a a@sddlZddlZddlZddlmZmZmZmZmZm Z m Z m Z m Z m Z mZmZmZmZmZmZmZmZddlmZddlmZmZmZmZmZmZmZm Z m!Z!m"Z"m#Z#m$Z$m%Z%m&Z&ddlm'Z'ddlm(Z(ddl)m*Z*gd Z+dd d Z,d dZ-ddZ.ddZ/ddZ0ddZ1ddZ2ddZ3dddZ4dddZ5dd d!Z6d"d#Z7dd$d%Z8dd&d'Z9dd(d)Z:dd*d+Z;dd,d-Zd2d3Z?d4d5Z@d6d7ZAd8d9ZBd:d;ZCdd=d>ZDeZEd?d@ZFdAdBZGdCdDZHdEdFZIddHdIZJdJdKZKdLdMZLdNdOZMdPdQZNdRdSZOdTdUZPdVdWZQdXdYZRdZd[ZSd\d]ZTd^d_ZUd`daZVdbdcZWdddeZXdfdgZYdhdiZZdjdkZ[dldmZ\dndoZ]dpdqZ^drdsZ_dtduZ`ddvdwZaddxdyZbdzd{Zcdd|d}Zddd~dZedddZfdddZgdS)N)piasarrayfloorisscalar iscomplexrealimagsqrtwheremgridsinplace issubdtypeextractinexactnanzerossinc)_ufuncs) mathieu_a mathieu_bivjvgammapsihankel1hankel2yvkvndtripochbinomhyp0f1)specfun) orthogonal) _comb_int)Fai_zerosassoc_laguerre bei_zeros beip_zeros ber_zeros bernoulli berp_zerosbi_zerosclpmncombdigammadiric erf_zeroseuler factorial factorial2 factorialk fresnel_zerosfresnelc_zerosfresnels_zerosrh1vph2vprrr#rivpjn_zeros jnjnp_zeros jnp_zeros jnyn_zerosrjvp kei_zeros keip_zeros kelvin_zeros ker_zeros kerp_zerosrkvplmbdalpmnlpnlqmnlqnrrmathieu_even_coefmathieu_odd_coefr obl_cv_seqpbdn_seqpbdv_seqpbvv_seqperm polygamma pro_cv_seqr riccati_jn riccati_ynry0_zerosy1_zeros y1p_zerosyn_zeros ynp_zerosryvpzetaTc Cs~z>|rt|}n|t|kr(t|}nt|dkr>> 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]) gC]r2gMbP?rr)rrdtyperfloatrshapenpZfinfoZepsrr rr absrrpowround) xrgZytypeyminvalZmask1ZdenomZmask2ZxsubnsubZzsubmaskZdsubrkrkrlr2ns6@         r2cCsnt|rt||ks|dkr$tdt|}t|\}}}}|d|d|d||d||d|fS)aCompute 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 izNumber must be integer <= 1200.rN)rrrcrbr$Zjdzo)ntrgmtZzorkrkrlr?s "r?cCsXt|rt|stdt||ks0t||kr8td|dkrHtdtt||S)aCompute nt zeros of Bessel functions Jn(x), Jn'(x), Yn(x), and Yn'(x). Returns 4 arrays of length `nt`, corresponding to the first `nt` zeros of Jn(x), Jn'(x), Yn(x), and Yn'(x), respectively. The zeros are returned in ascending order. Parameters ---------- n : int Order of the Bessel functions nt : int Number (<=1200) of zeros to compute Returns ------- Jn : ndarray First `nt` zeros of Jn Jnp : ndarray First `nt` zeros of Jn' Yn : ndarray First `nt` zeros of Yn Ynp : ndarray First `nt` zeros of Yn' See Also -------- jn_zeros, jnp_zeros, yn_zeros, ynp_zeros References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996, chapter 5. https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html Arguments must be scalars.zArguments must be integers.rznt > 0)rrcrr$Zjyzortrgr|rkrkrlrAs$rAcCst||dS)aCompute 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 `n` zeros of the Bessel function. See Also -------- jv 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 Examples -------- >>> import scipy.special as sc We can check that we are getting approximations of the zeros by evaluating them with `jv`. >>> n = 1 >>> x = sc.jn_zeros(n, 3) >>> x array([ 3.83170597, 7.01558667, 10.17346814]) >>> sc.jv(n, x) array([-0.00000000e+00, 1.72975330e-16, 2.89157291e-16]) Note that the zero at ``x = 0`` for ``n > 0`` is not included. >>> sc.jv(1, 0) 0.0 rrArrkrkrlr>)s2r>cCst||dS)aCompute 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 `n` zeros of the Bessel function. See Also -------- jvp, jv 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 Examples -------- >>> import scipy.special as sc We can check that we are getting approximations of the zeros by evaluating them with `jvp`. >>> n = 2 >>> x = sc.jnp_zeros(n, 3) >>> x