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# 

# Author: Travis Oliphant, 2002 

# 

 

from __future__ import division, print_function, absolute_import 

 

import operator 

import numpy as np 

import math 

from scipy._lib.six import xrange 

from numpy import (pi, asarray, floor, isscalar, iscomplex, real, 

imag, sqrt, where, mgrid, sin, place, issubdtype, 

extract, less, inexact, nan, zeros, sinc) 

from . import _ufuncs as ufuncs 

from ._ufuncs import (ellipkm1, mathieu_a, mathieu_b, iv, jv, gamma, 

psi, _zeta, hankel1, hankel2, yv, kv, ndtri, 

poch, binom, hyp0f1) 

from . import specfun 

from . import orthogonal 

from ._comb import _comb_int 

 

 

__all__ = ['ai_zeros', 'assoc_laguerre', 'bei_zeros', 'beip_zeros', 

'ber_zeros', 'bernoulli', 'berp_zeros', 

'bessel_diff_formula', 'bi_zeros', 'clpmn', 'comb', 

'digamma', 'diric', 'ellipk', 'erf_zeros', 'erfcinv', 

'erfinv', 'euler', 'factorial', 'factorialk', 'factorial2', 

'fresnel_zeros', 'fresnelc_zeros', 'fresnels_zeros', 

'gamma', 'h1vp', 'h2vp', 'hankel1', 'hankel2', 'hyp0f1', 

'iv', 'ivp', 'jn_zeros', 'jnjnp_zeros', 'jnp_zeros', 

'jnyn_zeros', 'jv', 'jvp', 'kei_zeros', 'keip_zeros', 

'kelvin_zeros', 'ker_zeros', 'kerp_zeros', 'kv', 'kvp', 

'lmbda', 'lpmn', 'lpn', 'lqmn', 'lqn', 'mathieu_a', 

'mathieu_b', 'mathieu_even_coef', 'mathieu_odd_coef', 

'ndtri', 'obl_cv_seq', 'pbdn_seq', 'pbdv_seq', 'pbvv_seq', 

'perm', 'polygamma', 'pro_cv_seq', 'psi', 'riccati_jn', 

'riccati_yn', 'sinc', 'y0_zeros', 'y1_zeros', 'y1p_zeros', 

'yn_zeros', 'ynp_zeros', 'yv', 'yvp', 'zeta'] 

 

 

def _nonneg_int_or_fail(n, var_name, strict=True): 

try: 

if strict: 

# Raises an exception if float 

n = operator.index(n) 

elif n == floor(n): 

n = int(n) 

else: 

raise ValueError() 

if n < 0: 

raise ValueError() 

except (ValueError, TypeError) as err: 

raise err.__class__("{} must be a non-negative integer".format(var_name)) 

return n 

 

 

def diric(x, n): 

"""Periodic sinc function, also called the Dirichlet function. 

 

The Dirichlet function is defined as:: 

 

diric(x, n) = sin(x * n/2) / (n * sin(x / 2)), 

 

where `n` is a positive integer. 

 

Parameters 

---------- 

x : array_like 

Input data 

n : int 

Integer defining the periodicity. 

 

Returns 

------- 

diric : ndarray 

 

Examples 

-------- 

>>> from scipy import special 

>>> import matplotlib.pyplot as plt 

 

>>> x = np.linspace(-8*np.pi, 8*np.pi, num=201) 

>>> plt.figure(figsize=(8, 8)); 

>>> for idx, n in enumerate([2, 3, 4, 9]): 

... plt.subplot(2, 2, idx+1) 

... plt.plot(x, special.diric(x, n)) 

... plt.title('diric, n={}'.format(n)) 

>>> plt.show() 

 

The following example demonstrates that `diric` gives the magnitudes 

(modulo the sign and scaling) of the Fourier coefficients of a 

rectangular pulse. 

 

Suppress output of values that are effectively 0: 

 

>>> np.set_printoptions(suppress=True) 

 

Create a signal `x` of length `m` with `k` ones: 

 

>>> m = 8 

>>> k = 3 

>>> x = np.zeros(m) 

>>> x[:k] = 1 

 

Use the FFT to compute the Fourier transform of `x`, and 

inspect the magnitudes of the coefficients: 

 

>>> np.abs(np.fft.fft(x)) 

array([ 3. , 2.41421356, 1. , 0.41421356, 1. , 

0.41421356, 1. , 2.41421356]) 

 

Now find the same values (up to sign) using `diric`. We multiply 

by `k` to account for the different scaling conventions of 

`numpy.fft.fft` and `diric`: 

 

>>> theta = np.linspace(0, 2*np.pi, m, endpoint=False) 

>>> k * special.diric(theta, k) 

array([ 3. , 2.41421356, 1. , -0.41421356, -1. , 

-0.41421356, 1. , 2.41421356]) 

""" 

x, n = asarray(x), asarray(n) 

n = asarray(n + (x-x)) 

x = asarray(x + (n-n)) 

if issubdtype(x.dtype, inexact): 

ytype = x.dtype 

else: 

ytype = float 

y = zeros(x.shape, ytype) 

 

# empirical minval for 32, 64 or 128 bit float computations 

# where sin(x/2) < minval, result is fixed at +1 or -1 

if np.finfo(ytype).eps < 1e-18: 

minval = 1e-11 

elif np.finfo(ytype).eps < 1e-15: 

minval = 1e-7 

else: 

minval = 1e-3 

 

mask1 = (n <= 0) | (n != floor(n)) 

place(y, mask1, nan) 

 

x = x / 2 

denom = sin(x) 

mask2 = (1-mask1) & (abs(denom) < minval) 

xsub = extract(mask2, x) 

nsub = extract(mask2, n) 

zsub = xsub / pi 

place(y, mask2, pow(-1, np.round(zsub)*(nsub-1))) 

 

mask = (1-mask1) & (1-mask2) 

xsub = extract(mask, x) 

nsub = extract(mask, n) 

dsub = extract(mask, denom) 

place(y, mask, sin(nsub*xsub)/(nsub*dsub)) 

return y 

 

 

def jnjnp_zeros(nt): 

"""Compute zeros of integer-order Bessel functions Jn and Jn'. 

 

Results are arranged in order of the magnitudes of the zeros. 

 

Parameters 

---------- 

nt : int 

Number (<=1200) of zeros to compute 

 

Returns 

------- 

zo[l-1] : ndarray 

Value of the lth zero of Jn(x) and Jn'(x). Of length `nt`. 

n[l-1] : ndarray 

Order of the Jn(x) or Jn'(x) associated with lth zero. Of length `nt`. 

m[l-1] : ndarray 

Serial number of the zeros of Jn(x) or Jn'(x) associated 

with lth zero. Of length `nt`. 

t[l-1] : ndarray 

0 if lth zero in zo is zero of Jn(x), 1 if it is a zero of Jn'(x). Of 

length `nt`. 

 

See Also 

-------- 

jn_zeros, jnp_zeros : to get separated arrays of zeros. 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996, chapter 5. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

if not isscalar(nt) or (floor(nt) != nt) or (nt > 1200): 

raise ValueError("Number must be integer <= 1200.") 

nt = int(nt) 

n, m, t, zo = specfun.jdzo(nt) 

return zo[1:nt+1], n[:nt], m[:nt], t[:nt] 

 

 

def jnyn_zeros(n, nt): 

"""Compute nt zeros of Bessel functions Jn(x), Jn'(x), Yn(x), and Yn'(x). 

