# # Author: Travis Oliphant, 2002 # import operator import numpy as np import math import warnings from collections import defaultdict from heapq import heapify, heappop from numpy import (pi, asarray, floor, isscalar, sqrt, where, sin, place, issubdtype, extract, inexact, nan, zeros, sinc) from . import _ufuncs from ._ufuncs import (mathieu_a, mathieu_b, iv, jv, gamma, psi, hankel1, hankel2, yv, kv, poch, binom, _stirling2_inexact) from ._gufuncs import (_lpn, _lpmn, _clpmn, _lqn, _lqmn, _rctj, _rcty, _sph_harm_all as _sph_harm_all_gufunc) from . import _specfun from ._comb import _comb_int __all__ = [ 'ai_zeros', 'assoc_laguerre', 'bei_zeros', 'beip_zeros', 'ber_zeros', 'bernoulli', 'berp_zeros', 'bi_zeros', 'clpmn', 'comb', 'digamma', 'diric', 'erf_zeros', 'euler', 'factorial', 'factorial2', 'factorialk', 'fresnel_zeros', 'fresnelc_zeros', 'fresnels_zeros', 'h1vp', 'h2vp', 'ivp', 'jn_zeros', 'jnjnp_zeros', 'jnp_zeros', 'jnyn_zeros', 'jvp', 'kei_zeros', 'keip_zeros', 'kelvin_zeros', 'ker_zeros', 'kerp_zeros', 'kvp', 'lmbda', 'lpmn', 'lpn', 'lqmn', 'lqn', 'mathieu_even_coef', 'mathieu_odd_coef', 'obl_cv_seq', 'pbdn_seq', 'pbdv_seq', 'pbvv_seq', 'perm', 'polygamma', 'pro_cv_seq', 'riccati_jn', 'riccati_yn', 'sinc', 'stirling2', 'y0_zeros', 'y1_zeros', 'y1p_zeros', 'yn_zeros', 'ynp_zeros', 'yvp', 'zeta' ] # mapping k to last n such that factorialk(n, k) < np.iinfo(np.int64).max _FACTORIALK_LIMITS_64BITS = {1: 20, 2: 33, 3: 44, 4: 54, 5: 65, 6: 74, 7: 84, 8: 93, 9: 101} # mapping k to last n such that factorialk(n, k) < np.iinfo(np.int32).max _FACTORIALK_LIMITS_32BITS = {1: 12, 2: 19, 3: 25, 4: 31, 5: 37, 6: 43, 7: 47, 8: 51, 9: 56} 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__(f"{var_name} must be a non-negative integer") from err 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 -------- >>> import numpy as np >>> 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/f77_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. The zeros are returned in ascending order. Parameters ---------- n : int Order of the Bessel functions nt : int Number (<=1200) of zeros to compute Returns ------- Jn : ndarray First `nt` zeros of Jn Jnp : ndarray First `nt` zeros of Jn' Yn : ndarray First `nt` zeros of Yn Ynp : ndarray First `nt` zeros of Yn' See Also -------- jn_zeros, jnp_zeros, yn_zeros, ynp_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/f77_src/special_functions/special_functions.html Examples -------- Compute the first three roots of :math:`J_1`, :math:`J_1'`, :math:`Y_1` and :math:`Y_1'`. >>> from scipy.special import jnyn_zeros >>> jn_roots, jnp_roots, yn_roots, ynp_roots = jnyn_zeros(1, 3) >>> jn_roots, yn_roots (array([ 3.83170597, 7.01558667, 10.17346814]), array([2.19714133, 5.42968104, 8.59600587])) Plot :math:`J_1`, :math:`J_1'`, :math:`Y_1`, :math:`Y_1'` and their roots. >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.special import jnyn_zeros, jvp, jn, yvp, yn >>> jn_roots, jnp_roots, yn_roots, ynp_roots = jnyn_zeros(1, 3) >>> fig, ax = plt.subplots() >>> xmax= 11 >>> x = np.linspace(0, xmax) >>> x[0] += 1e-15 >>> ax.plot(x, jn(1, x), label=r"$J_1$", c='r') >>> ax.plot(x, jvp(1, x, 1), label=r"$J_1'$", c='b') >>> ax.plot(x, yn(1, x), label=r"$Y_1$", c='y') >>> ax.plot(x, yvp(1, x, 1), label=r"$Y_1'$", c='c') >>> zeros = np.zeros((3, )) >>> ax.scatter(jn_roots, zeros, s=30, c='r', zorder=5, ... label=r"$J_1$ roots") >>> ax.scatter(jnp_roots, zeros, s=30, c='b', zorder=5, ... label=r"$J_1'$ roots") >>> ax.scatter(yn_roots, zeros, s=30, c='y', zorder=5, ... label=r"$Y_1$ roots") >>> ax.scatter(ynp_roots, zeros, s=30, c='c', zorder=5, ... label=r"$Y_1'$ roots") >>> ax.hlines(0, 0, xmax, color='k') >>> ax.set_ylim(-0.6, 0.6) >>> ax.set_xlim(0, xmax) >>> ax.legend(ncol=2, bbox_to_anchor=(1., 0.75)) >>> plt.tight_layout() >>> plt.show() """ 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): r"""Compute zeros of integer-order Bessel functions Jn. Compute `nt` zeros of the Bessel functions :math:`J_n(x)` on the interval :math:`(0, \infty)`. The zeros are returned in ascending order. Note that this interval excludes the zero at :math:`x = 0` that exists for :math:`n > 0`. Parameters ---------- n : int Order of Bessel function nt : int Number of zeros to return Returns ------- ndarray First `nt` zeros of the Bessel function. See Also -------- jv: Real-order Bessel functions of the first kind jnp_zeros: Zeros of :math:`Jn'` References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996, chapter 5. https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html Examples -------- Compute the first four positive roots of :math:`J_3`. >>> from scipy.special import jn_zeros >>> jn_zeros(3, 4) array([ 6.3801619 , 9.76102313, 13.01520072, 16.22346616]) Plot :math:`J_3` and its first four positive roots. Note that the root located at 0 is not returned by `jn_zeros`. >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.special import jn, jn_zeros >>> j3_roots = jn_zeros(3, 4) >>> xmax = 18 >>> xmin = -1 >>> x = np.linspace(xmin, xmax, 500) >>> fig, ax = plt.subplots() >>> ax.plot(x, jn(3, x), label=r'$J_3$') >>> ax.scatter(j3_roots, np.zeros((4, )), s=30, c='r', ... label=r"$J_3$_Zeros", zorder=5) >>> ax.scatter(0, 0, s=30, c='k', ... label=r"Root at 0", zorder=5) >>> ax.hlines(0, 0, xmax, color='k') >>> ax.set_xlim(xmin, xmax) >>> plt.legend() >>> plt.show() """ return jnyn_zeros(n, nt)[0] def jnp_zeros(n, nt): r"""Compute zeros of integer-order Bessel function derivatives Jn'. Compute `nt` zeros of the functions :math:`J_n'(x)` on the interval :math:`(0, \infty)`. The zeros are returned in ascending order. Note that this interval excludes the zero at :math:`x = 0` that exists for :math:`n > 1`. Parameters ---------- n : int Order of Bessel function nt : int Number of zeros to return Returns ------- ndarray First `nt` zeros of the Bessel function. See Also -------- jvp: Derivatives of integer-order Bessel functions of the first kind jv: Float-order Bessel functions of the first kind References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996, chapter 5. https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html Examples -------- Compute the first four roots of :math:`J_2'`. >>> from scipy.special import jnp_zeros >>> jnp_zeros(2, 4) array([ 3.05423693, 6.70613319, 9.96946782, 13.17037086]) As `jnp_zeros` yields the roots of :math:`J_n'`, it can be used to compute the locations of the peaks of :math:`J_n`. Plot :math:`J_2`, :math:`J_2'` and the locations of the roots of :math:`J_2'`. >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.special import jn, jnp_zeros, jvp >>> j2_roots = jnp_zeros(2, 4) >>> xmax = 15 >>> x = np.linspace(0, xmax, 500) >>> fig, ax = plt.subplots() >>> ax.plot(x, jn(2, x), label=r'$J_2$') >>> ax.plot(x, jvp(2, x, 1), label=r"$J_2'$") >>> ax.hlines(0, 0, xmax, color='k') >>> ax.scatter(j2_roots, np.zeros((4, )), s=30, c='r', ... label=r"Roots of $J_2'$", zorder=5) >>> ax.set_ylim(-0.4, 0.8) >>> ax.set_xlim(0, xmax) >>> plt.legend() >>> plt.show() """ return jnyn_zeros(n, nt)[1] def yn_zeros(n, nt): r"""Compute zeros of integer-order Bessel function Yn(x). Compute `nt` zeros of the functions :math:`Y_n(x)` on the interval :math:`(0, \infty)`. The zeros are returned in ascending order. Parameters ---------- n : int Order of Bessel function nt : int Number of zeros to return Returns ------- ndarray First `nt` zeros of the Bessel function. See Also -------- yn: Bessel function of the second kind for integer order yv: Bessel function of the second kind for real order References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996, chapter 5. https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html Examples -------- Compute the first four roots of :math:`Y_2`. >>> from scipy.special import yn_zeros >>> yn_zeros(2, 4) array([ 3.38424177, 6.79380751, 10.02347798, 13.20998671]) Plot :math:`Y_2` and its first four roots. >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.special import yn, yn_zeros >>> xmin = 2 >>> xmax = 15 >>> x = np.linspace(xmin, xmax, 500) >>> fig, ax = plt.subplots() >>> ax.hlines(0, xmin, xmax, color='k') >>> ax.plot(x, yn(2, x), label=r'$Y_2$') >>> ax.scatter(yn_zeros(2, 4), np.zeros((4, )), s=30, c='r', ... label='Roots', zorder=5) >>> ax.set_ylim(-0.4, 0.4) >>> ax.set_xlim(xmin, xmax) >>> plt.legend() >>> plt.show() """ return jnyn_zeros(n, nt)[2] def ynp_zeros(n, nt): r"""Compute zeros of integer-order Bessel function derivatives Yn'(x). Compute `nt` zeros of the functions :math:`Y_n'(x)` on the interval :math:`(0, \infty)`. The zeros are returned in ascending order. Parameters ---------- n : int Order of Bessel function nt : int Number of zeros to return Returns ------- ndarray First `nt` zeros of the Bessel derivative function. See Also -------- yvp References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996, chapter 5. https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html Examples -------- Compute the first four roots of the first derivative of the Bessel function of second kind for order 0 :math:`Y_0'`. >>> from scipy.special import ynp_zeros >>> ynp_zeros(0, 4) array([ 2.19714133, 5.42968104, 8.59600587, 11.74915483]) Plot :math:`Y_0`, :math:`Y_0'` and confirm visually that the roots of :math:`Y_0'` are located at local extrema of :math:`Y_0`. >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.special import yn, ynp_zeros, yvp >>> zeros = ynp_zeros(0, 4) >>> xmax = 13 >>> x = np.linspace(0, xmax, 500) >>> fig, ax = plt.subplots() >>> ax.plot(x, yn(0, x), label=r'$Y_0$') >>> ax.plot(x, yvp(0, x, 1), label=r"$Y_0'$") >>> ax.scatter(zeros, np.zeros((4, )), s=30, c='r', ... label=r"Roots of $Y_0'$", zorder=5) >>> for root in zeros: ... y0_extremum = yn(0, root) ... lower = min(0, y0_extremum) ... upper = max(0, y0_extremum) ... ax.vlines(root, lower, upper, color='r') >>> ax.hlines(0, 0, xmax, color='k') >>> ax.set_ylim(-0.6, 0.6) >>> ax.set_xlim(0, xmax) >>> plt.legend() >>> plt.show() """ 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/f77_src/special_functions/special_functions.html Examples -------- Compute the first 4 real roots and the derivatives at the roots of :math:`Y_0`: >>> import numpy as np >>> from scipy.special import y0_zeros >>> zeros, grads = y0_zeros(4) >>> with np.printoptions(precision=5): ... print(f"Roots: {zeros}") ... print(f"Gradients: {grads}") Roots: [ 0.89358+0.j 3.95768+0.j 7.08605+0.j 10.22235+0.j] Gradients: [-0.87942+0.j 0.40254+0.j -0.3001 +0.j 0.2497 +0.j] Plot the real part of :math:`Y_0` and the first four computed roots. >>> import matplotlib.pyplot as plt >>> from scipy.special import y0 >>> xmin = 0 >>> xmax = 11 >>> x = np.linspace(xmin, xmax, 500) >>> fig, ax = plt.subplots() >>> ax.hlines(0, xmin, xmax, color='k') >>> ax.plot(x, y0(x), label=r'$Y_0$') >>> zeros, grads = y0_zeros(4) >>> ax.scatter(zeros.real, np.zeros((4, )), s=30, c='r', ... label=r'$Y_0$_zeros', zorder=5) >>> ax.set_ylim(-0.5, 0.6) >>> ax.set_xlim(xmin, xmax) >>> plt.legend(ncol=2) >>> plt.show() Compute the first 4 complex roots and the derivatives at the roots of :math:`Y_0` by setting ``complex=True``: >>> y0_zeros(4, True) (array([ -2.40301663+0.53988231j, -5.5198767 +0.54718001j, -8.6536724 +0.54841207j, -11.79151203+0.54881912j]), array([ 0.10074769-0.88196771j, -0.02924642+0.5871695j , 0.01490806-0.46945875j, -0.00937368+0.40230454j])) """ 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/f77_src/special_functions/special_functions.html Examples -------- Compute the first 4 real roots and the derivatives at the roots of :math:`Y_1`: >>> import numpy as np >>> from scipy.special import y1_zeros >>> zeros, grads = y1_zeros(4) >>> with np.printoptions(precision=5): ... print(f"Roots: {zeros}") ... print(f"Gradients: {grads}") Roots: [ 2.19714+0.j 5.42968+0.j 8.59601+0.j 11.74915+0.j] Gradients: [ 0.52079+0.j -0.34032+0.j 0.27146+0.j -0.23246+0.j] Extract the real parts: >>> realzeros = zeros.real >>> realzeros array([ 2.19714133, 5.42968104, 8.59600587, 11.74915483]) Plot :math:`Y_1` and the first four computed roots. >>> import matplotlib.pyplot as plt >>> from scipy.special import y1 >>> xmin = 0 >>> xmax = 13 >>> x = np.linspace(xmin, xmax, 500) >>> zeros, grads = y1_zeros(4) >>> fig, ax = plt.subplots() >>> ax.hlines(0, xmin, xmax, color='k') >>> ax.plot(x, y1(x), label=r'$Y_1$') >>> ax.scatter(zeros.real, np.zeros((4, )), s=30, c='r', ... label=r'$Y_1$_zeros', zorder=5) >>> ax.set_ylim(-0.5, 0.5) >>> ax.set_xlim(xmin, xmax) >>> plt.legend() >>> plt.show() Compute the first 4 complex roots and the derivatives at the roots of :math:`Y_1` by setting ``complex=True``: >>> y1_zeros(4, True) (array([ -0.50274327+0.78624371j, -3.83353519+0.56235654j, -7.01590368+0.55339305j, -10.17357383+0.55127339j]), array([-0.45952768+1.31710194j, 0.04830191-0.69251288j, -0.02012695+0.51864253j, 0.011614 -0.43203296j])) """ 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/f77_src/special_functions/special_functions.html Examples -------- Compute the first four roots of :math:`Y_1'` and the values of :math:`Y_1` at these roots. >>> import numpy as np >>> from scipy.special import y1p_zeros >>> y1grad_roots, y1_values = y1p_zeros(4) >>> with np.printoptions(precision=5): ... print(f"Y1' Roots: {y1grad_roots.real}") ... print(f"Y1 values: {y1_values.real}") Y1' Roots: [ 3.68302 6.9415 10.1234 13.28576] Y1 values: [ 0.41673 -0.30317 0.25091 -0.21897] `y1p_zeros` can be used to calculate the extremal points of :math:`Y_1` directly. Here we plot :math:`Y_1` and the first four extrema. >>> import matplotlib.pyplot as plt >>> from scipy.special import y1, yvp >>> y1_roots, y1_values_at_roots = y1p_zeros(4) >>> real_roots = y1_roots.real >>> xmax = 15 >>> x = np.linspace(0, xmax, 500) >>> x[0] += 1e-15 >>> fig, ax = plt.subplots() >>> ax.plot(x, y1(x), label=r'$Y_1$') >>> ax.plot(x, yvp(1, x, 1), label=r"$Y_1'$") >>> ax.scatter(real_roots, np.zeros((4, )), s=30, c='r', ... label=r"Roots of $Y_1'$", zorder=5) >>> ax.scatter(real_roots, y1_values_at_roots.real, s=30, c='k', ... label=r"Extrema of $Y_1$", zorder=5) >>> ax.hlines(0, 0, xmax, color='k') >>> ax.set_ylim(-0.5, 0.5) >>> ax.set_xlim(0, xmax) >>> ax.legend(ncol=2, bbox_to_anchor=(1., 0.75)) >>> plt.tight_layout() >>> plt.show() """ 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 range(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) def jvp(v, z, n=1): """Compute derivatives of Bessel functions of the first kind. Compute the nth derivative of the Bessel function `Jv` with respect to `z`. Parameters ---------- v : array_like or float Order of Bessel function z : complex Argument at which to evaluate the derivative; can be real or complex. n : int, default 1 Order of derivative. For 0 returns the Bessel function `jv` itself. Returns ------- scalar or ndarray Values of the derivative of the Bessel function. 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/f77_src/special_functions/special_functions.html .. [2] NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/10.6.E7 Examples -------- Compute the Bessel function of the first kind of order 0 and its first two derivatives at 1. >>> from scipy.special import jvp >>> jvp(0, 1, 0), jvp(0, 1, 1), jvp(0, 1, 2) (0.7651976865579666, -0.44005058574493355, -0.3251471008130331) Compute the first derivative of the Bessel function of the first kind for several orders at 1 by providing an array for `v`. >>> jvp([0, 1, 2], 1, 1) array([-0.44005059, 0.3251471 , 0.21024362]) Compute the first derivative of the Bessel function of the first kind of order 0 at several points by providing an array for `z`. >>> import numpy as np >>> points = np.array([0., 1.5, 3.]) >>> jvp(0, points, 1) array([-0. , -0.55793651, -0.33905896]) Plot the Bessel function of the first kind of order 1 and its first three derivatives. >>> import matplotlib.pyplot as plt >>> x = np.linspace(-10, 10, 1000) >>> fig, ax = plt.subplots() >>> ax.plot(x, jvp(1, x, 0), label=r"$J_1$") >>> ax.plot(x, jvp(1, x, 1), label=r"$J_1'$") >>> ax.plot(x, jvp(1, x, 2), label=r"$J_1''$") >>> ax.plot(x, jvp(1, x, 3), label=r"$J_1'''$") >>> plt.legend() >>> plt.show() """ 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 derivatives of Bessel functions of the second kind. Compute the nth derivative of the Bessel function `Yv` with respect to `z`. Parameters ---------- v : array_like of float Order of Bessel function z : complex Argument at which to evaluate the derivative n : int, default 1 Order of derivative. For 0 returns the BEssel function `yv` Returns ------- scalar or ndarray nth derivative of the Bessel function. See Also -------- yv : Bessel functions of the second kind 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/f77_src/special_functions/special_functions.html .. [2] NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/10.6.E7 Examples -------- Compute the Bessel function of the second kind of order 0 and its first two derivatives at 1. >>> from scipy.special import yvp >>> yvp(0, 1, 0), yvp(0, 1, 1), yvp(0, 1, 2) (0.088256964215677, 0.7812128213002889, -0.8694697855159659) Compute the first derivative of the Bessel function of the second kind for several orders at 1 by providing an array for `v`. >>> yvp([0, 1, 2], 1, 1) array([0.78121282, 0.86946979, 2.52015239]) Compute the first derivative of the Bessel function of the second kind of order 0 at several points by providing an array for `z`. >>> import numpy as np >>> points = np.array([0.5, 1.5, 3.]) >>> yvp(0, points, 1) array([ 1.47147239, 0.41230863, -0.32467442]) Plot the Bessel function of the second kind of order 1 and its first three derivatives. >>> import matplotlib.pyplot as plt >>> x = np.linspace(0, 5, 1000) >>> x[0] += 1e-15 >>> fig, ax = plt.subplots() >>> ax.plot(x, yvp(1, x, 0), label=r"$Y_1$") >>> ax.plot(x, yvp(1, x, 1), label=r"$Y_1'$") >>> ax.plot(x, yvp(1, x, 2), label=r"$Y_1''$") >>> ax.plot(x, yvp(1, x, 3), label=r"$Y_1'''$") >>> ax.set_ylim(-10, 10) >>> plt.legend() >>> plt.show() """ 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 derivatives 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, default 1 Order of derivative. For 0 returns the Bessel function `kv` itself. Returns ------- out : ndarray The results See Also -------- kv 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/f77_src/special_functions/special_functions.html .. [2] NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/10.29.E5 Examples -------- Compute the modified bessel function of the second kind of order 0 and its first two derivatives at 1. >>> from scipy.special import kvp >>> kvp(0, 1, 0), kvp(0, 1, 1), kvp(0, 1, 2) (0.42102443824070834, -0.6019072301972346, 1.0229316684379428) Compute the first derivative of the modified Bessel function of the second kind for several orders at 1 by providing an array for `v`. >>> kvp([0, 1, 2], 1, 1) array([-0.60190723, -1.02293167, -3.85158503]) Compute the first derivative of the modified Bessel function of the second kind of order 0 at several points by providing an array for `z`. >>> import numpy as np >>> points = np.array([0.5, 1.5, 3.]) >>> kvp(0, points, 1) array([-1.65644112, -0.2773878 , -0.04015643]) Plot the modified bessel function of the second kind and its first three derivatives. >>> import matplotlib.pyplot as plt >>> x = np.linspace(0, 5, 1000) >>> fig, ax = plt.subplots() >>> ax.plot(x, kvp(1, x, 0), label=r"$K_1$") >>> ax.plot(x, kvp(1, x, 1), label=r"$K_1'$") >>> ax.plot(x, kvp(1, x, 2), label=r"$K_1''$") >>> ax.plot(x, kvp(1, x, 3), label=r"$K_1'''$") >>> ax.set_ylim(-2.5, 2.5) >>> plt.legend() >>> plt.show() """ 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 derivatives of modified Bessel functions of the first kind. Compute the nth derivative of the modified Bessel function `Iv` with respect to `z`. Parameters ---------- v : array_like or float Order of Bessel function z : array_like Argument at which to evaluate the derivative; can be real or complex. n : int, default 1 Order of derivative. For 0, returns the Bessel function `iv` itself. Returns ------- scalar or ndarray nth derivative of the modified Bessel function. See