array([3.05423693, 6.70613319, 9.96946782]) >>> sc.jvp(n, x) array([ 2.77555756e-17, 2.08166817e-16, -3.01841885e-16]) Note that the zero at ``x = 0`` for ``n > 1`` is not included. >>> sc.jvp(n, 0) 0.0 rrrrkrkrlr@^s2r@cCst||dS)aCompute 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 `n` zeros of the Bessel function. See Also -------- yn, yv 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 Examples -------- >>> import scipy.special as sc We can check that we are getting approximations of the zeros by evaluating them with `yn`. >>> n = 2 >>> x = sc.yn_zeros(n, 3) >>> x array([ 3.38424177, 6.79380751, 10.02347798]) >>> sc.yn(n, x) array([-1.94289029e-16, 8.32667268e-17, -1.52655666e-16]) rnrrrkrkrlr\s+r\cCst||dS)aCompute 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 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/f_src/special_functions/special_functions.html Examples -------- >>> import scipy.special as sc We can check that we are getting approximations of the zeros by evaluating them with `yvp`. >>> n = 2 >>> x = sc.ynp_zeros(n, 3) >>> x array([ 5.00258293, 8.3507247 , 11.57419547]) >>> sc.yvp(n, x) array([ 2.22044605e-16, -3.33066907e-16, 2.94902991e-16]) rrrkrkrlr]s'r]FcCs<t|rt||ks|dkr$tdd}| }t|||S)aCompute 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 r*Arguments must be scalar positive integer.rrrcr$Zcyzor|complexkfZkcrkrkrlrYs rYcCs<t|rt||ks|dkr$tdd}| }t|||S)aCompute 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 rrrrrrkrkrlrZs rZcCs<t|rt||ks|dkr$tdd}| }t|||S)aCompute 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 rrrnrrrkrkrlr[3s r[cCspt|}d}||||}td|dD]:}||||d|}||||||d|7}q(|d|S)N?rrn@)rrange)vzrgLphasepsirkrkrl_bessel_diff_formulaWs rcCs0t|d}|dkrt||St|||tdSdS)aCompute derivatives of Bessel functions of the first kind. Compute the nth derivative of the Bessel function `Jv` with respect to `z`. Parameters ---------- v : 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 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/f_src/special_functions/special_functions.html .. [2] NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/10.6.E7 rgrroN)rmrrrrrgrkrkrlrBes"  rBcCs0t|d}|dkrt||St|||tdSdS)awCompute derivatives of Bessel functions of the second kind. Compute the nth derivative of the Bessel function `Yv` 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 Returns ------- scalar or ndarray nth 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/f_src/special_functions/special_functions.html .. [2] NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/10.6.E7 rgrroN)rmrrrrkrkrlr^s!  r^cCs8t|d}|dkrt||Sd|t|||tdSdS)a)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. https://dlmf.nist.gov/10.29.E5 rgrrorN)rmrrrrkrkrlrHs0  rHcCs0t|d}|dkrt||St|||tdSdS)aCompute 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 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 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/f_src/special_functions/special_functions.html .. [2] NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/10.29.E5 rgrrN)rmrrrrkrkrlr=s&  r=cCs0t|d}|dkrt||St|||tdSdS)a_Compute nth derivative 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 Returns ------- scalar or ndarray Values of the derivative of the Hankel 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/f_src/special_functions/special_functions.html .. [2] NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/10.6.E7 