 

Returns 4 arrays of length `nt`, corresponding to the first `nt` zeros of 

Jn(x), Jn'(x), Yn(x), and Yn'(x), respectively. 

 

Parameters 

---------- 

n : int 

Order of the Bessel functions 

nt : int 

Number (<=1200) of zeros to compute 

 

See jn_zeros, jnp_zeros, yn_zeros, ynp_zeros to get separate arrays. 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996, chapter 5. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

if not (isscalar(nt) and isscalar(n)): 

raise ValueError("Arguments must be scalars.") 

if (floor(n) != n) or (floor(nt) != nt): 

raise ValueError("Arguments must be integers.") 

if (nt <= 0): 

raise ValueError("nt > 0") 

return specfun.jyzo(abs(n), nt) 

 

 

def jn_zeros(n, nt): 

"""Compute zeros of integer-order Bessel function Jn(x). 

 

Parameters 

---------- 

n : int 

Order of Bessel function 

nt : int 

Number of zeros to return 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996, chapter 5. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

return jnyn_zeros(n, nt)[0] 

 

 

def jnp_zeros(n, nt): 

"""Compute zeros of integer-order Bessel function derivative Jn'(x). 

 

Parameters 

---------- 

n : int 

Order of Bessel function 

nt : int 

Number of zeros to return 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996, chapter 5. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

return jnyn_zeros(n, nt)[1] 

 

 

def yn_zeros(n, nt): 

"""Compute zeros of integer-order Bessel function Yn(x). 

 

Parameters 

---------- 

n : int 

Order of Bessel function 

nt : int 

Number of zeros to return 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996, chapter 5. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

return jnyn_zeros(n, nt)[2] 

 

 

def ynp_zeros(n, nt): 

"""Compute zeros of integer-order Bessel function derivative Yn'(x). 

 

Parameters 

---------- 

n : int 

Order of Bessel function 

nt : int 

Number of zeros to return 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996, chapter 5. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

return jnyn_zeros(n, nt)[3] 

 

 

def y0_zeros(nt, complex=False): 

"""Compute nt zeros of Bessel function Y0(z), and derivative at each zero. 

 

The derivatives are given by Y0'(z0) = -Y1(z0) at each zero z0. 

 

Parameters 

---------- 

nt : int 

Number of zeros to return 

complex : bool, default False 

Set to False to return only the real zeros; set to True to return only 

the complex zeros with negative real part and positive imaginary part. 

Note that the complex conjugates of the latter are also zeros of the 

function, but are not returned by this routine. 

 

Returns 

------- 

z0n : ndarray 

Location of nth zero of Y0(z) 

y0pz0n : ndarray 

Value of derivative Y0'(z0) for nth zero 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996, chapter 5. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0): 

raise ValueError("Arguments must be scalar positive integer.") 

kf = 0 

kc = not complex 

return specfun.cyzo(nt, kf, kc) 

 

 

def y1_zeros(nt, complex=False): 

"""Compute nt zeros of Bessel function Y1(z), and derivative at each zero. 

 

The derivatives are given by Y1'(z1) = Y0(z1) at each zero z1. 

 

Parameters 

---------- 

nt : int 

Number of zeros to return 

complex : bool, default False 

Set to False to return only the real zeros; set to True to return only 

the complex zeros with negative real part and positive imaginary part. 

Note that the complex conjugates of the latter are also zeros of the 

function, but are not returned by this routine. 

 

Returns 

------- 

z1n : ndarray 

Location of nth zero of Y1(z) 

y1pz1n : ndarray 

Value of derivative Y1'(z1) for nth zero 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996, chapter 5. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0): 

raise ValueError("Arguments must be scalar positive integer.") 

kf = 1 

kc = not complex 

return specfun.cyzo(nt, kf, kc) 

 

 

def y1p_zeros(nt, complex=False): 

"""Compute nt zeros of Bessel derivative Y1'(z), and value at each zero. 

 

The values are given by Y1(z1) at each z1 where Y1'(z1)=0. 

 

Parameters 

---------- 

nt : int 

Number of zeros to return 

complex : bool, default False 

Set to False to return only the real zeros; set to True to return only 

the complex zeros with negative real part and positive imaginary part. 

Note that the complex conjugates of the latter are also zeros of the 

function, but are not returned by this routine. 

 

Returns 

------- 

z1pn : ndarray 

Location of nth zero of Y1'(z) 

y1z1pn : ndarray 

Value of derivative Y1(z1) for nth zero 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996, chapter 5. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0): 

raise ValueError("Arguments must be scalar positive integer.") 

kf = 2 

kc = not complex 

return specfun.cyzo(nt, kf, kc) 

 

 

def _bessel_diff_formula(v, z, n, L, phase): 

# from AMS55. 

# L(v, z) = J(v, z), Y(v, z), H1(v, z), H2(v, z), phase = -1 

# L(v, z) = I(v, z) or exp(v*pi*i)K(v, z), phase = 1 

# For K, you can pull out the exp((v-k)*pi*i) into the caller 

v = asarray(v) 

p = 1.0 

s = L(v-n, z) 

for i in xrange(1, n+1): 

p = phase * (p * (n-i+1)) / i # = choose(k, i) 

s += p*L(v-n + i*2, z) 

return s / (2.**n) 

 

 

bessel_diff_formula = np.deprecate(_bessel_diff_formula, 

message="bessel_diff_formula is a private function, do not use it!") 

 

 

def jvp(v, z, n=1): 

"""Compute nth derivative of Bessel function Jv(z) with respect to `z`. 

 

Parameters 

---------- 

v : float 

Order of Bessel function 

z : complex 

Argument at which to evaluate the derivative 

n : int, default 1 

Order of derivative 

 

Notes 

----- 

The derivative is computed using the relation DLFM 10.6.7 [2]_. 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996, chapter 5. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

.. [2] NIST Digital Library of Mathematical Functions. 

http://dlmf.nist.gov/10.6.E7 

 

""" 

n = _nonneg_int_or_fail(n, 'n') 

if n == 0: 

return jv(v, z) 

else: 

return _bessel_diff_formula(v, z, n, jv, -1) 

 

 

def yvp(v, z, n=1): 

"""Compute nth derivative of Bessel function Yv(z) with respect to `z`. 

 

Parameters 

---------- 

v : float 

Order of Bessel function 

z : complex 

Argument at which to evaluate the derivative 

n : int, default 1 

Order of derivative 

 

Notes 

----- 

The derivative is computed using the relation DLFM 10.6.7 [2]_. 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996, chapter 5. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

.. [2] NIST Digital Library of Mathematical Functions. 

http://dlmf.nist.gov/10.6.E7 

 

""" 

n = _nonneg_int_or_fail(n, 'n') 

if n == 0: 

return yv(v, z) 

else: 

return _bessel_diff_formula(v, z, n, yv, -1) 

 

 

def kvp(v, z, n=1): 

"""Compute nth derivative of real-order modified Bessel function Kv(z) 

 

Kv(z) is the modified Bessel function of the second kind. 

Derivative is calculated with respect to `z`. 