Also -------- iv 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/f77_src/special_functions/special_functions.html .. [2] NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/10.29.E5 Examples -------- Compute the modified Bessel function of the first kind of order 0 and its first two derivatives at 1. >>> from scipy.special import ivp >>> ivp(0, 1, 0), ivp(0, 1, 1), ivp(0, 1, 2) (1.2660658777520084, 0.565159103992485, 0.7009067737595233) Compute the first derivative of the modified Bessel function of the first kind for several orders at 1 by providing an array for `v`. >>> ivp([0, 1, 2], 1, 1) array([0.5651591 , 0.70090677, 0.29366376]) Compute the first derivative of the modified Bessel function of the first kind of order 0 at several points by providing an array for `z`. >>> import numpy as np >>> points = np.array([0., 1.5, 3.]) >>> ivp(0, points, 1) array([0. , 0.98166643, 3.95337022]) Plot the modified Bessel function of the first kind of order 1 and its first three derivatives. >>> import matplotlib.pyplot as plt >>> x = np.linspace(-5, 5, 1000) >>> fig, ax = plt.subplots() >>> ax.plot(x, ivp(1, x, 0), label=r"$I_1$") >>> ax.plot(x, ivp(1, x, 1), label=r"$I_1'$") >>> ax.plot(x, ivp(1, x, 2), label=r"$I_1''$") >>> ax.plot(x, ivp(1, x, 3), label=r"$I_1'''$") >>> plt.legend() >>> plt.show() """ 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 derivatives of Hankel function H1v(z) with respect to `z`. Parameters ---------- v : array_like Order of Hankel function z : array_like Argument at which to evaluate the derivative. Can be real or complex. n : int, default 1 Order of derivative. For 0 returns the Hankel function `h1v` itself. Returns ------- scalar or ndarray Values of the derivative of the Hankel function. See Also -------- hankel1 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/f77_src/special_functions/special_functions.html .. [2] NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/10.6.E7 Examples -------- Compute the Hankel function of the first kind of order 0 and its first two derivatives at 1. >>> from scipy.special import h1vp >>> h1vp(0, 1, 0), h1vp(0, 1, 1), h1vp(0, 1, 2) ((0.7651976865579664+0.088256964215677j), (-0.44005058574493355+0.7812128213002889j), (-0.3251471008130329-0.8694697855159659j)) Compute the first derivative of the Hankel function of the first kind for several orders at 1 by providing an array for `v`. >>> h1vp([0, 1, 2], 1, 1) array([-0.44005059+0.78121282j, 0.3251471 +0.86946979j, 0.21024362+2.52015239j]) Compute the first derivative of the Hankel function of the first kind of order 0 at several points by providing an array for `z`. >>> import numpy as np >>> points = np.array([0.5, 1.5, 3.]) >>> h1vp(0, points, 1) array([-0.24226846+1.47147239j, -0.55793651+0.41230863j, -0.33905896-0.32467442j]) """ 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 derivatives of Hankel function H2v(z) with respect to `z`. Parameters ---------- v : array_like Order of Hankel function z : array_like Argument at which to evaluate the derivative. Can be real or complex. n : int, default 1 Order of derivative. For 0 returns the Hankel function `h2v` itself. Returns ------- scalar or ndarray Values of the derivative of the Hankel function. See Also -------- hankel2 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/f77_src/special_functions/special_functions.html .. [2] NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/10.6.E7 Examples -------- Compute the Hankel function of the second kind of order 0 and its first two derivatives at 1. >>> from scipy.special import h2vp >>> h2vp(0, 1, 0), h2vp(0, 1, 1), h2vp(0, 1, 2) ((0.7651976865579664-0.088256964215677j), (-0.44005058574493355-0.7812128213002889j), (-0.3251471008130329+0.8694697855159659j)) Compute the first derivative of the Hankel function of the second kind for several orders at 1 by providing an array for `v`. >>> h2vp([0, 1, 2], 1, 1) array([-0.44005059-0.78121282j, 0.3251471 -0.86946979j, 0.21024362-2.52015239j]) Compute the first derivative of the Hankel function of the second kind of order 0 at several points by providing an array for `z`. >>> import numpy as np >>> points = np.array([0.5, 1.5, 3.]) >>> h2vp(0, points, 1) array([-0.24226846-1.47147239j, -0.55793651-0.41230863j, -0.33905896+0.32467442j]) """ 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/f77_src/special_functions/special_functions.html .. [2] NIST Digital Library of Mathematical Functions. https://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 jn = np.empty((n1 + 1,), dtype = np.float64) jnp = np.empty_like(jn) _rctj(x, out = (jn, jnp)) 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/f77_src/special_functions/special_functions.html .. [2] NIST Digital Library of Mathematical Functions. https://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 yn = np.empty((n1 + 1,), dtype = np.float64) ynp = np.empty_like(yn) _rcty(x, out = (yn, ynp)) return yn[:(n+1)], ynp[:(n+1)] def erf_zeros(nt): """Compute the first nt zero in the first quadrant, ordered by absolute value. Zeros in the other quadrants can be obtained by using the symmetries erf(-z) = erf(z) and erf(conj(z)) = conj(erf(z)). Parameters ---------- nt : int The number of zeros to compute Returns ------- The locations of the zeros of erf : ndarray (complex) Complex values at which zeros of erf(z) References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html Examples -------- >>> from scipy import special >>> special.erf_zeros(1) array([1.45061616+1.880943j]) Check that erf is (close to) zero for the value returned by erf_zeros >>> special.erf(special.erf_zeros(1)) array([4.95159469e-14-1.16407394e-16j]) """ 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). Parameters ---------- nt : int Number of zeros to compute Returns ------- fresnelc_zeros: ndarray Zeros of the cosine Fresnel integral References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f77_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). Parameters ---------- nt : int Number of zeros to compute Returns ------- fresnels_zeros: ndarray Zeros of the sine Fresnel integral References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f77_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). Parameters ---------- nt : int Number of zeros to compute Returns ------- zeros_sine: ndarray Zeros of the sine Fresnel integral zeros_cosine : ndarray Zeros of the cosine Fresnel integral References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f77_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``. Parameters ---------- x : float or ndarray Points where to evaluate the Laguerre polynomial n : int Degree of the Laguerre polynomial k : int Order