rgrroN)rmrrrrkrkrlr;s  r;cCs0t|d}|dkrt||St|||tdSdS)a_Compute nth derivative 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 Returns ------- scalar or ndarray Values of the derivative of the Hankel 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/f_src/special_functions/special_functions.html .. [2] NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/10.6.E7 rgrroN)rmrrrrkrkrlr<@s  r<cCsjt|rt|stdt|ddd}|dkr4d}n|}t||\}}}|d|d|d|dfS)aCompute 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. https://dlmf.nist.gov/10.51.E1 arguments must be scalars.rgFrirrN)rrcrmr$Zrctjrgrwn1nmZjnZjnprkrkrlrWfs)rWcCsjt|rt|stdt|ddd}|dkr4d}n|}t||\}}}|d|d|d|dfS)aCompute 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. https://dlmf.nist.gov/10.51.E1 rrgFrrrN)rrcrmr$ZrctyrrkrkrlrXs)rXcCs.t||ks|dkst|s$tdt|S)aCompute 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) 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]) 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 r)Argument must be positive scalar integer.)rrrcr$Zcerzor|rkrkrlr3s#r3cCs0t||ks|dkst|s$tdtd|S)a4Compute 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 rrrrrrcr$Zfcszorrkrkrlr9s r9cCs0t||ks|dkst|s$tdtd|S)a2Compute 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 rrrnrrrkrkrlr:s r:cCs<t||ks|dkst|s$tdtd|td|fS)aGCompute 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 rrrnrrrrkrkrlr8s r8cCst|||S)a|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)``. )r%Zeval_genlaguerre)rwrgkrkrkrlr(#s r(cCsLt|t|}}d|dt|dt|d|}t|dkt||S)a.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) grrr)rrr_r r)rgrwZfac2rkrkrlrU5s&&rUcCst|rt|std|dkr(td|t|ks<|dkrDtd|dkrvddt|d|d t||}n(d d t|d |d t||}t|d|}|dkrtdd}tt|}|drd}t||}t||||}|d|S)aFourier 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 https://dlmf.nist.gov/28.4#i m and q must be scalars.rq >=0zm must be an integer >=0.r@ L@fffff`@̬V@1@@T㥛 ?笭_vOn??)Warning, too many predicted coefficients.rnN) rrcrr rbprintrr$fcoef)r}qqmkmkdafcrkrkrlrN`s&#*(  rNcCst|rt|std|dkr(td|t|ks<|dkrDtd|dkrvddt|d|d t||}n(d d t|d |d t||}t|d|}|dkrtdd}tt|}|drd}t||}t||||}|d|S)a?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 rrrzm must be an integer > 0rrrrrrrrrrrrrnrN) rrcrr rbrrr$r)r}rrrrbrrkrkrlrOs&!*(  rOc CsZt|rt||krtdt|r,|dkr4tdt|sDtdt|rTtd|dkr"| }td|dd|df\}}tjdd~t|dkrt||kd d |t||dt||d}n,t||kd t||dt||d}Wd n1s0Yn|}t |||\}}|dkrR||}||}||fS) aYSequence of associated Legendre functions of the first kind. Computes the associated Legendre function of the first kind of order m and degree n, ``Pmn(z)`` = :math:`P_n^m(z)`, and its derivative, ``Pmn'(z)``. Returns two arrays of size ``(m+1, n+1)`` containing ``Pmn(z)`` and ``Pmn'(z)`` for all orders from ``0..m`` and degrees from ``0..n``. This function takes a real argument ``z``. For complex arguments ``z`` use clpmn instead. Parameters ---------- m : int ``|m| <= n``; the order of the Legendre function. n : int where ``n >= 0``; the degree of the Legendre function. Often called ``l`` (lower case L) in descriptions of the associated Legendre function z : float Input value. Returns ------- Pmn_z : (m+1, n+1) array Values for all orders 0..m and degrees 0..n Pmn_d_z : (m+1, n+1) array Derivatives for all orders 0..m and degrees 0..n See Also -------- clpmn: associated Legendre functions of the first kind for complex z Notes ----- In the interval (-1, 1), Ferrer's function of the first kind is returned. The