprc -- Degenerate symmetric integral RC. |
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elliprd -- Symmetric elliptic integral of the second kind. |
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elliprf -- Completely-symmetric elliptic integral of the first kind. |
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elliprg -- Completely-symmetric elliptic integral of the second kind. |
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elliprj -- Symmetric elliptic integral of the third kind. |
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Bessel functions |
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---------------- |
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.. autosummary:: |
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:toctree: generated/ |
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jv -- Bessel function of the first kind of real order and \ |
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complex argument. |
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jve -- Exponentially scaled Bessel function of order `v`. |
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yn -- Bessel function of the second kind of integer order and \ |
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real argument. |
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yv -- Bessel function of the second kind of real order and \ |
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complex argument. |
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yve -- Exponentially scaled Bessel function of the second kind \ |
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of real order. |
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kn -- Modified Bessel function of the second kind of integer \ |
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order `n` |
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kv -- Modified Bessel function of the second kind of real order \ |
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`v` |
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kve -- Exponentially scaled modified Bessel function of the \ |
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second kind. |
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iv -- Modified Bessel function of the first kind of real order. |
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ive -- Exponentially scaled modified Bessel function of the \ |
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first kind. |
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hankel1 -- Hankel function of the first kind. |
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hankel1e -- Exponentially scaled Hankel function of the first kind. |
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hankel2 -- Hankel function of the second kind. |
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hankel2e -- Exponentially scaled Hankel function of the second kind. |
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wright_bessel -- Wright's generalized Bessel function. |
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log_wright_bessel -- Logarithm of Wright's generalized Bessel function. |
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The following function does not accept NumPy arrays (it is not a |
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universal function): |
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.. autosummary:: |
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:toctree: generated/ |
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lmbda -- Jahnke-Emden Lambda function, Lambdav(x). |
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Zeros of Bessel functions |
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^^^^^^^^^^^^^^^^^^^^^^^^^ |
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The following functions do not accept NumPy arrays (they are not |
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universal functions): |
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.. autosummary:: |
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:toctree: generated/ |
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jnjnp_zeros -- Compute zeros of integer-order Bessel functions Jn and Jn'. |
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jnyn_zeros -- Compute nt zeros of Bessel functions Jn(x), Jn'(x), Yn(x), and Yn'(x). |
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jn_zeros -- Compute zeros of integer-order Bessel function Jn(x). |
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jnp_zeros -- Compute zeros of integer-order Bessel function derivative Jn'(x). |
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yn_zeros -- Compute zeros of integer-order Bessel function Yn(x). |
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ynp_zeros -- Compute zeros of integer-order Bessel function derivative Yn'(x). |
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y0_zeros -- Compute nt zeros of Bessel function Y0(z), and derivative at each zero. |
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y1_zeros -- Compute nt zeros of Bessel function Y1(z), and derivative at each zero. |
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y1p_zeros -- Compute nt zeros of Bessel derivative Y1'(z), and value at each zero. |
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Faster versions of common Bessel functions |
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^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ |
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.. autosummary:: |
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:toctree: generated/ |
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j0 -- Bessel function of the first kind of order 0. |
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j1 -- Bessel function of the first kind of order 1. |