of the Laguerre polynomial Returns ------- assoc_laguerre: float or ndarray Associated laguerre polynomial values Notes ----- `assoc_laguerre` is a simple wrapper around `eval_genlaguerre`, with reversed argument order ``(x, n, k=0.0) --> (n, k, x)``. """ return _ufuncs.eval_genlaguerre(n, k, x) digamma = psi def polygamma(n, x): r"""Polygamma functions. Defined as :math:`\psi^{(n)}(x)` where :math:`\psi` is the `digamma` function. See [dlmf]_ for details. Parameters ---------- n : array_like The order of the derivative of the digamma function; must be integral x : array_like Real valued input Returns ------- ndarray Function results See Also -------- digamma References ---------- .. [dlmf] NIST, Digital Library of Mathematical Functions, https://dlmf.nist.gov/5.15 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/f77_src/special_functions/special_functions.html .. [2] NIST Digital Library of Mathematical Functions https://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: warnings.warn("Too many predicted coefficients.", RuntimeWarning, stacklevel=2) 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/f77_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: warnings.warn("Too many predicted coefficients.", RuntimeWarning, stacklevel=2) 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 : array_like 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/f77_src/special_functions/special_functions.html .. [2] NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/14.3 """ n = _nonneg_int_or_fail(n, 'n', strict=False) 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 np.iscomplexobj(z): raise ValueError("Argument must be real. Use clpmn instead.") m, n = int(m), int(n) # Convert to int to maintain backwards compatibility. if (m < 0): m_signbit = True m_abs = -m else: m_signbit = False m_abs = m z = np.asarray(z) if (not np.issubdtype(z.dtype, np.inexact)): z = z.astype(np.float64) p = np.empty((m_abs + 1, n + 1) + z.shape, dtype=np.float64) pd = np.empty_like(p) if (z.ndim == 0): _lpmn(z, m_signbit, out = (p, pd)) else: _lpmn(z, m_signbit, out = (np.moveaxis(p, (0, 1), (-2, -1)), np.moveaxis(pd, (0, 1), (-2, -1)))) # new axes must be last for the ufunc 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 : array_like, 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/f77_src/special_functions/special_functions.html .. [2] NIST Digital Library of Mathematical Functions https://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 (type == 2 or type == 3): raise ValueError("type must be either 2 or 3.") m, n = int(m), int(n) # Convert to int to maintain backwards compatibility. if (m < 0): mp = -m m_signbit = True else: mp = m m_signbit = False z = np.asarray(z) if (not np.issubdtype(z.dtype, np.inexact)): z = z.astype(np.complex128) p = np.empty((mp + 1, n + 1) + z.shape, dtype=np.complex128) pd = np.empty_like(p) if (z.ndim == 0): _clpmn(z, type, m_signbit, out = (p, pd)) else: _clpmn(z, type, m_signbit, out = (np.moveaxis(p, (0, 1), (-2, -1)), np.moveaxis(pd, (0, 1), (-2, -1)))) # new axes must be last for the ufunc 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 : array_like, 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/f77_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.") m, n = int(m), int(n) # Convert to int to maintain backwards compatibility. # Ensure neither m nor n == 0 mm = max(1, m) nn = max(1, n) z = np.asarray(z) if (not np.issubdtype(z.dtype, np.inexact)): z = z.astype(np.float64) if np.iscomplexobj(z): q = np.empty((mm + 1, nn + 1) + z.shape, dtype = np.complex128) else: q = np.empty((mm + 1, nn + 1) + z.shape, dtype = np.float64) qd = np.empty_like(q) if (z.ndim == 0): _lqmn(z, out = (q, qd)) else: _lqmn(z, out = (np.moveaxis(q, (0, 1), (-2, -1)), np.moveaxis(qd, (0, 1), (-2, -1)))) # new axes must be last for the ufunc return q[:(m+1), :(n+1)], qd[:(m+1), :(n+1)] def bernoulli(n): """Bernoulli numbers B0..Bn (inclusive). Parameters ---------- n : int Indicated the number of terms in the Bernoulli series to generate. Returns ------- ndarray The Bernoulli numbers ``[B(0), B(1), ..., B(n)]``. References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html .. [2] "Bernoulli number", Wikipedia, https://en.wikipedia.org/wiki/Bernoulli_number Examples -------- >>> import numpy as np >>> from scipy.special import bernoulli, zeta >>> bernoulli(4) array([ 1. , -0.5 , 0.16666667, 0. , -0.03333333]) The Wikipedia article ([2]_) points out the relationship between the Bernoulli numbers and the zeta function, ``B_n^+ = -n * zeta(1 - n)`` for ``n > 0``: >>> n = np.arange(1, 5) >>> -n * zeta(1 - n) array([ 0.5 , 0.16666667, -0. , -0.03333333]) Note that, in the notation used in the wikipedia article, `bernoulli` computes ``B_n^-`` (i.e. it used the convention that ``B_1`` is -1/2). The relation given above is for ``B_n^+``, so the sign of 0.5 does not match the output of ``bernoulli(4)``. """ 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/f77_src/special_functions/special_functions.html Examples -------- >>> import numpy as np >>> 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/f77_src/special_functions/special_functions.html """ n = _nonneg_int_or_fail(n, 'n', strict=False) z = np.asarray(z) if (not np.issubdtype(z.dtype, np.inexact)): z = z.astype(np.float64) pn = np.empty((n + 1,) + z.shape, dtype=z.dtype) pd = np.empty_like(pn) if (z.ndim == 0): _lpn(z, out = (pn, pd)) else: _lpn(z, out = (np.moveaxis(pn, 0, -1), np.moveaxis(pd, 0, -1))) # new axes must be last for the ufunc return pn, pd 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/f77_src/special_functions/special_functions.html """ n = _nonneg_int_or_fail(n, 'n', strict=False) if (n < 1): n1 = 1 else: n1 = n z = np.asarray(z) if (not np.issubdtype(z.dtype, np.inexact)): z = z.astype(float) if np.iscomplexobj(z): qn = np.empty((n1 + 1,) + z.shape, dtype=np.complex128) else: qn = np.empty((n1 + 1,) + z.shape, dtype=np.float64) qd = np.empty_like(qn) if (z.ndim == 0): _lqn(z, out = (qn, qd)) else: _lqn(z, out = (np.moveaxis(qn, 0, -1), np.moveaxis(qd, 0, -1))) # new axes must be last for the ufunc 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/f77_src/special_functions/special_functions.html Examples -------- >>> from scipy import special >>> a, ap, ai, aip = special.ai_zeros(3) >>> a array([-2.33810741, -4.08794944, -5.52055983]) >>> ap array([-1.01879297, -3.24819758, -4.82009921]) >>> ai array([ 0.53565666, -0.41901548, 0.38040647]) >>> aip array([ 0.70121082, -0.80311137, 0.86520403]) """ 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/f77_src/special_functions/special_functions.html Examples -------- >>> from scipy import special >>> b, bp, bi, bip = special.bi_zeros(3) >>> b array([-1.17371322, -3.2710933 , -4.83073784]) >>> bp array([-2.29443968, -4.07315509, -5.51239573]) >>> bi array([-0.45494438, 0.39652284, -0.36796916]) >>> bip array([ 0.60195789, -0.76031014, 0.83699101]) """ 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/f77_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/f77_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/f77_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/f77_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. Parameters ---------- nt : int Number of zeros to compute. Must be positive. Returns ------- ndarray First `nt` zeros of the Kelvin function. See Also -------- ber References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f77_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. Parameters ---------- nt : int Number of zeros to compute. Must be positive. Returns ------- ndarray First `nt` zeros of the Kelvin function. See Also -------- bei References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f77_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. Parameters ---------- nt : int Number of zeros to compute. Must be positive. Returns ------- ndarray First `nt` zeros of the Kelvin function. See Also -------- ker References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f77_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. Parameters ---------- nt : int Number of zeros to compute. Must be positive. Returns ------- ndarray First `nt` zeros of the Kelvin function. See Also -------- kei References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f77_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, 4) def berp_zeros(nt): """Compute nt zeros of the derivative of the Kelvin function ber. Parameters ---------- nt : int Number of zeros to compute. Must be positive. Returns ------- ndarray First `nt` zeros of the derivative of the Kelvin function. See Also -------- ber, berp References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f77_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 derivative of the Kelvin function bei. Parameters ---------- nt : int Number of zeros to compute. Must be positive. Returns ------- ndarray First `nt` zeros of the derivative of the Kelvin function. See Also -------- bei, beip References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f77_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 derivative of the Kelvin function ker. Parameters ---------- nt : int Number of zeros to compute. Must be positive. Returns ------- ndarray First `nt` zeros of the derivative of the Kelvin function. See Also -------- ker, kerp References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f77_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 derivative of the Kelvin function kei. Parameters ---------- nt : int Number of zeros to compute. Must be positive. Returns ------- ndarray First `nt` zeros of the derivative of the Kelvin function. See Also -------- kei, keip References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f77_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/f77_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/f77_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/f77_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 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 For integers, if `exact` is False, then floating point precision is used, otherwise the result is computed exactly. .. deprecated:: 1.14.0 ``exact=True`` is deprecated for non-integer `N` and `k` and will raise an error in SciPy 1.16.0 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 considered as a function of two real variables. Notes ----- - Array arguments accepted only for exact=False case. - If N < 0, or k < 0, then 0 is returned. - If k > N and repetition=False, then 0 is returned. Examples -------- >>> import numpy as np >>> 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) 120 >>> comb(10, 3, exact=True, repetition=True) 220 """ if repetition: return comb(N + k - 1, k, exact=exact) if exact: if int(N) == N and int(k) == k: # _comb_int casts inputs to integers, which is safe & intended here return _comb_int(N, k) # otherwise, we disregard `exact=True`; it makes no sense for # non-integral arguments msg = ("`exact=True` is deprecated for non-integer `N` and `k` and will raise " "an error in SciPy 1.16.0") warnings.warn(msg, DeprecationWarning, stacklevel=2) return comb(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 ``True``, calculate the answer exactly using long integer arithmetic (`N` and `k` must be scalar integers). If ``False``, a floating point approximation is calculated (more rapidly) using `poch`. Default is ``False``. 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 -------- >>> import numpy as np >>> 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: N = np.squeeze(N)[()] # for backward compatibility (accepted size 1 arrays) k = np.squeeze(k)[()] if not (isscalar(N) and isscalar(k)): raise ValueError("`N` and `k` must scalar integers be with `exact=True`.") floor_N, floor_k = int(N), int(k) non_integral = not (floor_N == N and floor_k == k) if (k > N) or (N < 0) or (k < 0): if non_integral: msg = ("Non-integer `N` and `k` with `exact=True` is deprecated and " "will raise an error in SciPy 1.16.0.") warnings.warn(msg, DeprecationWarning, stacklevel=2) return 0 if non_integral: raise ValueError("Non-integer `N` and `k` with `exact=True` is not " "supported.") val = 1 for i in range(floor_N - floor_k + 1, floor_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 # https://stackoverflow.com/a/16327037 def _range_prod(lo, hi, k=1): """ Product of a range of numbers spaced k apart (from hi). For k=1, this returns the product of lo * (lo+1) * (lo+2) * ... * (hi-2) * (hi-1) * hi = hi! / (lo-1)! For k>1, it correspond to taking only every k'th number when counting down from hi - e.g. 18!!!! = _range_prod(1, 18, 4). Breaks into smaller products first for speed: _range_prod(2, 9) = ((2*3)*(4*5))*((6*7)*(8*9)) """ if lo + k < hi: mid = (hi + lo) // 2 if k > 1: # make sure mid is a multiple of k away from hi mid = mid - ((mid - hi) % k) return _range_prod(lo, mid, k) * _range_prod(mid + k, hi, k) elif lo + k == hi: return lo * hi else: return hi def _factorialx_array_exact(n, k=1): """ Exact computation of factorial for an array. The factorials are computed in incremental fashion, by taking the sorted unique values of n and multiplying the intervening numbers between the different unique values. In other words, the factorial for the largest input is only computed once, with each other result computed in the process. k > 1 corresponds to the multifactorial. """ un = np.unique(n) # numpy