# Special functions (scipy.special)¶

Nearly all of the functions below are universal functions and follow broadcasting and automatic array-looping rules. Exceptions are noted.

## Error handling¶

Errors are handled by returning nans, or other appropriate values. Some of the special function routines will emit warnings when an error occurs. By default this is disabled. To enable such messages use errprint(1), and to disable such messages use errprint(0).

Example:

>>> print scipy.special.bdtr(-1,10,0.3)
>>> scipy.special.errprint(1)
>>> print scipy.special.bdtr(-1,10,0.3)

 errprint([inflag]) Sets or returns the error printing flag for special functions. SpecialFunctionWarning Warning that can be issued with errprint(True)

## Available functions¶

### Airy functions¶

 airy(z) Airy functions and their derivatives. airye(z) Exponentially scaled Airy functions and their derivatives. ai_zeros(nt) Compute nt zeros and values of the Airy function Ai and its derivative. bi_zeros(nt) Compute nt zeros and values of the Airy function Bi and its derivative. itairy(x) Integrals of Airy functions

### Elliptic Functions and Integrals¶

 ellipj(u, m) Jacobian elliptic functions Calculates the Jacobian elliptic functions of parameter m between 0 and 1, and real u. ellipk(m) Complete elliptic integral of the first kind. ellipkm1(p) Complete elliptic integral of the first kind around m = 1 This function is defined as .. ellipkinc(phi, m) Incomplete elliptic integral of the first kind This function is defined as .. ellipe(m) Complete elliptic integral of the second kind This function is defined as .. ellipeinc(phi, m) Incomplete elliptic integral of the second kind This function is defined as ..

### Bessel Functions¶

 jv(v, z) Bessel function of the first kind of real order v jn(v, z) Bessel function of the first kind of real order v jve(v, z) Exponentially scaled Bessel function of order v yn(n, x) Bessel function of the second kind of integer order Returns the Bessel function of the second kind of integer order n at x. yv(v, z) Bessel function of the second kind of real order Returns the Bessel function of the second kind of real order v at complex z. yve(v, z) Exponentially scaled Bessel function of the second kind of real order kn(n, x) Modified Bessel function of the second kind of integer order n Returns the modified Bessel function of the second kind for integer order n at real z. kv(v, z) Modified Bessel function of the second kind of real order v Returns the modified Bessel function of the second kind for real order v at complex z. kve(v, z) Exponentially scaled modified Bessel function of the second kind. iv(v, z) Modified Bessel function of the first kind of real order :Parameters: v Order. ive(v, z) Exponentially scaled modified Bessel function of the first kind hankel1(v, z) Hankel function of the first kind hankel1e(v, z) Exponentially scaled Hankel function of the first kind hankel2(v, z) Hankel function of the second kind hankel2e(v, z) Exponentially scaled Hankel function of the second kind

The following is not an universal function:

 lmbda(v, x) Jahnke-Emden Lambda function, Lambdav(x).

#### Zeros of Bessel Functions¶

These are not universal functions:

 jnjnp_zeros(nt) Compute zeros of integer-order Bessel functions Jn and Jn’. jnyn_zeros(n, nt) Compute nt zeros of Bessel functions Jn(x), Jn’(x), Yn(x), and Yn’(x). jn_zeros(n, nt) Compute zeros of integer-order Bessel function Jn(x). jnp_zeros(n, nt) Compute zeros of integer-order Bessel function derivative Jn’(x). yn_zeros(n, nt) Compute zeros of integer-order Bessel function Yn(x). ynp_zeros(n, nt) Compute zeros of integer-order Bessel function derivative Yn’(x). y0_zeros(nt[, complex]) Compute nt zeros of Bessel function Y0(z), and derivative at each zero. y1_zeros(nt[, complex]) Compute nt zeros of Bessel function Y1(z), and derivative at each zero. y1p_zeros(nt[, complex]) Compute nt zeros of Bessel derivative Y1’(z), and value at each zero.