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y0 -- Bessel function of the second kind of order 0. |
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y1 -- Bessel function of the second kind of order 1. |
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i0 -- Modified Bessel function of order 0. |
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i0e -- Exponentially scaled modified Bessel function of order 0. |
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i1 -- Modified Bessel function of order 1. |
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i1e -- Exponentially scaled modified Bessel function of order 1. |
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k0 -- Modified Bessel function of the second kind of order 0, :math:`K_0`. |
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k0e -- Exponentially scaled modified Bessel function K of order 0 |
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k1 -- Modified Bessel function of the second kind of order 1, :math:`K_1(x)`. |
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k1e -- Exponentially scaled modified Bessel function K of order 1. |
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Integrals of Bessel functions |
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^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ |
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.. autosummary:: |
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:toctree: generated/ |
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itj0y0 -- Integrals of Bessel functions of order 0. |
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it2j0y0 -- Integrals related to Bessel functions of order 0. |
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iti0k0 -- Integrals of modified Bessel functions of order 0. |
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it2i0k0 -- Integrals related to modified Bessel functions of order 0. |
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besselpoly -- Weighted integral of a Bessel function. |
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Derivatives of Bessel functions |
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^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ |
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.. autosummary:: |
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:toctree: generated/ |
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jvp -- Compute nth derivative of Bessel function Jv(z) with respect to `z`. |
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yvp -- Compute nth derivative of Bessel function Yv(z) with respect to `z`. |
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kvp -- Compute nth derivative of real-order modified Bessel function Kv(z) |
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ivp -- Compute nth derivative of modified Bessel function Iv(z) with respect to `z`. |
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h1vp -- Compute nth derivative of Hankel function H1v(z) with respect to `z`. |
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h2vp -- Compute nth derivative of Hankel function H2v(z) with respect to `z`. |
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Spherical Bessel functions |
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^^^^^^^^^^^^^^^^^^^^^^^^^^ |
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.. autosummary:: |
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:toctree: generated/ |
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spherical_jn -- Spherical Bessel function of the first kind or its derivative. |
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spherical_yn -- Spherical Bessel function of the second kind or its derivative. |
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spherical_in -- Modified spherical Bessel function of the first kind or its derivative. |
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spherical_kn -- Modified spherical Bessel function of the second kind or its derivative. |
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Riccati-Bessel functions |
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^^^^^^^^^^^^^^^^^^^^^^^^ |
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The following functions do not accept NumPy arrays (they are not |
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universal functions): |
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.. autosummary:: |
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:toctree: generated/ |
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riccati_jn -- Compute Ricatti-Bessel function of the first kind and its derivative. |
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riccati_yn -- Compute Ricatti-Bessel function of the second kind and its derivative. |
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Struve functions |
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---------------- |
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.. autosummary:: |
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:toctree: generated/ |
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struve -- Struve function. |
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modstruve -- Modified Struve function. |
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itstruve0 -- Integral of the Struve function of order 0. |
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it2struve0 -- Integral related to the Struve function of order 0. |
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itmodstruve0 -- Integral of the modified Struve function of order 0. |
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Raw statistical functions |
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------------------------- |