phase convention used for the intervals (1, inf) and (-inf, -1) is such that the result is always real. References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html .. [2] NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/14.3 m must be <= n.r!n must be a non-negative integer.z must be scalar.z)Argument must be real. Use clpmn instead.rignoreallrroN) rrtrcrr ufuncserrstater rr$rJ) r}rgrmpmfnffixarrrpdrkrkrlrJs.0    &N rJrc Csjt|rt||krtdt|r,|dkr4tdt|sDtd|dks\|dks\td|dkr&| }td|dd|df\}}tjd d z|dkrt||kd d |t||dt||d}n,t||kd t||dt||d}Wd n1s0Yn|}t ||t |t ||\}} |dkrb||}| |} || fS)aAssociated 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 https://dlmf.nist.gov/14.21 rrrrrnrztype must be either 2 or 3.rrrrroN) rrtrcr rrr rr$r/rr) r}rgrtyperrrrrrrkrkrlr/ s.7   &N r/cCst|r|dkrtdt|r(|dkr0tdt|s@tdt|}t|}td|}td|}t|rt|||\}}nt|||\}}|d|dd|df|d|dd|dffS)a_Sequence of associated Legendre functions of the second kind. Computes the associated Legendre function of the second kind of order m and degree n, ``Qmn(z)`` = :math:`Q_n^m(z)`, and its derivative, ``Qmn'(z)``. Returns two arrays of size ``(m+1, n+1)`` containing ``Qmn(z)`` and ``Qmn'(z)`` for all orders from ``0..m`` and degrees from ``0..n``. Parameters ---------- m : int ``|m| <= n``; the order of the Legendre function. n : int where ``n >= 0``; the degree of the Legendre function. Often called ``l`` (lower case L) in descriptions of the associated Legendre function z : complex Input value. Returns ------- Qmn_z : (m+1, n+1) array Values for all orders 0..m and degrees 0..n Qmn_d_z : (m+1, n+1) array Derivatives for all orders 0..m and degrees 0..n References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html rz!m must be a non-negative integer.rrrN)rrcrbmaxrr$ZclqmnrL)r}rgrmmnnrqdrkrkrlrLqs!  rLcCsLt|r|dkrtdt|}|dkr.d}n|}tt|d|dS)a'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/f_src/special_functions/special_functions.html .. [2] "Bernoulli number", Wikipedia, https://en.wikipedia.org/wiki/Bernoulli_number Examples -------- >>> 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)``. rrrnNr)rrcrbr$Zbernobrgrrkrkrlr,s(r,cCsHt|r|dkrtdt|}|dkr.d}n|}t|d|dS)aEuler numbers E(0), E(1), ..., E(n). The Euler numbers [1]_ are also known as the secant numbers. Because ``euler(n)`` returns floating point values, it does not give exact values for large `n`. The first inexact value is E(22). Parameters ---------- n : int The highest index of the Euler number to be returned. Returns ------- ndarray The Euler numbers [E(0), E(1), ..., E(n)]. The odd Euler numbers, which are all zero, are included. References ---------- .. [1] Sequence A122045, The On-Line Encyclopedia of Integer Sequences, https://oeis.org/A122045 .. [2] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html Examples -------- >>> from scipy.special import euler >>> euler(6) array([ 1., 0., -1., 0., 5., 0., -61.]) >>> euler(13).astype(np.int64) array([ 1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521, 0, 2702765, 0]) >>> euler(22)[-1] # Exact value of E(22) is -69348874393137901. -69348874393137976.0 rrrnNr)rrcrbr$Zeulerbrrkrkrlr4s)r4cCst|rt|stdt|ddd}|dkr4d}n|}t|rRt||\}}nt||\}}|d|d|d|dfS)aLegendre 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 rrgFrrN)rrcrmrr$ZclpnrK)rgrrZpnrrkrkrlrK srKcCst|rt|stdt|ddd}|dkr4d}n|}t|rRt||\}}nt||\}}|d|d|d|dfS)aLegendre function of the second kind. Compute sequence of Legendre