changed nan-sorting behaviour with 1.21, see numpy/numpy#18070; # to unify the behaviour, we remove the nan's here; the respective # values will be set separately at the end un = un[~np.isnan(un)] # Convert to object array if np.int64 can't handle size if np.isnan(n).any(): dt = float elif k in _FACTORIALK_LIMITS_64BITS.keys(): if un[-1] > _FACTORIALK_LIMITS_64BITS[k]: # e.g. k=1: 21! > np.iinfo(np.int64).max dt = object elif un[-1] > _FACTORIALK_LIMITS_32BITS[k]: # e.g. k=3: 26!!! > np.iinfo(np.int32).max dt = np.int64 else: dt = np.dtype("long") else: # for k >= 10, we always use object dt = object 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 # we can only multiply incrementally if the values are k apart; # therefore we partition `un` into "lanes", i.e. its residues modulo k for lane in range(0, k): ul = un[(un % k) == lane] if k > 1 else un if ul.size: # after np.unique, un resp. ul are sorted, ul[0] is the smallest; # cast to python ints to avoid overflow with np.int-types val = _range_prod(1, int(ul[0]), k=k) out[n == ul[0]] = val for i in range(len(ul) - 1): # by the filtering above, we have ensured that prev & current # are a multiple of k apart prev = ul[i] current = ul[i + 1] # we already multiplied all factors until prev; continue # building the full factorial from the following (`prev + 1`); # use int() for the same reason as above val *= _range_prod(int(prev + 1), int(current), k=k) out[n == current] = val if np.isnan(n).any(): out = out.astype(np.float64) out[np.isnan(n)] = np.nan return out def _factorialx_array_approx(n, k): """ Calculate approximation to multifactorial for array n and integer k. Ensure we only call _factorialx_approx_core where necessary/required. """ result = zeros(n.shape) # keep nans as nans place(result, np.isnan(n), np.nan) # only compute where n >= 0 (excludes nans), everything else is 0 cond = (n >= 0) n_to_compute = extract(cond, n) place(result, cond, _factorialx_approx_core(n_to_compute, k=k)) return result def _factorialx_approx_core(n, k): """ Core approximation to multifactorial for array n and integer k. """ if k == 1: # shortcut for k=1 result = gamma(n + 1) if isinstance(n, np.ndarray): # gamma does not maintain 0-dim arrays result = np.array(result) return result n_mod_k = n % k # scalar case separately, unified handling would be inefficient for arrays; # don't use isscalar due to numpy/numpy#23574; 0-dim arrays treated below if not isinstance(n, np.ndarray): return ( np.power(k, (n - n_mod_k) / k) * gamma(n / k + 1) / gamma(n_mod_k / k + 1) * max(n_mod_k, 1) ) # factor that's independent of the residue class (see factorialk docstring) result = np.power(k, n / k) * gamma(n / k + 1) # factor dependent on residue r (for `r=0` it's 1, so we skip `r=0` # below and thus also avoid evaluating `max(r, 1)`) def corr(k, r): return np.power(k, -r / k) / gamma(r / k + 1) * r for r in np.unique(n_mod_k): if r == 0: continue # cast to int because uint types break on `-r` result[n_mod_k == r] *= corr(k, int(r)) return result 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 -------- >>> import numpy as np >>> 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) 120 """ # don't use isscalar due to numpy/numpy#23574; 0-dim arrays treated below if np.ndim(n) == 0 and not isinstance(n, np.ndarray): # scalar cases if n is None or np.isnan(n): return np.nan elif not (np.issubdtype(type(n), np.integer) or np.issubdtype(type(n), np.floating)): raise ValueError( f"Unsupported datatype for factorial: {type(n)}\n" "Permitted data types are integers and floating point numbers" ) elif n < 0: return 0 elif exact and np.issubdtype(type(n), np.integer): return math.factorial(n) elif exact: msg = ("Non-integer values of `n` together with `exact=True` are " "deprecated. Either ensure integer `n` or use `exact=False`.") warnings.warn(msg, DeprecationWarning, stacklevel=2) return _factorialx_approx_core(n, k=1) # arrays & array-likes n = asarray(n) if n.size == 0: # return empty arrays unchanged return n if not (np.issubdtype(n.dtype, np.integer) or np.issubdtype(n.dtype, np.floating)): raise ValueError( f"Unsupported datatype for factorial: {n.dtype}\n" "Permitted data types are integers and floating point numbers" ) if exact and not np.issubdtype(n.dtype, np.integer): msg = ("factorial with `exact=True` does not " "support non-integral arrays") raise ValueError(msg) if exact: return _factorialx_array_exact(n, k=1) return _factorialx_array_approx(n, k=1) 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!! = 2 ** (n / 2) * gamma(n / 2 + 1) * sqrt(2 / pi) n odd = 2 ** (n / 2) * gamma(n / 2 + 1) n even = 2 ** (n / 2) * (n / 2)! n even Parameters ---------- n : int or array_like Calculate ``n!!``. 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) 105 """ # don't use isscalar due to numpy/numpy#23574; 0-dim arrays treated below if np.ndim(n) == 0 and not isinstance(n, np.ndarray): # scalar cases if n is None or np.isnan(n): return np.nan elif not np.issubdtype(type(n), np.integer): msg = "factorial2 does not support non-integral scalar arguments" raise ValueError(msg) elif n < 0: return 0 elif n in {0, 1}: return 1 # general integer case if exact: return _range_prod(1, n, k=2) return _factorialx_approx_core(n, k=2) # arrays & array-likes n = asarray(n) if n.size == 0: # return empty arrays unchanged return n if not np.issubdtype(n.dtype, np.integer): raise ValueError("factorial2 does not support non-integral arrays") if exact: return _factorialx_array_exact(n, k=2) return _factorialx_array_approx(n, k=2) def factorialk(n, k, exact=None): """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 or array_like 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, otherwise use an approximation (faster, but yields floats instead of integers) .. warning:: The default value for ``exact`` will be changed to ``False`` in SciPy 1.15.0. Returns ------- val : int Multifactorial of `n`. Examples -------- >>> from scipy.special import factorialk >>> factorialk(5, k=1, exact=True) 120 >>> factorialk(5, k=3, exact=True) 10 >>> factorialk([5, 7, 9], k=3, exact=True) array([ 10, 28, 162]) >>> factorialk([5, 7, 9], k=3, exact=False) array([ 10., 28., 162.]) Notes ----- While less straight-forward than for the double-factorial, it's possible to calculate a general approximation formula of n!(k) by studying ``n`` for a given remainder ``r < k`` (thus ``n = m * k + r``, resp. ``r = n % k``), which can be put together into something valid for all integer values ``n >= 0`` & ``k > 0``:: n!