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.. seealso:: :mod:`scipy.stats`: Friendly versions of these functions. |
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Binomial distribution |
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^^^^^^^^^^^^^^^^^^^^^ |
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.. autosummary:: |
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:toctree: generated/ |
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bdtr -- Binomial distribution cumulative distribution function. |
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bdtrc -- Binomial distribution survival function. |
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bdtri -- Inverse function to `bdtr` with respect to `p`. |
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bdtrik -- Inverse function to `bdtr` with respect to `k`. |
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bdtrin -- Inverse function to `bdtr` with respect to `n`. |
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Beta distribution |
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^^^^^^^^^^^^^^^^^ |
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.. autosummary:: |
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:toctree: generated/ |
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btdtria -- Inverse of `betainc` with respect to `a`. |
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btdtrib -- Inverse of `betainc` with respect to `b`. |
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F distribution |
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^^^^^^^^^^^^^^ |
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.. autosummary:: |
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:toctree: generated/ |
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fdtr -- F cumulative distribution function. |
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fdtrc -- F survival function. |
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fdtri -- The `p`-th quantile of the F-distribution. |
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fdtridfd -- Inverse to `fdtr` vs dfd. |
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Gamma distribution |
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^^^^^^^^^^^^^^^^^^ |
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.. autosummary:: |
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:toctree: generated/ |
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gdtr -- Gamma distribution cumulative distribution function. |
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gdtrc -- Gamma distribution survival function. |
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gdtria -- Inverse of `gdtr` vs a. |
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gdtrib -- Inverse of `gdtr` vs b. |
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gdtrix -- Inverse of `gdtr` vs x. |
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Negative binomial distribution |
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^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ |
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.. autosummary:: |
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:toctree: generated/ |
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nbdtr -- Negative binomial cumulative distribution function. |
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nbdtrc -- Negative binomial survival function. |
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nbdtri -- Inverse of `nbdtr` vs `p`. |
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nbdtrik -- Inverse of `nbdtr` vs `k`. |
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nbdtrin -- Inverse of `nbdtr` vs `n`. |
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Noncentral F distribution |
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^^^^^^^^^^^^^^^^^^^^^^^^^ |
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.. autosummary:: |
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:toctree: generated/ |
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ncfdtr -- Cumulative distribution function of the non-central F distribution. |
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ncfdtridfd -- Calculate degrees of freedom (denominator) for the noncentral F-distribution. |
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ncfdtridfn -- Calculate degrees of freedom (numerator) for the noncentral F-distribution. |
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ncfdtri -- Inverse cumulative distribution function of the non-central F distribution. |
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ncfdtrinc -- Calculate non-centrality parameter for non-central F distribution. |
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Noncentral t distribution |
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^^^^^^^^^^^^^^^^^^^^^^^^^ |
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.. autosummary:: |
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:toctree: generated/ |
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nctdtr -- Cumulative distribution function of the non-central `t` distribution. |
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nctdtridf -- Calculate degrees of freedom for non-central t distribution. |
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nctdtrit -- Inverse cumulative distribution function of the non-central t distribution. |
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nctdtrinc -- Calculate non-centrality parameter for non-central t distribution. |
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Normal distribution |
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^^^^^^^^^^^^^^^^^^^ |
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.. autosummary:: |
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:toctree: generated/ |