 

Parameters 

---------- 

v : array_like of float 

Order of Bessel function 

z : array_like of complex 

Argument at which to evaluate the derivative 

n : int 

Order of derivative. Default is first derivative. 

 

Returns 

------- 

out : ndarray 

The results 

 

Examples 

-------- 

Calculate multiple values at order 5: 

 

>>> from scipy.special import kvp 

>>> kvp(5, (1, 2, 3+5j)) 

array([-1.84903536e+03+0.j , -2.57735387e+01+0.j , 

-3.06627741e-02+0.08750845j]) 

 

 

Calculate for a single value at multiple orders: 

 

>>> kvp((4, 4.5, 5), 1) 

array([ -184.0309, -568.9585, -1849.0354]) 

 

Notes 

----- 

The derivative is computed using the relation DLFM 10.29.5 [2]_. 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996, chapter 6. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

.. [2] NIST Digital Library of Mathematical Functions. 

http://dlmf.nist.gov/10.29.E5 

 

""" 

n = _nonneg_int_or_fail(n, 'n') 

if n == 0: 

return kv(v, z) 

else: 

return (-1)**n * _bessel_diff_formula(v, z, n, kv, 1) 

 

 

def ivp(v, z, n=1): 

"""Compute nth derivative of modified Bessel function Iv(z) with respect 

to `z`. 

 

Parameters 

---------- 

v : array_like of float 

Order of Bessel function 

z : array_like of complex 

Argument at which to evaluate the derivative 

n : int, default 1 

Order of derivative 

 

Notes 

----- 

The derivative is computed using the relation DLFM 10.29.5 [2]_. 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996, chapter 6. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

.. [2] NIST Digital Library of Mathematical Functions. 

http://dlmf.nist.gov/10.29.E5 

 

""" 

n = _nonneg_int_or_fail(n, 'n') 

if n == 0: 

return iv(v, z) 

else: 

return _bessel_diff_formula(v, z, n, iv, 1) 

 

 

def h1vp(v, z, n=1): 

"""Compute nth derivative of Hankel function H1v(z) with respect to `z`. 

 

Parameters 

---------- 

v : float 

Order of Hankel function 

z : complex 

Argument at which to evaluate the derivative 

n : int, default 1 

Order of derivative 

 

Notes 

----- 

The derivative is computed using the relation DLFM 10.6.7 [2]_. 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996, chapter 5. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

.. [2] NIST Digital Library of Mathematical Functions. 

http://dlmf.nist.gov/10.6.E7 

 

""" 

n = _nonneg_int_or_fail(n, 'n') 

if n == 0: 

return hankel1(v, z) 

else: 

return _bessel_diff_formula(v, z, n, hankel1, -1) 

 

 

def h2vp(v, z, n=1): 

"""Compute nth derivative of Hankel function H2v(z) with respect to `z`. 

 

Parameters 

---------- 

v : float 

Order of Hankel function 

z : complex 

Argument at which to evaluate the derivative 

n : int, default 1 

Order of derivative 

 

Notes 

----- 

The derivative is computed using the relation DLFM 10.6.7 [2]_. 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996, chapter 5. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

.. [2] NIST Digital Library of Mathematical Functions. 

http://dlmf.nist.gov/10.6.E7 

 

""" 

n = _nonneg_int_or_fail(n, 'n') 

if n == 0: 

return hankel2(v, z) 

else: 

return _bessel_diff_formula(v, z, n, hankel2, -1) 

 

 

def riccati_jn(n, x): 

r"""Compute Ricatti-Bessel function of the first kind and its derivative. 

 

The Ricatti-Bessel function of the first kind is defined as :math:`x 

j_n(x)`, where :math:`j_n` is the spherical Bessel function of the first 

kind of order :math:`n`. 

 

This function computes the value and first derivative of the 

Ricatti-Bessel function for all orders up to and including `n`. 

 

Parameters 

---------- 

n : int 

Maximum order of function to compute 

x : float 

Argument at which to evaluate 

 

Returns 

------- 

jn : ndarray 

Value of j0(x), ..., jn(x) 

jnp : ndarray 

First derivative j0'(x), ..., jn'(x) 

 

Notes 

----- 

The computation is carried out via backward recurrence, using the 

relation DLMF 10.51.1 [2]_. 

 

Wrapper for a Fortran routine created by Shanjie Zhang and Jianming 

Jin [1]_. 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

.. [2] NIST Digital Library of Mathematical Functions. 

http://dlmf.nist.gov/10.51.E1 

 

""" 

if not (isscalar(n) and isscalar(x)): 

raise ValueError("arguments must be scalars.") 

n = _nonneg_int_or_fail(n, 'n', strict=False) 

if (n == 0): 

n1 = 1 

else: 

n1 = n 

nm, jn, jnp = specfun.rctj(n1, x) 

return jn[:(n+1)], jnp[:(n+1)] 

 

 

def riccati_yn(n, x): 

"""Compute Ricatti-Bessel function of the second kind and its derivative. 

 

The Ricatti-Bessel function of the second kind is defined as :math:`x 

y_n(x)`, where :math:`y_n` is the spherical Bessel function of the second 

kind of order :math:`n`. 

 

This function computes the value and first derivative of the function for 

all orders up to and including `n`. 

 

Parameters 

---------- 

n : int 

Maximum order of function to compute 

x : float 

Argument at which to evaluate 

 

Returns 

------- 

yn : ndarray 

Value of y0(x), ..., yn(x) 

ynp : ndarray 

First derivative y0'(x), ..., yn'(x) 

 

Notes 

----- 

The computation is carried out via ascending recurrence, using the 

relation DLMF 10.51.1 [2]_. 

 

Wrapper for a Fortran routine created by Shanjie Zhang and Jianming 

Jin [1]_. 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

.. [2] NIST Digital Library of Mathematical Functions. 

http://dlmf.nist.gov/10.51.E1 

 

""" 

if not (isscalar(n) and isscalar(x)): 

raise ValueError("arguments must be scalars.") 

n = _nonneg_int_or_fail(n, 'n', strict=False) 

if (n == 0): 

n1 = 1 

else: 

n1 = n 

nm, jn, jnp = specfun.rcty(n1, x) 

return jn[:(n+1)], jnp[:(n+1)] 

 

 

def erfinv(y): 

"""Inverse function for erf. 

""" 

return ndtri((y+1)/2.0)/sqrt(2) 

 

 

def erfcinv(y): 

"""Inverse function for erfc. 

""" 

return -ndtri(0.5*y)/sqrt(2) 

 

 

def erf_zeros(nt): 

"""Compute nt complex zeros of error function erf(z). 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

if (floor(nt) != nt) or (nt <= 0) or not isscalar(nt): 

raise ValueError("Argument must be positive scalar integer.") 

return specfun.cerzo(nt) 

 

 

def fresnelc_zeros(nt): 

"""Compute nt complex zeros of cosine Fresnel integral C(z). 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

if (floor(nt) != nt) or (nt <= 0) or not isscalar(nt): 

raise ValueError("Argument must be positive scalar integer.") 

return specfun.fcszo(1, nt) 

 

 

def fresnels_zeros(nt): 

"""Compute nt complex zeros of sine Fresnel integral S(z). 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

if (floor(nt) != nt) or (nt <= 0) or not isscalar(nt): 

raise ValueError("Argument must be positive scalar integer.") 

return specfun.fcszo(2, nt) 

 

 

def fresnel_zeros(nt): 

"""Compute nt complex zeros of sine and cosine Fresnel integrals S(z) and C(z). 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

if (floor(nt) != nt) or (nt <= 0) or not isscalar(nt): 

raise ValueError("Argument must be positive scalar integer.") 

return specfun.fcszo(2, nt), specfun.fcszo(1, nt) 

 

 

def assoc_laguerre(x, n, k=0.0): 

"""Compute the generalized (associated) Laguerre polynomial of degree n and order k. 