(k) = k ** ((n - r)/k) * gamma(n/k + 1) / gamma(r/k + 1) * max(r, 1) This is the basis of the approximation when ``exact=False``. Compare also [1]. References ---------- .. [1] Complex extension to multifactorial https://en.wikipedia.org/wiki/Double_factorial#Alternative_extension_of_the_multifactorial """ if not np.issubdtype(type(k), np.integer) or k < 1: raise ValueError(f"k must be a positive integer, received: {k}") if exact is None: msg = ( "factorialk will default to `exact=False` starting from SciPy " "1.15.0. To avoid behaviour changes due to this, explicitly " "specify either `exact=False` (faster, returns floats), or the " "past default `exact=True` (slower, lossless result as integer)." ) warnings.warn(msg, DeprecationWarning, stacklevel=2) exact = True helpmsg = "" if k in {1, 2}: func = "factorial" if k == 1 else "factorial2" helpmsg = f"\nYou can try to use {func} instead" # don't use isscalar due to numpy/numpy#23574; 0-dim arrays treated below if np.ndim(n) == 0 and not isinstance(n, np.ndarray): # scalar cases if n is None or np.isnan(n): return np.nan elif not np.issubdtype(type(n), np.integer): msg = "factorialk does not support non-integral scalar arguments!" raise ValueError(msg + helpmsg) elif n < 0: return 0 elif n in {0, 1}: return 1 # general integer case if exact: return _range_prod(1, n, k=k) return _factorialx_approx_core(n, k=k) # arrays & array-likes n = asarray(n) if n.size == 0: # return empty arrays unchanged return n if not np.issubdtype(n.dtype, np.integer): msg = "factorialk does not support non-integral arrays!" raise ValueError(msg + helpmsg) if exact: return _factorialx_array_exact(n, k=k) return _factorialx_array_approx(n, k=k) def stirling2(N, K, *, exact=False): r"""Generate Stirling number(s) of the second kind. Stirling numbers of the second kind count the number of ways to partition a set with N elements into K non-empty subsets. The values this function returns are calculated using a dynamic program which avoids redundant computation across the subproblems in the solution. For array-like input, this implementation also avoids redundant computation across the different Stirling number calculations. The numbers are sometimes denoted .. math:: {N \brace{K}} see [1]_ for details. This is often expressed-verbally-as "N subset K". Parameters ---------- N : int, ndarray Number of things. K : int, ndarray Number of non-empty subsets taken. exact : bool, optional Uses dynamic programming (DP) with floating point numbers for smaller arrays and uses a second order approximation due to Temme for larger entries of `N` and `K` that allows trading speed for accuracy. See [2]_ for a description. Temme approximation is used for values `n>50`. The max error from the DP has max relative error `4.5*10^-16` for `n<=50` and the max error from the Temme approximation has max relative error `5*10^-5` for `51 <= n < 70` and `9*10^-6` for `70 <= n < 101`. Note that these max relative errors will decrease further as `n` increases. Returns ------- val : int, float, ndarray The number of partitions. See Also -------- comb : The number of combinations of N things taken k at a time. Notes ----- - If N < 0, or K < 0, then 0 is returned. - If K > N, then 0 is returned. The output type will always be `int` or ndarray of `object`. The input must contain either numpy or python integers otherwise a TypeError is raised. References ---------- .. [1] R. L. Graham, D. E. Knuth and O. Patashnik, "Concrete Mathematics: A Foundation for Computer Science," Addison-Wesley Publishing Company, Boston, 1989. Chapter 6, page 258. .. [2] Temme, Nico M. "Asymptotic estimates of Stirling numbers." Studies in Applied Mathematics 89.3 (1993): 233-243. Examples -------- >>> import numpy as np >>> from scipy.special import stirling2 >>> k = np.array([3, -1, 3]) >>> n = np.array([10, 10, 9]) >>> stirling2(n, k) array([9330, 0, 3025], dtype=object) """ output_is_scalar = np.isscalar(N) and np.isscalar(K) # make a min-heap of unique (n,k) pairs N, K = asarray(N), asarray(K) if not np.issubdtype(N.dtype, np.integer): raise TypeError("Argument `N` must contain only integers") if not np.issubdtype(K.dtype, np.integer): raise TypeError("Argument `K` must contain only integers") if not exact: # NOTE: here we allow np.uint via casting to double types prior to # passing to private ufunc dispatcher. All dispatched functions # take double type for (n,k) arguments and return double. return _stirling2_inexact(N.astype(float), K.astype(float)) nk_pairs = list( set([(n.take(0), k.take(0)) for n, k in np.nditer([N, K], ['refs_ok'])]) ) heapify(nk_pairs) # base mapping for small values snsk_vals = defaultdict(int) for pair in [(0, 0), (1, 1), (2, 1), (2, 2)]: snsk_vals[pair] = 1 # for each pair in the min-heap, calculate the value, store for later n_old, n_row = 2, [0, 1, 1] while nk_pairs: n, k = heappop(nk_pairs) if n < 2 or k > n or k <= 0: continue elif k == n or k == 1: snsk_vals[(n, k)] = 1 continue elif n != n_old: num_iters = n - n_old while num_iters > 0: n_row.append(1) # traverse from back to remove second row for j in range(len(n_row)-2, 1, -1): n_row[j] = n_row[j]*j + n_row[j-1] num_iters -= 1 snsk_vals[(n, k)] = n_row[k] else: snsk_vals[(n, k)] = n_row[k] n_old, n_row = n, n_row out_types = [object, object, object] if exact else [float, float, float] # for each pair in the map, fetch the value, and populate the array it = np.nditer( [N, K, None], ['buffered', 'refs_ok'], [['readonly'], ['readonly'], ['writeonly', 'allocate']], op_dtypes=out_types, ) with it: while not it.finished: it[2] = snsk_vals[(int(it[0]), int(it[1]))] it.iternext() output = it.operands[2] # If N and K were both scalars, convert output to scalar. if output_is_scalar: output = output.take(0) return output 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). See Also -------- zetac Notes ----- The two-argument version is the Hurwitz zeta function .. math:: \zeta(x, q) = \sum_{k=0}^{\infty} \frac{1}{(k + q)^x}; see [dlmf]_ for details. The Riemann zeta function corresponds to the case when ``q = 1``. References ---------- .. [dlmf] NIST, Digital Library of Mathematical Functions, https://dlmf.nist.gov/25.11#i Examples -------- >>> import numpy as np >>> 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: return _ufuncs._riemann_zeta(x, out) else: return _ufuncs._zeta(x, q, out) def _sph_harm_all(m, n, theta, phi): """Private function. This may be removed or modified at any time.""" theta = np.asarray(theta) if (not np.issubdtype(theta.dtype, np.inexact)): theta = theta.astype(np.float64) phi = np.asarray(phi) if (not np.issubdtype(phi.dtype, np.inexact)): phi = phi.astype(np.float64) out = np.empty((2 * m + 1, n + 1) + np.broadcast_shapes(theta.shape, phi.shape), dtype = np.result_type(1j, theta.dtype, phi.dtype)) _sph_harm_all_gufunc(theta, phi, out = np.moveaxis(out, (0, 1), (-2, -1))) return out