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nrdtrimn -- Calculate mean of normal distribution given other params. |
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nrdtrisd -- Calculate standard deviation of normal distribution given other params. |
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ndtr -- Normal cumulative distribution function. |
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log_ndtr -- Logarithm of normal cumulative distribution function. |
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ndtri -- Inverse of `ndtr` vs x. |
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ndtri_exp -- Inverse of `log_ndtr` vs x. |
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Poisson distribution |
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^^^^^^^^^^^^^^^^^^^^ |
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.. autosummary:: |
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:toctree: generated/ |
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pdtr -- Poisson cumulative distribution function. |
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pdtrc -- Poisson survival function. |
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pdtri -- Inverse to `pdtr` vs m. |
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pdtrik -- Inverse to `pdtr` vs k. |
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Student t distribution |
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^^^^^^^^^^^^^^^^^^^^^^ |
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.. autosummary:: |
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:toctree: generated/ |
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stdtr -- Student t distribution cumulative distribution function. |
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stdtridf -- Inverse of `stdtr` vs df. |
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stdtrit -- Inverse of `stdtr` vs `t`. |
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Chi square distribution |
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^^^^^^^^^^^^^^^^^^^^^^^ |
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.. autosummary:: |
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:toctree: generated/ |
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chdtr -- Chi square cumulative distribution function. |
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chdtrc -- Chi square survival function. |
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chdtri -- Inverse to `chdtrc`. |
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chdtriv -- Inverse to `chdtr` vs `v`. |
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Non-central chi square distribution |
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^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ |
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.. autosummary:: |
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:toctree: generated/ |
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chndtr -- Non-central chi square cumulative distribution function. |
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chndtridf -- Inverse to `chndtr` vs `df`. |
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chndtrinc -- Inverse to `chndtr` vs `nc`. |
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chndtrix -- Inverse to `chndtr` vs `x`. |
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Kolmogorov distribution |
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^^^^^^^^^^^^^^^^^^^^^^^ |
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.. autosummary:: |
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:toctree: generated/ |
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smirnov -- Kolmogorov-Smirnov complementary cumulative distribution function. |
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smirnovi -- Inverse to `smirnov`. |
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kolmogorov -- Complementary cumulative distribution function of Kolmogorov distribution. |
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kolmogi -- Inverse function to `kolmogorov`. |
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Box-Cox transformation |
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^^^^^^^^^^^^^^^^^^^^^^ |
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.. autosummary:: |
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:toctree: generated/ |
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boxcox -- Compute the Box-Cox transformation. |
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boxcox1p -- Compute the Box-Cox transformation of 1 + `x`. |
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inv_boxcox -- Compute the inverse of the Box-Cox transformation. |
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inv_boxcox1p -- Compute the inverse of the Box-Cox transformation. |
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Sigmoidal functions |
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^^^^^^^^^^^^^^^^^^^ |
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.. autosummary:: |
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:toctree: generated/ |
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logit -- Logit ufunc for ndarrays. |
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expit -- Logistic sigmoid function. |
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log_expit -- Logarithm of the logistic sigmoid function. |
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Miscellaneous |
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^^^^^^^^^^^^^ |
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.. autosummary:: |
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:toctree: generated/ |
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tklmbda -- Tukey-Lambda cumulative distribution function. |