#### Faster versions of common Bessel Functions¶

 j0(x) Bessel function the first kind of order 0 j1(x) Bessel function of the first kind of order 1 y0(x) Bessel function of the second kind of order 0 Returns the Bessel function of the second kind of order 0 at x. y1(x) Bessel function of the second kind of order 1 Returns the Bessel function of the second kind of order 1 at x. i0(x) Modified Bessel function of order 0 i0e(x) Exponentially scaled modified Bessel function of order 0. i1(x) Modified Bessel function of order 1 i1e(x) Exponentially scaled modified Bessel function of order 1. k0(x) Modified Bessel function K of order 0 Modified Bessel function of the second kind (sometimes called the third kind) of order 0. k0e(x) Exponentially scaled modified Bessel function K of order 0 Defined as:: k0e(x) = exp(x) * k0(x). k1(x) Modified Bessel function of the first kind of order 1 k1e(x) Exponentially scaled modified Bessel function K of order 1

#### Integrals of Bessel Functions¶

 itj0y0(x) Integrals of Bessel functions of order 0 Returns simple integrals from 0 to x of the zeroth order Bessel functions j0 and y0. it2j0y0(x) Integrals related to Bessel functions of order 0 iti0k0(x) Integrals of modified Bessel functions of order 0 Returns simple integrals from 0 to x of the zeroth order modified Bessel functions i0 and k0. it2i0k0(x) Integrals related to modified Bessel functions of order 0 besselpoly(a, lmb, nu) Weighted integral of a Bessel function.

#### Derivatives of Bessel Functions¶

 jvp(v, z[, n]) Compute nth derivative of Bessel function Jv(z) with respect to z. yvp(v, z[, n]) Compute nth derivative of Bessel function Yv(z) with respect to z. kvp(v, z[, n]) Compute nth derivative of real-order modified Bessel function Kv(z) Kv(z) is the modified Bessel function of the second kind. ivp(v, z[, n]) Compute nth derivative of modified Bessel function Iv(z) with respect to z. h1vp(v, z[, n]) Compute nth derivative of Hankel function H1v(z) with respect to z. h2vp(v, z[, n]) Compute nth derivative of Hankel function H2v(z) with respect to z.

#### Spherical Bessel Functions¶

These are not universal functions:

 sph_jn(n, z) Compute spherical Bessel function jn(z) and derivative. sph_yn(n, z) Compute spherical Bessel function yn(z) and derivative. sph_jnyn(n, z) Compute spherical Bessel functions jn(z) and yn(z) and derivatives. sph_in(n, z) Compute spherical Bessel function in(z) and derivative. sph_kn(n, z) Compute spherical Bessel function kn(z) and derivative. sph_inkn(n, z) Compute spherical Bessel functions in(z), kn(z), and derivatives.

#### Riccati-Bessel Functions¶

These are not universal functions:

 riccati_jn(n, x) Compute Ricatti-Bessel function of the first kind and derivative. riccati_yn(n, x) Compute Ricatti-Bessel function of the second kind and derivative.

### Struve Functions¶

 struve(v, x) Struve function Computes the struve function Hv(x) of order v at x, x must be positive unless v is an integer. modstruve(v, x) Modified Struve function Returns the modified Struve function Lv(x) of order v at x, x must be positive unless v is an integer. itstruve0(x) Integral of the Struve function of order 0 it2struve0(x) Integral related to Struve function of order 0 itmodstruve0(x) Integral of the modified Struve function of order 0

### Raw Statistical Functions¶

See also

scipy.stats: Friendly versions of these functions.