 

The polynomial :math:`L^{(k)}_n(x)` is orthogonal over ``[0, inf)``, 

with weighting function ``exp(-x) * x**k`` with ``k > -1``. 

 

Notes 

----- 

`assoc_laguerre` is a simple wrapper around `eval_genlaguerre`, with 

reversed argument order ``(x, n, k=0.0) --> (n, k, x)``. 

 

""" 

return orthogonal.eval_genlaguerre(n, k, x) 

 

 

digamma = psi 

 

 

def polygamma(n, x): 

"""Polygamma function n. 

 

This is the nth derivative of the digamma (psi) function. 

 

Parameters 

---------- 

n : array_like of int 

The order of the derivative of `psi`. 

x : array_like 

Where to evaluate the polygamma function. 

 

Returns 

------- 

polygamma : ndarray 

The result. 

 

Examples 

-------- 

>>> from scipy import special 

>>> x = [2, 3, 25.5] 

>>> special.polygamma(1, x) 

array([ 0.64493407, 0.39493407, 0.03999467]) 

>>> special.polygamma(0, x) == special.psi(x) 

array([ True, True, True], dtype=bool) 

 

""" 

n, x = asarray(n), asarray(x) 

fac2 = (-1.0)**(n+1) * gamma(n+1.0) * zeta(n+1, x) 

return where(n == 0, psi(x), fac2) 

 

 

def mathieu_even_coef(m, q): 

r"""Fourier coefficients for even Mathieu and modified Mathieu functions. 

 

The Fourier series of the even solutions of the Mathieu differential 

equation are of the form 

 

.. math:: \mathrm{ce}_{2n}(z, q) = \sum_{k=0}^{\infty} A_{(2n)}^{(2k)} \cos 2kz 

 

.. math:: \mathrm{ce}_{2n+1}(z, q) = \sum_{k=0}^{\infty} A_{(2n+1)}^{(2k+1)} \cos (2k+1)z 

 

This function returns the coefficients :math:`A_{(2n)}^{(2k)}` for even 

input m=2n, and the coefficients :math:`A_{(2n+1)}^{(2k+1)}` for odd input 

m=2n+1. 

 

Parameters 

---------- 

m : int 

Order of Mathieu functions. Must be non-negative. 

q : float (>=0) 

Parameter of Mathieu functions. Must be non-negative. 

 

Returns 

------- 

Ak : ndarray 

Even or odd Fourier coefficients, corresponding to even or odd m. 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

.. [2] NIST Digital Library of Mathematical Functions 

http://dlmf.nist.gov/28.4#i 

 

""" 

if not (isscalar(m) and isscalar(q)): 

raise ValueError("m and q must be scalars.") 

if (q < 0): 

raise ValueError("q >=0") 

if (m != floor(m)) or (m < 0): 

raise ValueError("m must be an integer >=0.") 

 

if (q <= 1): 

qm = 7.5 + 56.1*sqrt(q) - 134.7*q + 90.7*sqrt(q)*q 

else: 

qm = 17.0 + 3.1*sqrt(q) - .126*q + .0037*sqrt(q)*q 

km = int(qm + 0.5*m) 

if km > 251: 

print("Warning, too many predicted coefficients.") 

kd = 1 

m = int(floor(m)) 

if m % 2: 

kd = 2 

 

a = mathieu_a(m, q) 

fc = specfun.fcoef(kd, m, q, a) 

return fc[:km] 

 

 

def mathieu_odd_coef(m, q): 

r"""Fourier coefficients for even Mathieu and modified Mathieu functions. 

 

The Fourier series of the odd solutions of the Mathieu differential 

equation are of the form 

 

.. math:: \mathrm{se}_{2n+1}(z, q) = \sum_{k=0}^{\infty} B_{(2n+1)}^{(2k+1)} \sin (2k+1)z 

 

.. math:: \mathrm{se}_{2n+2}(z, q) = \sum_{k=0}^{\infty} B_{(2n+2)}^{(2k+2)} \sin (2k+2)z 

 

This function returns the coefficients :math:`B_{(2n+2)}^{(2k+2)}` for even 

input m=2n+2, and the coefficients :math:`B_{(2n+1)}^{(2k+1)}` for odd 

input m=2n+1. 

 

Parameters 

---------- 

m : int 

Order of Mathieu functions. Must be non-negative. 

q : float (>=0) 

Parameter of Mathieu functions. Must be non-negative. 

 

Returns 

------- 

Bk : ndarray 

Even or odd Fourier coefficients, corresponding to even or odd m. 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

if not (isscalar(m) and isscalar(q)): 

raise ValueError("m and q must be scalars.") 

if (q < 0): 

raise ValueError("q >=0") 

if (m != floor(m)) or (m <= 0): 

raise ValueError("m must be an integer > 0") 

 

if (q <= 1): 

qm = 7.5 + 56.1*sqrt(q) - 134.7*q + 90.7*sqrt(q)*q 

else: 

qm = 17.0 + 3.1*sqrt(q) - .126*q + .0037*sqrt(q)*q 

km = int(qm + 0.5*m) 

if km > 251: 

print("Warning, too many predicted coefficients.") 

kd = 4 

m = int(floor(m)) 

if m % 2: 

kd = 3 

 

b = mathieu_b(m, q) 

fc = specfun.fcoef(kd, m, q, b) 

return fc[:km] 

 

 

def lpmn(m, n, z): 

"""Sequence of associated Legendre functions of the first kind. 

 

Computes the associated Legendre function of the first kind of order m and 

degree n, ``Pmn(z)`` = :math:`P_n^m(z)`, and its derivative, ``Pmn'(z)``. 

Returns two arrays of size ``(m+1, n+1)`` containing ``Pmn(z)`` and 

``Pmn'(z)`` for all orders from ``0..m`` and degrees from ``0..n``. 

 

This function takes a real argument ``z``. For complex arguments ``z`` 

use clpmn instead. 

 

Parameters 

---------- 

m : int 

``|m| <= n``; the order of the Legendre function. 

n : int 

where ``n >= 0``; the degree of the Legendre function. Often 

called ``l`` (lower case L) in descriptions of the associated 

Legendre function 

z : float 

Input value. 