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owens_t -- Owen's T Function. |
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Information Theory functions |
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---------------------------- |
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.. autosummary:: |
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:toctree: generated/ |
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entr -- Elementwise function for computing entropy. |
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rel_entr -- Elementwise function for computing relative entropy. |
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kl_div -- Elementwise function for computing Kullback-Leibler divergence. |
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huber -- Huber loss function. |
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pseudo_huber -- Pseudo-Huber loss function. |
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Gamma and related functions |
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--------------------------- |
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.. autosummary:: |
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:toctree: generated/ |
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gamma -- Gamma function. |
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gammaln -- Logarithm of the absolute value of the Gamma function for real inputs. |
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loggamma -- Principal branch of the logarithm of the Gamma function. |
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gammasgn -- Sign of the gamma function. |
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gammainc -- Regularized lower incomplete gamma function. |
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gammaincinv -- Inverse to `gammainc`. |
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gammaincc -- Regularized upper incomplete gamma function. |
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gammainccinv -- Inverse to `gammaincc`. |
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beta -- Beta function. |
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betaln -- Natural logarithm of absolute value of beta function. |
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betainc -- Incomplete beta integral. |
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betaincc -- Complemented incomplete beta integral. |
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betaincinv -- Inverse function to beta integral. |
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betainccinv -- Inverse of the complemented incomplete beta integral. |
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psi -- The digamma function. |
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rgamma -- Gamma function inverted. |
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polygamma -- Polygamma function n. |
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multigammaln -- Returns the log of multivariate gamma, also sometimes called the generalized gamma. |
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digamma -- psi(x[, out]). |
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poch -- Rising factorial (z)_m. |
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Error function and Fresnel integrals |
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------------------------------------ |
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.. autosummary:: |
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:toctree: generated/ |
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erf -- Returns the error function of complex argument. |
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erfc -- Complementary error function, ``1 - erf(x)``. |
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erfcx -- Scaled complementary error function, ``exp(x**2) * erfc(x)``. |
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erfi -- Imaginary error function, ``-i erf(i z)``. |
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erfinv -- Inverse function for erf. |
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erfcinv -- Inverse function for erfc. |
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wofz -- Faddeeva function. |
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dawsn -- Dawson's integral. |
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fresnel -- Fresnel sin and cos integrals. |
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fresnel_zeros -- Compute nt complex zeros of sine and cosine Fresnel integrals S(z) and C(z). |
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modfresnelp -- Modified Fresnel positive integrals. |
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modfresnelm -- Modified Fresnel negative integrals. |
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voigt_profile -- Voigt profile. |
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The following functions do not accept NumPy arrays (they are not |
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universal functions): |
|
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|
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|
.. autosummary:: |
|
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:toctree: generated/ |
|
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|
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erf_zeros -- Compute nt complex zeros of error function erf(z). |
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fresnelc_zeros -- Compute nt complex zeros of cosine Fresnel integral C(z). |
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fresnels_zeros -- Compute nt complex zeros of sine Fresnel integral S(z). |
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Legendre functions |