functions of the second kind, Qn(z) and derivatives for all degrees from 0 to n (inclusive). References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html rrgFrrN)rrcrmrr$ZclqnZlqnb)rgrrqnrrkrkrlrM(s rMcCs4d}t|r t||ks |dkr(tdt||S)a 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) 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]) 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 rr%nt must be a positive integer scalar.rrrcr$Zairyzor|rrkrkrlr'Cs-r'cCs4d}t|r t||ks |dkr(tdt||S)a 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) 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]) 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 rnrrrrrkrkrlr.vs-r.c Cst|rt|std|dkr(tdt|}||}|dkrFd}n|}||}|t|krrt||\}}}nt||\}}}|d|d|d|dfS)aJahnke-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 rrzargument must be > 0.rN)rrcrbrr$ZlamvZlamn) rrwrgv0rv1ZvmZvldlrkrkrlrIs  rIc Csvt|rt|stdt|}||}|dkr6d}n|}||}t||\}}}} |d|d|d|dfS)aParabolic 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 rrN)rrcrbr$Zpbdv rrwrgrrrZdvZdpZpdfZpddrkrkrlrRsrRc Csvt|rt|stdt|}||}|dkr6d}n|}||}t||\}}}} |d|d|d|dfS)aParabolic cylinder functions Vv(x) and derivatives. Parameters ---------- v : float Order of the parabolic cylinder function x : float Value at which to evaluate the function and derivatives Returns ------- dv : ndarray Values of V_vi(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v. dp : ndarray Derivatives V_vi'(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v. References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996, chapter 13. https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html rrN)rrcrbr$ZpbvvrrkrkrlrSsrScCsrt|rt|stdt||kr,tdt|dkr>d}n|}t||\}}|d|d|d|dfS)a~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 rzn must be an integer.rN)rrcrrtr$Zcpbdn)rgrrZcpbZcpdrkrkrlrQ%s  rQcCs0t|rt||ks|dkr$tdt|dS)aCompute 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/f_src/special_functions/special_functions.html r#nt must be positive integer scalar.rrrrcr$Zklvnzorrkrkrlr+Isr+cCs0t|rt||ks|dkr$tdt|dS)aCompute 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/f_src/special_functions/special_functions.html rrrnrrrkrkrlr)fsr)cCs0t|rt||ks|dkr$tdt|dS)aCompute nt zeros of the Kelvin function ker. Parameters ---------- nt : int Number of zeros to compute. Must be positive. Returns ------- ndarray First `nt` zeros of the Kelvin function. See Also -------- ker References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html rrrrrrkrkrlrFsrFcCs0t|rt||ks|dkr$tdt|dS)aCompute 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/f_src/special_functions/special_functions.html rrrrrrkrkrlrCsrCcCs0t|rt||ks|dkr$tdt|dS)a,Compute nt zeros of the derivative of the Kelvin function ber. Parameters ---------- nt : int Number of zeros to compute. Must be positive. Returns ------- ndarray First `nt` zeros of the derivative of the Kelvin function. See Also -------- ber, berp References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html rrrrrkrkrlr-sr-cCs0t|rt||ks|dkr$tdt|dS)a,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/f_src/special_functions/special_functions.html rrrrrkrkrlr*sr*cCs0t|rt||ks|dkr$tdt|dS)a,Compute nt zeros of the derivative of the Kelvin function ker. Parameters ---------- nt : int Number of zeros to compute. Must be positive. Returns ------- ndarray First `nt` zeros of the derivative of the Kelvin function. See Also -------- ker, kerp References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html rrrrrkrkrlrGsrGcCs0t|rt||ks|dkr$tdt|dS)a,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/f_src/special_functions/special_functions.html rrrrrkrkrlrDsrDc Csxt|rt||ks|dkr$tdt|dt|dt|dt|dt|dt|dt|d t|d fS) aCompute nt zeros of all Kelvin functions. Returned in a length-8 tuple of arrays of length nt. The tuple contains the arrays of zeros of (ber, bei, ker, kei, ber', bei', ker', kei'). References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html rrrrnrrrrrrrrrkrkrlrE1s         rEcCs|t|rt|rt|s td|t|ks8|t|kr@td||dkrTtd||d}t|||ddd|S)aCharacteristic values for prolate spheroidal wave functions. Compute a sequence of characteristic values for the prolate spheroidal wave functions for mode m and n'=m..n and spheroidal parameter c. References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html rModes must be integers.(Difference between n and m is too large.rNrrcrr$Zsegvr}rgcZmaxLrkrkrlrVJs  rVcCs|t|rt|rt|s td|t|ks8|t|kr@td||dkrTtd||d}t|||ddd|S)aCharacteristic 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 rrrrrroNrrrkrkrlrPbs  rPcCs|rt||d||S|r&t||St|t|}}||k|dk@|dk@}t||}t|tjrrd||<n|std}|SdS)a{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 N < 0, or k < 0, then 0 is returned. - If k > N and repetition=False, then 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) 120 >>> comb(10, 3, exact=True, repetition=True) 220 rrN)r0r&rr" isinstancersndarrayfloat64)NrexactZ repetitioncondvalsrkrkrlr0zs.     r0cCs|rL||ks|dks|dkr dSd}t||d|dD] }||9}q:|St|t|}}||k|dk@|dk@}t||d|}t|tjrd||<n|std}|SdS)aHPermutations 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 rrN)rrr!rrsrr)rrrvalrrrrkrkrlrTs$    rTcCsD|d|kr0||d}t||t|d|S||kr<|S||S)z 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)) rrn) _range_prod)lohiZmidrkrkrlrs  rc Cs|rt|dkr8t|r"|S|dkr.dSt|St|}t|t}t| rdt }n,|ddkrvt}n|ddkrtj }ntj }tj ||d}tjdd4||dk}d||d k<d||dk<Wd n1s0Y|jr^t|d}||||dk<tt|dD]8}||d}||d}|t||9}||||k<q$t| r|tj}|t||t|<|Snt|}|Sd S) aA 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) 120 rro )rprrrrnN)rsndimisnanmathr5runiqueZastypeobjectanyrqZint64int_Z empty_likersizerlenrrrZ _factorial) rgrundtoutrrprevcurrentrkrkrlr5 sB/     *    r5c Cs|r>|dkrdS|dkrdSd}t|ddD] }||9}q,|St|}t|jd}|d|dk@}d|d|dk@}t||}t||}|d} |d} t||t| dttt d| dt||t| dt d| |Sd S) aDouble 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) 105 rorrdrnrrN) rrrrrrr rr rru) rgrrrrZcond1Zcond2ZoddnZevennZnd2oZnd2erkrkrlr6_ s("    *r6cCsL|rD|d|krdS|dkr dSd}t|d| D] }||}q2|StdS)aMultifactorial 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) 120 >>> factorialk(5, 3, exact=True) 10 rrN)rNotImplementedError)rgrrrjrkrkrlr7 s*  r7cCs&|durt||St|||SdS)a 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}; see [dlmf]_ for details. The Riemann zeta function corresponds to the case when ``q = 1``. See Also -------- zetac References ---------- .. [dlmf] NIST, Digital Library of Mathematical Functions, https://dlmf.nist.gov/25.11#i 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 N)rZ _riemann_zetaZ_zeta)rwrrrkrkrlr_ s< r_)T)F)F)F)r)r)r)r)r)r)r)r)FF)F)F)F)T)NN)hr`Znumpyrsrrrrrrrrr r r r r rrrrrrrrrrrrrrrrrrr r!r"r#r$r%Z_combr&__all__rmr2r?rAr>r@r\r]rYrZr[rrBr^rHr=r;r<rWrXr3r9r:r8r(r1rUrNrOrJr/rLr,r4rKrMr'r.rIrRrSrQr+r)rFrCr-r*rGrDrErVrPr0rTrr5r6r7r_rkrkrkrlsP @   J e)-55.* $ $ $ ) ( 7 - & &44( +;9L Q523332%%$ = 7 ^ 9 7