 

Returns 

------- 

Pmn_z : (m+1, n+1) array 

Values for all orders 0..m and degrees 0..n 

Pmn_d_z : (m+1, n+1) array 

Derivatives for all orders 0..m and degrees 0..n 

 

See Also 

-------- 

clpmn: associated Legendre functions of the first kind for complex z 

 

Notes 

----- 

In the interval (-1, 1), Ferrer's function of the first kind is 

returned. The phase convention used for the intervals (1, inf) 

and (-inf, -1) is such that the result is always real. 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

.. [2] NIST Digital Library of Mathematical Functions 

http://dlmf.nist.gov/14.3 

 

""" 

if not isscalar(m) or (abs(m) > n): 

raise ValueError("m must be <= n.") 

if not isscalar(n) or (n < 0): 

raise ValueError("n must be a non-negative integer.") 

if not isscalar(z): 

raise ValueError("z must be scalar.") 

if iscomplex(z): 

raise ValueError("Argument must be real. Use clpmn instead.") 

if (m < 0): 

mp = -m 

mf, nf = mgrid[0:mp+1, 0:n+1] 

with ufuncs.errstate(all='ignore'): 

if abs(z) < 1: 

# Ferrer function; DLMF 14.9.3 

fixarr = where(mf > nf, 0.0, 

(-1)**mf * gamma(nf-mf+1) / gamma(nf+mf+1)) 

else: 

# Match to clpmn; DLMF 14.9.13 

fixarr = where(mf > nf, 0.0, gamma(nf-mf+1) / gamma(nf+mf+1)) 

else: 

mp = m 

p, pd = specfun.lpmn(mp, n, z) 

if (m < 0): 

p = p * fixarr 

pd = pd * fixarr 

return p, pd 

 

 

def clpmn(m, n, z, type=3): 

"""Associated Legendre function of the first kind for complex arguments. 

 

Computes the associated Legendre function of the first kind of order m and 

degree n, ``Pmn(z)`` = :math:`P_n^m(z)`, and its derivative, ``Pmn'(z)``. 

Returns two arrays of size ``(m+1, n+1)`` containing ``Pmn(z)`` and 

``Pmn'(z)`` for all orders from ``0..m`` and degrees from ``0..n``. 

 

Parameters 

---------- 

m : int 

``|m| <= n``; the order of the Legendre function. 

n : int 

where ``n >= 0``; the degree of the Legendre function. Often 

called ``l`` (lower case L) in descriptions of the associated 

Legendre function 

z : float or complex 

Input value. 

type : int, optional 

takes values 2 or 3 

2: cut on the real axis ``|x| > 1`` 

3: cut on the real axis ``-1 < x < 1`` (default) 

 

Returns 

------- 

Pmn_z : (m+1, n+1) array 

Values for all orders ``0..m`` and degrees ``0..n`` 

Pmn_d_z : (m+1, n+1) array 

Derivatives for all orders ``0..m`` and degrees ``0..n`` 

 

See Also 

-------- 

lpmn: associated Legendre functions of the first kind for real z 

 

Notes 

----- 

By default, i.e. for ``type=3``, phase conventions are chosen according 

to [1]_ such that the function is analytic. The cut lies on the interval 

(-1, 1). Approaching the cut from above or below in general yields a phase 

factor with respect to Ferrer's function of the first kind 

(cf. `lpmn`). 

 

For ``type=2`` a cut at ``|x| > 1`` is chosen. Approaching the real values 

on the interval (-1, 1) in the complex plane yields Ferrer's function 

of the first kind. 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

.. [2] NIST Digital Library of Mathematical Functions 

http://dlmf.nist.gov/14.21 

 

""" 

if not isscalar(m) or (abs(m) > n): 

raise ValueError("m must be <= n.") 

if not isscalar(n) or (n < 0): 

raise ValueError("n must be a non-negative integer.") 

if not isscalar(z): 

raise ValueError("z must be scalar.") 

if not(type == 2 or type == 3): 

raise ValueError("type must be either 2 or 3.") 

if (m < 0): 

mp = -m 

mf, nf = mgrid[0:mp+1, 0:n+1] 

with ufuncs.errstate(all='ignore'): 

if type == 2: 

fixarr = where(mf > nf, 0.0, 

(-1)**mf * gamma(nf-mf+1) / gamma(nf+mf+1)) 

else: 

fixarr = where(mf > nf, 0.0, gamma(nf-mf+1) / gamma(nf+mf+1)) 

else: 

mp = m 

p, pd = specfun.clpmn(mp, n, real(z), imag(z), type) 

if (m < 0): 

p = p * fixarr 

pd = pd * fixarr 

return p, pd 

 

 

def lqmn(m, n, z): 

"""Sequence of associated Legendre functions of the second kind. 

 

Computes the associated Legendre function of the second kind of order m and 

degree n, ``Qmn(z)`` = :math:`Q_n^m(z)`, and its derivative, ``Qmn'(z)``. 

Returns two arrays of size ``(m+1, n+1)`` containing ``Qmn(z)`` and 

``Qmn'(z)`` for all orders from ``0..m`` and degrees from ``0..n``. 

 

Parameters 

---------- 

m : int 

``|m| <= n``; the order of the Legendre function. 

n : int 

where ``n >= 0``; the degree of the Legendre function. Often 

called ``l`` (lower case L) in descriptions of the associated 

Legendre function 

z : complex 

Input value. 

 

Returns 

------- 

Qmn_z : (m+1, n+1) array 

Values for all orders 0..m and degrees 0..n 

Qmn_d_z : (m+1, n+1) array 

Derivatives for all orders 0..m and degrees 0..n 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

if not isscalar(m) or (m < 0): 

raise ValueError("m must be a non-negative integer.") 

if not isscalar(n) or (n < 0): 

raise ValueError("n must be a non-negative integer.") 

if not isscalar(z): 

raise ValueError("z must be scalar.") 

m = int(m) 

n = int(n) 

 

# Ensure neither m nor n == 0 

mm = max(1, m) 

nn = max(1, n) 

 

if iscomplex(z): 

q, qd = specfun.clqmn(mm, nn, z) 

else: 

q, qd = specfun.lqmn(mm, nn, z) 

return q[:(m+1), :(n+1)], qd[:(m+1), :(n+1)] 

 

 

def bernoulli(n): 

"""Bernoulli numbers B0..Bn (inclusive). 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

if not isscalar(n) or (n < 0): 

raise ValueError("n must be a non-negative integer.") 

n = int(n) 

if (n < 2): 

n1 = 2 

else: 

n1 = n 

return specfun.bernob(int(n1))[:(n+1)] 

 

 

def euler(n): 

"""Euler numbers E(0), E(1), ..., E(n). 

 

The Euler numbers [1]_ are also known as the secant numbers. 

 

Because ``euler(n)`` returns floating point values, it does not give 

exact values for large `n`. The first inexact value is E(22). 

 

Parameters 

---------- 

n : int 

The highest index of the Euler number to be returned. 

 

Returns 

------- 

ndarray 

The Euler numbers [E(0), E(1), ..., E(n)]. 

The odd Euler numbers, which are all zero, are included. 

 

References 

---------- 

.. [1] Sequence A122045, The On-Line Encyclopedia of Integer Sequences, 

https://oeis.org/A122045 

.. [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.html 

 

Examples 

-------- 

>>> from scipy.special import euler 

>>> euler(6) 

array([ 1., 0., -1., 0., 5., 0., -61.]) 

 

>>> euler(13).astype(np.int64) 

array([ 1, 0, -1, 0, 5, 0, -61, 

0, 1385, 0, -50521, 0, 2702765, 0]) 

 

>>> euler(22)[-1] # Exact value of E(22) is -69348874393137901. 