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------------------ |
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.. autosummary:: |
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:toctree: generated/ |
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legendre_p -- Legendre polynomials of the first kind. |
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legendre_p_all -- All Legendre polynomials of the first kind up to a specified order. |
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assoc_legendre_p -- Associated Legendre polynomials of the first kind. |
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|
assoc_legendre_p_all -- All associated Legendre polynomials of the first kind up to a specified order and degree. |
|
|
sph_legendre_p -- Spherical Legendre polynomials of the first kind. |
|
|
sph_legendre_p_all -- All spherical Legendre polynomials of the first kind up to a specified order and degree. |
|
|
sph_harm_y -- Spherical harmonics. |
|
|
sph_harm_y_all -- All spherical harmonics up to a specified order and degree. |
|
|
|
|
|
The following functions are in the process of being deprecated in favor of the above, |
|
|
which provide a more flexible and consistent interface. |
|
|
|
|
|
.. autosummary:: |
|
|
:toctree: generated/ |
|
|
|
|
|
lpmv -- Associated Legendre function of integer order and real degree. |
|
|
sph_harm -- Compute spherical harmonics. |
|
|
clpmn -- Associated Legendre function of the first kind for complex arguments. |
|
|
lpn -- Legendre function of the first kind. |
|
|
lqn -- Legendre function of the second kind. |
|
|
lpmn -- Sequence of associated Legendre functions of the first kind. |
|
|
lqmn -- Sequence of associated Legendre functions of the second kind. |
|
|
|
|
|
Ellipsoidal harmonics |
|
|
--------------------- |
|
|
|
|
|
.. autosummary:: |
|
|
:toctree: generated/ |
|
|
|
|
|
ellip_harm -- Ellipsoidal harmonic functions E^p_n(l). |
|
|
ellip_harm_2 -- Ellipsoidal harmonic functions F^p_n(l). |
|
|
ellip_normal -- Ellipsoidal harmonic normalization constants gamma^p_n. |
|
|
|
|
|
Orthogonal polynomials |
|
|
---------------------- |
|
|
|
|
|
The following functions evaluate values of orthogonal polynomials: |
|
|
|
|
|
.. autosummary:: |
|
|
:toctree: generated/ |
|
|
|
|
|
assoc_laguerre -- Compute the generalized (associated) Laguerre polynomial of degree n and order k. |
|
|
eval_legendre -- Evaluate Legendre polynomial at a point. |
|
|
eval_chebyt -- Evaluate Chebyshev polynomial of the first kind at a point. |
|
|
eval_chebyu -- Evaluate Chebyshev polynomial of the second kind at a point. |
|
|
eval_chebyc -- Evaluate Chebyshev polynomial of the first kind on [-2, 2] at a point. |
|
|
eval_chebys -- Evaluate Chebyshev polynomial of the second kind on [-2, 2] at a point. |
|
|
eval_jacobi -- Evaluate Jacobi polynomial at a point. |
|
|
eval_laguerre -- Evaluate Laguerre polynomial at a point. |
|
|
eval_genlaguerre -- Evaluate generalized Laguerre polynomial at a point. |
|
|
eval_hermite -- Evaluate physicist's Hermite polynomial at a point. |
|
|
eval_hermitenorm -- Evaluate probabilist's (normalized) Hermite polynomial at a point. |
|
|
eval_gegenbauer -- Evaluate Gegenbauer polynomial at a point. |
|
|
eval_sh_legendre -- Evaluate shifted Legendre polynomial at a point. |
|
|
eval_sh_chebyt -- Evaluate shifted Chebyshev polynomial of the first kind at a point. |
|
|
eval_sh_chebyu -- Evaluate shifted Chebyshev polynomial of the second kind at a point. |
|
|
eval_sh_jacobi -- Evaluate shifted Jacobi polynomial at a point. |
|
|
|
|
|
The following functions compute roots and quadrature weights for |
|
|
orthogonal polynomials: |
|
|
|
|
|
.. autosummary:: |
|
|
:toctree: generated/ |
|
|
|
|
|
roots_legendre -- Gauss-Legendre quadrature. |
|
|
roots_chebyt -- Gauss-Chebyshev (first kind) quadrature. |
|
|
roots_chebyu -- Gauss-Chebyshev (second kind) quadrature. |
|
|
roots_chebyc -- Gauss-Chebyshev (first kind) quadrature. |
|
|
roots_chebys -- Gauss-Chebyshev (second kind) quadrature. |
|
|
roots_jacobi -- Gauss-Jacobi quadrature. |
|
|
roots_laguerre -- Gauss-Laguerre quadrature. |
|
|
roots_genlaguerre -- Gauss-generalized Laguerre quadrature. |
|
|
roots_hermite -- Gauss-Hermite (physicist's) quadrature. |
|
|
roots_hermitenorm -- Gauss-Hermite (statistician's) quadrature. |
|
|
roots_gegenbauer -- Gauss-Gegenbauer quadrature. |
|
|
roots_sh_legendre -- Gauss-Legendre (shifted) quadrature. |
|
|
roots_sh_chebyt -- Gauss-Chebyshev (first kind, shifted) quadrature. |
|
|
roots_sh_chebyu -- Gauss-Chebyshev (second kind, shifted) quadrature. |
|
|
roots_sh_jacobi -- Gauss-Jacobi (shifted) quadrature. |
|
|
|
|
|
The functions below, in turn, return the polynomial coefficients in |
|
|
``orthopoly1d`` objects, which function similarly as `numpy.poly1d`. |
|
|
The ``orthopoly1d`` class also has an attribute ``weights``, which returns |
|
|
the roots, weights, and total weights for the appropriate form of Gaussian |
|
|
quadrature. These are returned in an ``n x 3`` array with roots in the first |
|
|
column, weights in the second column, and total weights in the final column. |
|
|
Note that ``orthopoly1d`` objects are converted to `~numpy.poly1d` when doing |
|
|
arithmetic, and lose information of the original orthogonal polynomial. |
|
|
|
|
|
.. autosummary:: |
|
|
:toctree: generated/ |
|
|
|
|
|
legendre -- Legendre polynomial. |
|
|
chebyt -- Chebyshev polynomial of the first kind. |
|
|
chebyu -- Chebyshev polynomial of the second kind. |
|
|
chebyc -- Chebyshev polynomial of the first kind on :math:`[-2, 2]`. |
|
|
chebys -- Chebyshev polynomial of the second kind on :math:`[-2, 2]`. |
|
|
jacobi -- Jacobi polynomial. |
|
|
laguerre -- Laguerre polynomial. |
|
|
genlaguerre -- Generalized (associated) Laguerre polynomial. |
|
|
hermite -- Physicist's Hermite polynomial. |
|
|
hermitenorm -- Normalized (probabilist's) Hermite polynomial. |
|
|
gegenbauer -- Gegenbauer (ultraspherical) polynomial. |
|
|
sh_legendre -- Shifted Legendre polynomial. |
|
|
sh_chebyt -- Shifted Chebyshev polynomial of the first kind. |
|
|
sh_chebyu -- Shifted Chebyshev polynomial of the second kind. |
|
|
sh_jacobi -- Shifted Jacobi polynomial. |
|
|
|
|
|
.. warning:: |
|
|
|
|
|
Computing values of high-order polynomials (around ``order > 20``) using |
|
|
polynomial coefficients is numerically unstable. To evaluate polynomial |
|
|
values, the ``eval_*`` functions should be used instead. |
|
|
|
|
|
|
|
|
Hypergeometric functions |
|
|
------------------------ |
|
|
|
|
|
.. autosummary:: |
|
|
:toctree: generated/ |
|
|
|
|
|
hyp2f1 -- Gauss hypergeometric function 2F1(a, b; c; z). |
|
|
hyp1f1 -- Confluent hypergeometric function 1F1(a, b; x). |
|
|
hyperu -- Confluent hypergeometric function U(a, b, x) of the second kind. |
|
|
hyp0f1 -- Confluent hypergeometric limit function 0F1. |
|
|
|
|
|
|
|
|
Parabolic cylinder functions |
|
|
---------------------------- |
|
|
|
|
|
.. autosummary:: |
|
|
:toctree: generated/ |
|
|
|
|
|
pbdv -- Parabolic cylinder function D. |
|
|
pbvv -- Parabolic cylinder function V. |
|
|
pbwa -- Parabolic cylinder function W. |
|
|
|
|
|
The following functions do not accept NumPy arrays (they are not |
|
|
universal functions): |
|
|
|
|
|
.. autosummary:: |
|
|
:toctree: generated/ |
|
|
|
|
|
pbdv_seq -- Parabolic cylinder functions Dv(x) and derivatives. |
|
|
pbvv_seq -- Parabolic cylinder functions Vv(x) and derivatives. |
|
|
pbdn_seq -- Parabolic cylinder functions Dn(z) and derivatives. |
|
|
|
|
|
Mathieu and related functions |
|
|
----------------------------- |
|
|
|
|
|
.. autosummary:: |
|
|
:toctree: generated/ |
|
|
|
|
|
mathieu_a -- Characteristic value of even Mathieu functions. |
|
|
mathieu_b -- Characteristic value of odd Mathieu functions. |
|
|
|
|
|
The following functions do not accept NumPy arrays (they are not |
|
|
universal functions): |
|
|
|
|
|
.. autosummary:: |
|
|
:toctree: generated/ |
|
|
|
|
|
mathieu_even_coef -- Fourier coefficients for even Mathieu and modified Mathieu functions. |
|
|
mathieu_odd_coef -- Fourier coefficients for even Mathieu and modified Mathieu functions. |
|
|
|
|
|
The following return both function and first derivative: |
|
|
|
|
|
.. autosummary:: |
|
|
:toctree: generated/ |
|
|
|
|
|
mathieu_cem -- Even Mathieu function and its derivative. |
|
|
mathieu_sem -- Odd Mathieu function and its derivative. |
|
|
mathieu_modcem1 -- Even modified Mathieu function of the first kind and its derivative. |
|
|
mathieu_modcem2 -- Even modified Mathieu function of the second kind and its derivative. |
|
|
mathieu_modsem1 -- Odd modified Mathieu function of the first kind and its derivative. |
|
|
mathieu_modsem2 -- Odd modified Mathieu function of the second kind and its derivative. |
|
|
|
|
|
Spheroidal wave functions |
|
|
------------------------- |
|
|
|
|
|
.. autosummary:: |
|
|
:toctree: generated/ |
|
|
|
|
|
pro_ang1 -- Prolate spheroidal angular function of the first kind and its derivative. |
|
|
pro_rad1 -- Prolate spheroidal radial function of the first kind and its derivative. |
|
|
pro_rad2 -- Prolate spheroidal radial function of the second kind and its derivative. |
|
|
obl_ang1 -- Oblate spheroidal angular function of the first kind and its derivative. |
|
|
obl_rad1 -- Oblate spheroidal radial function of the first kind and its derivative. |
|
|
obl_rad2 -- Oblate spheroidal radial function of the second kind and its derivative. |
|
|
pro_cv -- Characteristic value of prolate spheroidal function. |
|
|
obl_cv -- Characteristic value of oblate spheroidal function. |
|
|
pro_cv_seq -- Characteristic values for prolate spheroidal wave functions. |
|
|
obl_cv_seq -- Characteristic values for oblate spheroidal wave functions. |
|
|
|
|
|
The following functions require pre-computed characteristic value: |
|
|
|
|
|
.. autosummary:: |
|
|
:toctree: generated/ |
|
|
|
|
|
pro_ang1_cv -- Prolate spheroidal angular function pro_ang1 for precomputed characteristic value. |
|
|
pro_rad1_cv -- Prolate spheroidal radial function pro_rad1 for precomputed characteristic value. |
|
|
pro_rad2_cv -- Prolate spheroidal radial function pro_rad2 for precomputed characteristic value. |
|
|
obl_ang1_cv -- Oblate spheroidal angular function obl_ang1 for precomputed characteristic value. |
|
|
obl_rad1_cv -- Oblate spheroidal radial function obl_rad1 for precomputed characteristic value. |
|
|
obl_rad2_cv -- Oblate spheroidal radial function obl_rad2 for precomputed characteristic value. |
|
|
|
|
|
Kelvin functions |
|
|
---------------- |
|
|
|
|
|
.. autosummary:: |
|
|
:toctree: generated/ |
|
|
|
|
|
kelvin -- Kelvin functions as complex numbers. |
|
|
kelvin_zeros -- Compute nt zeros of all Kelvin functions. |
|
|
ber -- Kelvin function ber. |
|
|
bei -- Kelvin function bei |
|
|
berp -- Derivative of the Kelvin function `ber`. |
|
|
beip -- Derivative of the Kelvin function `bei`. |
|
|
ker -- Kelvin function ker. |
|
|
kei -- Kelvin function ker. |
|
|
kerp -- Derivative of the Kelvin function ker. |
|
|
keip -- Derivative of the Kelvin function kei. |
|
|
|
|
|
The following functions do not accept NumPy arrays (they are not |
|
|
universal functions): |
|
|
|
|
|
.. autosummary:: |
|
|
:toctree: generated/ |
|
|
|
|
|
ber_zeros -- Compute nt zeros of the Kelvin function ber(x). |
|
|
bei_zeros -- Compute nt zeros of the Kelvin function bei(x). |
|
|
berp_zeros -- Compute nt zeros of the Kelvin function ber'(x). |
|
|
beip_zeros -- Compute nt zeros of the Kelvin function bei'(x). |
|
|
ker_zeros -- Compute nt zeros of the Kelvin function ker(x). |
|
|
kei_zeros -- Compute nt zeros of the Kelvin function kei(x). |
|
|
kerp_zeros -- Compute nt zeros of the Kelvin function ker'(x). |
|
|
keip_zeros -- Compute nt zeros of the Kelvin function kei'(x). |