-69348874393137976.0 

 

""" 

if not isscalar(n) or (n < 0): 

raise ValueError("n must be a non-negative integer.") 

n = int(n) 

if (n < 2): 

n1 = 2 

else: 

n1 = n 

return specfun.eulerb(n1)[:(n+1)] 

 

 

def lpn(n, z): 

"""Legendre function of the first kind. 

 

Compute sequence of Legendre functions of the first kind (polynomials), 

Pn(z) and derivatives for all degrees from 0 to n (inclusive). 

 

See also special.legendre for polynomial class. 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

if not (isscalar(n) and isscalar(z)): 

raise ValueError("arguments must be scalars.") 

n = _nonneg_int_or_fail(n, 'n', strict=False) 

if (n < 1): 

n1 = 1 

else: 

n1 = n 

if iscomplex(z): 

pn, pd = specfun.clpn(n1, z) 

else: 

pn, pd = specfun.lpn(n1, z) 

return pn[:(n+1)], pd[:(n+1)] 

 

 

def lqn(n, z): 

"""Legendre function of the second kind. 

 

Compute sequence of Legendre functions of the second kind, Qn(z) and 

derivatives for all degrees from 0 to n (inclusive). 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

if not (isscalar(n) and isscalar(z)): 

raise ValueError("arguments must be scalars.") 

n = _nonneg_int_or_fail(n, 'n', strict=False) 

if (n < 1): 

n1 = 1 

else: 

n1 = n 

if iscomplex(z): 

qn, qd = specfun.clqn(n1, z) 

else: 

qn, qd = specfun.lqnb(n1, z) 

return qn[:(n+1)], qd[:(n+1)] 

 

 

def ai_zeros(nt): 

""" 

Compute `nt` zeros and values of the Airy function Ai and its derivative. 

 

Computes the first `nt` zeros, `a`, of the Airy function Ai(x); 

first `nt` zeros, `ap`, of the derivative of the Airy function Ai'(x); 

the corresponding values Ai(a'); 

and the corresponding values Ai'(a). 

 

Parameters 

---------- 

nt : int 

Number of zeros to compute 

 

Returns 

------- 

a : ndarray 

First `nt` zeros of Ai(x) 

ap : ndarray 

First `nt` zeros of Ai'(x) 

ai : ndarray 

Values of Ai(x) evaluated at first `nt` zeros of Ai'(x) 

aip : ndarray 

Values of Ai'(x) evaluated at first `nt` zeros of Ai(x) 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

kf = 1 

if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0): 

raise ValueError("nt must be a positive integer scalar.") 

return specfun.airyzo(nt, kf) 

 

 

def bi_zeros(nt): 

""" 

Compute `nt` zeros and values of the Airy function Bi and its derivative. 

 

Computes the first `nt` zeros, b, of the Airy function Bi(x); 

first `nt` zeros, b', of the derivative of the Airy function Bi'(x); 

the corresponding values Bi(b'); 

and the corresponding values Bi'(b). 

 

Parameters 

---------- 

nt : int 

Number of zeros to compute 

 

Returns 

------- 

b : ndarray 

First `nt` zeros of Bi(x) 

bp : ndarray 

First `nt` zeros of Bi'(x) 

bi : ndarray 

Values of Bi(x) evaluated at first `nt` zeros of Bi'(x) 

bip : ndarray 

Values of Bi'(x) evaluated at first `nt` zeros of Bi(x) 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

kf = 2 

if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0): 

raise ValueError("nt must be a positive integer scalar.") 

return specfun.airyzo(nt, kf) 

 

 

def lmbda(v, x): 

r"""Jahnke-Emden Lambda function, Lambdav(x). 

 

This function is defined as [2]_, 

 

.. math:: \Lambda_v(x) = \Gamma(v+1) \frac{J_v(x)}{(x/2)^v}, 

 

where :math:`\Gamma` is the gamma function and :math:`J_v` is the 

Bessel function of the first kind. 

 

Parameters 

---------- 

v : float 

Order of the Lambda function 

x : float 

Value at which to evaluate the function and derivatives 

 

Returns 

------- 

vl : ndarray 

Values of Lambda_vi(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v. 

dl : ndarray 

Derivatives Lambda_vi'(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v. 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

.. [2] Jahnke, E. and Emde, F. "Tables of Functions with Formulae and 

Curves" (4th ed.), Dover, 1945 

""" 

if not (isscalar(v) and isscalar(x)): 

raise ValueError("arguments must be scalars.") 

if (v < 0): 

raise ValueError("argument must be > 0.") 

n = int(v) 

v0 = v - n 

if (n < 1): 

n1 = 1 

else: 

n1 = n 

v1 = n1 + v0 

if (v != floor(v)): 

vm, vl, dl = specfun.lamv(v1, x) 

else: 

vm, vl, dl = specfun.lamn(v1, x) 

return vl[:(n+1)], dl[:(n+1)] 

 

 

def pbdv_seq(v, x): 

"""Parabolic cylinder functions Dv(x) and derivatives. 

 

Parameters 

---------- 

v : float 

Order of the parabolic cylinder function 

x : float 

Value at which to evaluate the function and derivatives 

 

Returns 

------- 

dv : ndarray 

Values of D_vi(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v. 

dp : ndarray 

Derivatives D_vi'(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v. 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996, chapter 13. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

if not (isscalar(v) and isscalar(x)): 

raise ValueError("arguments must be scalars.") 

n = int(v) 

v0 = v-n 

if (n < 1): 

n1 = 1 

else: 

n1 = n 

v1 = n1 + v0 

dv, dp, pdf, pdd = specfun.pbdv(v1, x) 

return dv[:n1+1], dp[:n1+1] 

 

 

def pbvv_seq(v, x): 

"""Parabolic cylinder functions Vv(x) and derivatives. 

 

Parameters 

---------- 

v : float 

Order of the parabolic cylinder function 

x : float 

Value at which to evaluate the function and derivatives 

 

Returns 

------- 

dv : ndarray 

Values of V_vi(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v. 

dp : ndarray 

Derivatives V_vi'(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v. 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996, chapter 13. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

if not (isscalar(v) and isscalar(x)): 

raise ValueError("arguments must be scalars.") 

n = int(v) 

v0 = v-n 

if (n <= 1): 

n1 = 1 

else: 

n1 = n 

v1 = n1 + v0 

dv, dp, pdf, pdd = specfun.pbvv(v1, x) 

return dv[:n1+1], dp[:n1+1] 

 

 

def pbdn_seq(n, z): 

"""Parabolic cylinder functions Dn(z) and derivatives. 