|
|
|
|
|
Combinatorics |
|
|
------------- |
|
|
|
|
|
.. autosummary:: |
|
|
:toctree: generated/ |
|
|
|
|
|
comb -- The number of combinations of N things taken k at a time. |
|
|
perm -- Permutations of N things taken k at a time, i.e., k-permutations of N. |
|
|
stirling2 -- Stirling numbers of the second kind. |
|
|
|
|
|
Lambert W and related functions |
|
|
------------------------------- |
|
|
|
|
|
.. autosummary:: |
|
|
:toctree: generated/ |
|
|
|
|
|
lambertw -- Lambert W function. |
|
|
wrightomega -- Wright Omega function. |
|
|
|
|
|
Other special functions |
|
|
----------------------- |
|
|
|
|
|
.. autosummary:: |
|
|
:toctree: generated/ |
|
|
|
|
|
agm -- Arithmetic, Geometric Mean. |
|
|
bernoulli -- Bernoulli numbers B0..Bn (inclusive). |
|
|
binom -- Binomial coefficient |
|
|
diric -- Periodic sinc function, also called the Dirichlet function. |
|
|
euler -- Euler numbers E0..En (inclusive). |
|
|
expn -- Exponential integral E_n. |
|
|
exp1 -- Exponential integral E_1 of complex argument z. |
|
|
expi -- Exponential integral Ei. |
|
|
factorial -- The factorial of a number or array of numbers. |
|
|
factorial2 -- Double factorial. |
|
|
factorialk -- Multifactorial of n of order k, n(!!...!). |
|
|
shichi -- Hyperbolic sine and cosine integrals. |
|
|
sici -- Sine and cosine integrals. |
|
|
softmax -- Softmax function. |
|
|
log_softmax -- Logarithm of softmax function. |
|
|
spence -- Spence's function, also known as the dilogarithm. |
|
|
zeta -- Riemann zeta function. |
|
|
zetac -- Riemann zeta function minus 1. |
|
|
softplus -- Softplus function. |
|
|
|
|
|
Convenience functions |
|
|
--------------------- |
|
|
|
|
|
.. autosummary:: |
|
|
:toctree: generated/ |
|
|
|
|
|
cbrt -- Cube root of `x`. |
|
|
exp10 -- 10**x. |
|
|
exp2 -- 2**x. |
|
|
radian -- Convert from degrees to radians. |
|
|
cosdg -- Cosine of the angle `x` given in degrees. |
|
|
sindg -- Sine of angle given in degrees. |
|
|
tandg -- Tangent of angle x given in degrees. |
|
|
cotdg -- Cotangent of the angle `x` given in degrees. |
|
|
log1p -- Calculates log(1+x) for use when `x` is near zero. |
|
|
expm1 -- ``exp(x) - 1`` for use when `x` is near zero. |
|
|
cosm1 -- ``cos(x) - 1`` for use when `x` is near zero. |
|
|
powm1 -- ``x**y - 1`` for use when `y` is near zero or `x` is near 1. |
|
|
round -- Round to nearest integer. |
|
|
xlogy -- Compute ``x*log(y)`` so that the result is 0 if ``x = 0``. |
|
|
xlog1py -- Compute ``x*log1p(y)`` so that the result is 0 if ``x = 0``. |
|
|
logsumexp -- Compute the log of the sum of exponentials of input elements. |
|
|
exprel -- Relative error exponential, (exp(x)-1)/x, for use when `x` is near zero. |
|
|
sinc -- Return the sinc function. |
|
|
|
|
|
""" |
|
|
|
|
|
import os |
|
|
import warnings |
|
|
|
|
|
|
|
|
def _load_libsf_error_state(): |
|
|
"""Load libsf_error_state.dll shared library on Windows |
|
|
|
|
|
libsf_error_state manages shared state used by |
|
|
``scipy.special.seterr`` and ``scipy.special.geterr`` so that these |
|
|
can work consistently between special functions provided by different |
|
|
extension modules. This shared library is installed in scipy/special |
|
|
alongside this __init__.py file. Due to lack of rpath support, Windows |
|
|
cannot find shared libraries installed within wheels. To circumvent this, |
|
|
we pre-load ``lib_sf_error_state.dll`` when on Windows. |
|
|
|
|
|
The logic for this function was borrowed from the function ``make_init`` |
|
|
in `scipy/tools/openblas_support.py`: |
|
|
https://github.com/scipy/scipy/blob/bb92c8014e21052e7dde67a76b28214dd1dcb94a/tools/openblas_support.py#L239-L274 |
|
|
""" |
|
|
if os.name == "nt": |
|
|
try: |
|
|
from ctypes import WinDLL |
|
|
basedir = os.path.dirname(__file__) |
|
|
except: |
|
|
pass |
|
|
else: |
|
|
dll_path = os.path.join(basedir, "libsf_error_state.dll") |
|
|
if os.path.exists(dll_path): |
|
|
WinDLL(dll_path) |
|
|
|
|
|
|
|
|
_load_libsf_error_state() |
|
|
|
|
|
|
|
|
from ._sf_error import SpecialFunctionWarning, SpecialFunctionError |
|
|
|
|
|
from . import _ufuncs |
|
|
from ._ufuncs import * |
|
|
|
|
|
|
|
|
from ._support_alternative_backends import ( |
|
|
log_ndtr, ndtr, ndtri, erf, erfc, i0, i0e, i1, i1e, gammaln, |
|
|
gammainc, gammaincc, logit, expit, entr, rel_entr, xlogy, |
|
|
chdtr, chdtrc, betainc, betaincc, stdtr) |
|
|
|
|
|
from . import _basic |
|
|
from ._basic import * |
|
|
|
|
|
from ._logsumexp import logsumexp, softmax, log_softmax |
|
|
|
|
|
from . import _multiufuncs |
|
|
from ._multiufuncs import * |
|
|
|
|
|
from . import _orthogonal |
|
|
from ._orthogonal import * |
|
|
|
|
|
from ._spfun_stats import multigammaln |
|
|
from ._ellip_harm import ( |
|
|
ellip_harm, |
|
|
ellip_harm_2, |
|
|
ellip_normal |
|
|
) |
|
|
from ._lambertw import lambertw |
|
|
from ._spherical_bessel import ( |
|
|
spherical_jn, |
|
|
spherical_yn, |
|
|
spherical_in, |
|
|
spherical_kn |
|
|
) |
|
|
|
|
|
|
|
|
from . import add_newdocs, basic, orthogonal, specfun, sf_error, spfun_stats |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
__all__ = _ufuncs.__all__ + _basic.__all__ + _orthogonal.__all__ + _multiufuncs.__all__ |
|
|
__all__ += [ |
|
|
'SpecialFunctionWarning', |
|
|
'SpecialFunctionError', |
|
|
'logsumexp', |
|
|
'softmax', |
|
|
'log_softmax', |
|
|
'multigammaln', |
|
|
'ellip_harm', |
|
|
'ellip_harm_2', |
|
|
'ellip_normal', |
|
|
'lambertw', |
|
|
'spherical_jn', |
|
|
'spherical_yn', |
|
|
'spherical_in', |
|
|
'spherical_kn', |
|
|
] |
|
|
|
|
|
from scipy._lib._testutils import PytestTester |
|
|
test = PytestTester(__name__) |
|
|
del PytestTester |
|
|
|
|
|
|
|
|
def _get_include(): |
|
|
"""This function is for development purposes only. |
|
|
|
|
|
This function could disappear or its behavior could change at any time. |
|
|
""" |
|
|
import os |
|
|
return os.path.dirname(__file__) |
|
|
|
|
|
|