 

Parameters 

---------- 

n : int 

Order of the parabolic cylinder function 

z : complex 

Value at which to evaluate the function and derivatives 

 

Returns 

------- 

dv : ndarray 

Values of D_i(z), for i=0, ..., i=n. 

dp : ndarray 

Derivatives D_i'(z), for i=0, ..., i=n. 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996, chapter 13. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

if not (isscalar(n) and isscalar(z)): 

raise ValueError("arguments must be scalars.") 

if (floor(n) != n): 

raise ValueError("n must be an integer.") 

if (abs(n) <= 1): 

n1 = 1 

else: 

n1 = n 

cpb, cpd = specfun.cpbdn(n1, z) 

return cpb[:n1+1], cpd[:n1+1] 

 

 

def ber_zeros(nt): 

"""Compute nt zeros of the Kelvin function ber(x). 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0): 

raise ValueError("nt must be positive integer scalar.") 

return specfun.klvnzo(nt, 1) 

 

 

def bei_zeros(nt): 

"""Compute nt zeros of the Kelvin function bei(x). 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0): 

raise ValueError("nt must be positive integer scalar.") 

return specfun.klvnzo(nt, 2) 

 

 

def ker_zeros(nt): 

"""Compute nt zeros of the Kelvin function ker(x). 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0): 

raise ValueError("nt must be positive integer scalar.") 

return specfun.klvnzo(nt, 3) 

 

 

def kei_zeros(nt): 

"""Compute nt zeros of the Kelvin function kei(x). 

""" 

if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0): 

raise ValueError("nt must be positive integer scalar.") 

return specfun.klvnzo(nt, 4) 

 

 

def berp_zeros(nt): 

"""Compute nt zeros of the Kelvin function ber'(x). 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0): 

raise ValueError("nt must be positive integer scalar.") 

return specfun.klvnzo(nt, 5) 

 

 

def beip_zeros(nt): 

"""Compute nt zeros of the Kelvin function bei'(x). 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0): 

raise ValueError("nt must be positive integer scalar.") 

return specfun.klvnzo(nt, 6) 

 

 

def kerp_zeros(nt): 

"""Compute nt zeros of the Kelvin function ker'(x). 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0): 

raise ValueError("nt must be positive integer scalar.") 

return specfun.klvnzo(nt, 7) 

 

 

def keip_zeros(nt): 

"""Compute nt zeros of the Kelvin function kei'(x). 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0): 

raise ValueError("nt must be positive integer scalar.") 

return specfun.klvnzo(nt, 8) 

 

 

def kelvin_zeros(nt): 

"""Compute nt zeros of all Kelvin functions. 

 

Returned in a length-8 tuple of arrays of length nt. The tuple contains 

the arrays of zeros of (ber, bei, ker, kei, ber', bei', ker', kei'). 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0): 

raise ValueError("nt must be positive integer scalar.") 

return (specfun.klvnzo(nt, 1), 

specfun.klvnzo(nt, 2), 

specfun.klvnzo(nt, 3), 

specfun.klvnzo(nt, 4), 

specfun.klvnzo(nt, 5), 

specfun.klvnzo(nt, 6), 

specfun.klvnzo(nt, 7), 

specfun.klvnzo(nt, 8)) 

 

 

def pro_cv_seq(m, n, c): 

"""Characteristic values for prolate spheroidal wave functions. 

 

Compute a sequence of characteristic values for the prolate 

spheroidal wave functions for mode m and n'=m..n and spheroidal 

parameter c. 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

if not (isscalar(m) and isscalar(n) and isscalar(c)): 

raise ValueError("Arguments must be scalars.") 

if (n != floor(n)) or (m != floor(m)): 

raise ValueError("Modes must be integers.") 

if (n-m > 199): 

raise ValueError("Difference between n and m is too large.") 

maxL = n-m+1 

return specfun.segv(m, n, c, 1)[1][:maxL] 

 

 

def obl_cv_seq(m, n, c): 

"""Characteristic values for oblate spheroidal wave functions. 

 

Compute a sequence of characteristic values for the oblate 

spheroidal wave functions for mode m and n'=m..n and spheroidal 

parameter c. 

 

References 

---------- 

.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special 

Functions", John Wiley and Sons, 1996. 

https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html 

 

""" 

if not (isscalar(m) and isscalar(n) and isscalar(c)): 

raise ValueError("Arguments must be scalars.") 

if (n != floor(n)) or (m != floor(m)): 

raise ValueError("Modes must be integers.") 

if (n-m > 199): 

raise ValueError("Difference between n and m is too large.") 

maxL = n-m+1 

return specfun.segv(m, n, c, -1)[1][:maxL] 

 

 

def ellipk(m): 

r"""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. 

 

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 

 

References 

---------- 

.. [1] Milton Abramowitz and Irene A. Stegun, eds. 

Handbook of Mathematical Functions with Formulas, 

Graphs, and Mathematical Tables. New York: Dover, 1972. 

 

""" 

return ellipkm1(1 - asarray(m)) 

 

 

def comb(N, k, exact=False, repetition=False): 

"""The number of combinations of N things taken k at a time. 

 

This is often expressed as "N choose k". 

 

Parameters 

---------- 

N : int, ndarray 

Number of things. 

k : int, ndarray 

Number of elements taken. 

exact : bool, optional 

If `exact` is False, then floating point precision is used, otherwise 

exact long integer is computed. 

repetition : bool, optional 

If `repetition` is True, then the number of combinations with 

repetition is computed. 

 

Returns 

------- 

val : int, float, ndarray 

The total number of combinations. 

 

See Also 

-------- 

binom : Binomial coefficient ufunc 

 

Notes 

----- 

- Array arguments accepted only for exact=False case. 

- If k > N, N < 0, or k < 0, then a 0 is returned. 

 

Examples 

-------- 

>>> from scipy.special import comb 

>>> k = np.array([3, 4]) 

>>> n = np.array([10, 10]) 

>>> comb(n, k, exact=False) 

array([ 120., 210.]) 

>>> comb(10, 3, exact=True) 

120L 

>>> comb(10, 3, exact=True, repetition=True) 

220L 

 

""" 

if repetition: 

return comb(N + k - 1, k, exact) 

if exact: 

return _comb_int(N, k) 

else: 

k, N = asarray(k), asarray(N) 

cond = (k <= N) & (N >= 0) & (k >= 0) 

vals = binom(N, k) 

if isinstance(vals, np.ndarray): 

vals[~cond] = 0 

elif not cond: 

vals = np.float64(0) 

return vals 

 

 

def perm(N, k, exact=False): 

"""Permutations of N things taken k at a time, i.e., k-permutations of N. 

 

It's also known as "partial permutations". 

 

Parameters 

---------- 

N : int, ndarray 

Number of things. 

k : int, ndarray 

Number of elements taken. 

exact : bool, optional 

If `exact` is False, then floating point precision is used, otherwise 

exact long integer is computed. 

 

Returns 

------- 

val : int, ndarray 

The number of k-permutations of N. 

 

Notes 

----- 

- Array arguments accepted only for exact=False case. 

- If k > N, N < 0, or k < 0, then a 0 is returned. 

 

Examples 

-------- 

>>> from scipy.special import perm 

>>> k = np.array([3, 4]) 

>>> n = np.array([10, 10]) 

>>> perm(n, k) 

array([ 720., 5040.]) 

>>> perm(10, 3, exact=True) 

720 

 

""" 

if exact: 

if (k > N) or (N < 0) or (k < 0): 

return 0 

val = 1 

for i in xrange(N - k + 1, N + 1): 

val *= i 

return val 

else: 

k, N = asarray(k), asarray(N) 

cond = (k <= N) & (N >= 0) & (k >= 0) 

vals = poch(N - k + 1, k) 

if isinstance(vals, np.ndarray): 

vals[~cond] = 0 

elif not cond: 

vals = np.float64(0) 

return vals 

 

 

# http://stackoverflow.com/a/16327037/125507 

def _range_prod(lo, hi): 

""" 

Product of a range of numbers. 

 

Returns the product of 

lo * (lo+1) * (lo+2) * ... * (hi-2) * (hi-1) * hi 

= hi! / (lo-1)! 

 

Breaks into smaller products first for speed: 

_range_prod(2, 9) = ((2*3)*(4*5))*((6*7)*(8*9)) 

""" 

if lo + 1 < hi: 

mid = (hi + lo) // 2 

return _range_prod(lo, mid) * _range_prod(mid + 1, hi) 

if lo == hi: 

return lo 

return lo * hi 

 

 

def factorial(n, exact=False): 

""" 

The factorial of a number or array of numbers. 

 

The factorial of non-negative integer `n` is the product of all 

positive integers less than or equal to `n`:: 

 

n! = n * (n - 1) * (n - 2) * ... * 1 

 

Parameters 

---------- 

n : int or array_like of ints 

Input values. If ``n < 0``, the return value is 0. 

exact : bool, optional 

If True, calculate the answer exactly using long integer arithmetic. 

If False, result is approximated in floating point rapidly using the 

`gamma` function. 

Default is False. 

 

Returns 

------- 

nf : float or int or ndarray 

Factorial of `n`, as integer or float depending on `exact`. 

 

Notes 

----- 

For arrays with ``exact=True``, the factorial is computed only once, for 

the largest input, with each other result computed in the process. 

The output dtype is increased to ``int64`` or ``object`` if necessary. 

 

With ``exact=False`` the factorial is approximated using the gamma 

function: 

 

.. math:: n! = \\Gamma(n+1) 

 

Examples 

-------- 

>>> from scipy.special import factorial 

>>> arr = np.array([3, 4, 5]) 

>>> factorial(arr, exact=False) 

array([ 6., 24., 120.]) 

>>> factorial(arr, exact=True) 

array([ 6, 24, 120]) 

>>> factorial(5, exact=True) 

120L 

 

""" 

if exact: 

if np.ndim(n) == 0: 

return 0 if n < 0 else math.factorial(n) 

else: 

n = asarray(n) 

un = np.unique(n).astype(object) 

 

# Convert to object array of long ints if np.int can't handle size 

if un[-1] > 20: 

dt = object 

elif un[-1] > 12: 

dt = np.int64 

else: 

dt = np.int 

 

out = np.empty_like(n, dtype=dt) 

 

# Handle invalid/trivial values 

un = un[un > 1] 

out[n < 2] = 1 

out[n < 0] = 0 

 

# Calculate products of each range of numbers 

if un.size: 

val = math.factorial(un[0]) 

out[n == un[0]] = val 

for i in xrange(len(un) - 1): 

prev = un[i] + 1 

current = un[i + 1] 

val *= _range_prod(prev, current) 

out[n == current] = val 

return out 

else: 

n = asarray(n) 

vals = gamma(n + 1) 

return where(n >= 0, vals, 0) 

 

 

def factorial2(n, exact=False): 

"""Double factorial. 

 

This is the factorial with every second value skipped. E.g., ``7!! = 7 * 5 

* 3 * 1``. It can be approximated numerically as:: 

 

n!! = special.gamma(n/2+1)*2**((m+1)/2)/sqrt(pi) n odd 

= 2**(n/2) * (n/2)! n even 

 

Parameters 

---------- 

n : int or array_like 

Calculate ``n!!``. Arrays are only supported with `exact` set 

to False. If ``n < 0``, the return value is 0. 

exact : bool, optional 

The result can be approximated rapidly using the gamma-formula 

above (default). If `exact` is set to True, calculate the 

answer exactly using integer arithmetic. 

 

Returns 

------- 

nff : float or int 

Double factorial of `n`, as an int or a float depending on 

`exact`. 

 

Examples 

-------- 

>>> from scipy.special import factorial2 

>>> factorial2(7, exact=False) 

array(105.00000000000001) 

>>> factorial2(7, exact=True) 

105L 

 

""" 

if exact: 

if n < -1: 

return 0 

if n <= 0: 

return 1 

val = 1 

for k in xrange(n, 0, -2): 

val *= k 

return val 

else: 

n = asarray(n) 

vals = zeros(n.shape, 'd') 

cond1 = (n % 2) & (n >= -1) 

cond2 = (1-(n % 2)) & (n >= -1) 

oddn = extract(cond1, n) 

evenn = extract(cond2, n) 

nd2o = oddn / 2.0 

nd2e = evenn / 2.0 

place(vals, cond1, gamma(nd2o + 1) / sqrt(pi) * pow(2.0, nd2o + 0.5)) 

place(vals, cond2, gamma(nd2e + 1) * pow(2.0, nd2e)) 

return vals 

 

 

def factorialk(n, k, exact=True): 

"""Multifactorial of n of order k, n(!!...!). 

 

This is the multifactorial of n skipping k values. For example, 

 

factorialk(17, 4) = 17!!!! = 17 * 13 * 9 * 5 * 1 

 

In particular, for any integer ``n``, we have 

 

factorialk(n, 1) = factorial(n) 

 

factorialk(n, 2) = factorial2(n) 

 

Parameters 

---------- 

n : int 

Calculate multifactorial. If `n` < 0, the return value is 0. 

k : int 

Order of multifactorial. 

exact : bool, optional 

If exact is set to True, calculate the answer exactly using 

integer arithmetic. 

 

Returns 

------- 

val : int 

Multifactorial of `n`. 

 

Raises 

------ 

NotImplementedError 

Raises when exact is False 

 

Examples 

-------- 

>>> from scipy.special import factorialk 

>>> factorialk(5, 1, exact=True) 

120L 

>>> factorialk(5, 3, exact=True) 

10L 

 

""" 

if exact: 

if n < 1-k: 

return 0 

if n <= 0: 

return 1 

val = 1 

for j in xrange(n, 0, -k): 

val = val*j 

return val 

else: 

raise NotImplementedError 

 

 

def zeta(x, q=None, out=None): 

r""" 

Riemann or Hurwitz zeta function. 

 

Parameters 

---------- 

x : array_like of float 

Input data, must be real 

q : array_like of float, optional 

Input data, must be real. Defaults to Riemann zeta. 

out : ndarray, optional 

Output array for the computed values. 

 

Returns 

------- 

out : array_like 

Values of zeta(x). 

 

Notes 

----- 

The two-argument version is the Hurwitz zeta function: 

 

.. math:: \zeta(x, q) = \sum_{k=0}^{\infty} \frac{1}{(k + q)^x}, 

 

Riemann zeta function corresponds to ``q = 1``. 

 

See Also 

-------- 

zetac 

 

Examples 

-------- 

>>> from scipy.special import zeta, polygamma, factorial 

 

Some specific values: 

 

>>> zeta(2), np.pi**2/6 

(1.6449340668482266, 1.6449340668482264) 

 

>>> zeta(4), np.pi**4/90 

(1.0823232337111381, 1.082323233711138) 

 

Relation to the `polygamma` function: 

 

>>> m = 3 

>>> x = 1.25 

>>> polygamma(m, x) 

array(2.782144009188397) 

>>> (-1)**(m+1) * factorial(m) * zeta(m+1, x) 

2.7821440091883969 

 

""" 

if q is None: 

q = 1 

return _zeta(x, q, out)