 bdtr(k, n, p) Binomial distribution cumulative distribution function. bdtrc(k, n, p) Binomial distribution survival function. bdtri(k, n, y) Inverse function to bdtr vs. bdtrik(y, n, p) Inverse function to bdtr vs k bdtrin(k, y, p) Inverse function to bdtr vs n btdtr(a, b, x) Cumulative beta distribution. btdtri(a, b, p) p-th quantile of the beta distribution. btdtria(p, b, x) Inverse of btdtr vs a btdtrib(a, p, x) Inverse of btdtr vs b fdtr(dfn, dfd, x) F cumulative distribution function Returns the area from zero to x under the F density function (also known as Snedcor’s density or the variance ratio density). fdtrc(dfn, dfd, x) F survival function Returns the complemented F distribution function. fdtri(dfn, dfd, p) Inverse to fdtr vs x Finds the F density argument x such that fdtr(dfn, dfd, x) == p. fdtridfd(dfn, p, x) Inverse to fdtr vs dfd Finds the F density argument dfd such that fdtr(dfn, dfd, x) == p. gdtr(a, b, x) Gamma distribution cumulative density function. gdtrc(a, b, x) Gamma distribution survival function. gdtria(p, b, x[, out]) Inverse of gdtr vs a. gdtrib(a, p, x[, out]) Inverse of gdtr vs b. gdtrix(a, b, p[, out]) Inverse of gdtr vs x. nbdtr(k, n, p) Negative binomial cumulative distribution function Returns the sum of the terms 0 through k of the negative binomial distribution:: sum((n+j-1)Cj p**n (1-p)**j, j=0..k). nbdtrc(k, n, p) Negative binomial survival function Returns the sum of the terms k+1 to infinity of the negative binomial distribution. nbdtri(k, n, y) Inverse of nbdtr vs p Finds the argument p such that nbdtr(k, n, p) = y. nbdtrik(y, n, p) Inverse of nbdtr vs k Finds the argument k such that nbdtr(k, n, p) = y. nbdtrin(k, y, p) Inverse of nbdtr vs n Finds the argument n such that nbdtr(k, n, p) = y. ncfdtr(dfn, dfd, nc, f) Cumulative distribution function of the non-central F distribution. ncfdtridfd(p, f, dfn, nc) Calculate degrees of freedom (denominator) for the noncentral F-distribution. ncfdtridfn(p, f, dfd, nc) Calculate degrees of freedom (numerator) for the noncentral F-distribution. ncfdtri(p, dfn, dfd, nc) Inverse cumulative distribution function of the non-central F distribution. ncfdtrinc(p, f, dfn, dfd) Calculate non-centrality parameter for non-central F distribution. nctdtr(df, nc, t) Cumulative distribution function of the non-central t distribution. nctdtridf(p, nc, t) Calculate degrees of freedom for non-central t distribution. nctdtrit(df, nc, p) Inverse cumulative distribution function of the non-central t distribution. nctdtrinc(df, p, t) Calculate non-centrality parameter for non-central t distribution. nrdtrimn(p, x, std) Calculate mean of normal distribution given other params. nrdtrisd(p, x, mn) Calculate standard deviation of normal distribution given other params. pdtr(k, m) Poisson cumulative distribution function Returns the sum of the first k terms of the Poisson distribution: sum(exp(-m) * m**j / j!, j=0..k) = gammaincc( k+1, m). pdtrc(k, m) Poisson survival function Returns the sum of the terms from k+1 to infinity of the Poisson distribution: sum(exp(-m) * m**j / j!, j=k+1..inf) = gammainc( k+1, m). pdtri(k, y) Inverse to pdtr vs m Returns the Poisson variable m such that the sum from 0 to k of the Poisson density is equal to the given probability y: calculated by gammaincinv(k+1, y). pdtrik(p, m) Inverse to pdtr vs k stdtr(df, t) Student t distribution cumulative density function stdtridf(p, t) Inverse of stdtr vs df Returns the argument df such that stdtr(df, t) is equal to p. stdtrit(df, p) Inverse of stdtr vs t Returns the argument t such that stdtr(df, t) is equal to p. chdtr(v, x) Chi square cumulative distribution function chdtrc(v, x) Chi square survival function chdtri(v, p) Inverse to chdtrc Returns the argument x such that chdtrc(v, x) == p. chdtriv(p, x) Inverse to chdtr vs v Returns the argument v such that chdtr(v, x) == p. ndtr(x) Gaussian cumulative distribution function log_ndtr(x) Logarithm of Gaussian cumulative distribution function ndtri(y) Inverse of ndtr vs x Returns the argument x for which the area under the Gaussian probability density function (integrated from minus infinity to x) is equal to y. chndtr(x, df, nc) Non-central chi square cumulative distribution function chndtridf(x, p, nc) Inverse to chndtr vs df chndtrinc(x, df, p) Inverse to chndtr vs nc chndtrix(p, df, nc) Inverse to chndtr vs x smirnov(n, e) Kolmogorov-Smirnov complementary cumulative distribution function Returns the exact Kolmogorov-Smirnov complementary cumulative distribution function (Dn+ or Dn-) for a one-sided test of equality between an empirical and a theoretical distribution. smirnovi(n, y) Inverse to smirnov Returns e such that smirnov(n, e) = y. kolmogorov(y) Complementary cumulative distribution function of Kolmogorov distribution Returns the complementary cumulative distribution function of Kolmogorov’s limiting distribution (Kn* for large n) of a two-sided test for equality between an empirical and a theoretical distribution. kolmogi(p) Inverse function to kolmogorov Returns y such that kolmogorov(y) == p. tklmbda(x, lmbda) Tukey-Lambda cumulative distribution function logit(x) Logit ufunc for ndarrays. expit(x) Expit ufunc for ndarrays. boxcox(x, lmbda) Compute the Box-Cox transformation. boxcox1p(x, lmbda) Compute the Box-Cox transformation of 1 + x. inv_boxcox(y, lmbda) Compute the inverse of the Box-Cox transformation. inv_boxcox1p(y, lmbda) Compute the inverse of the Box-Cox transformation.

### Information Theory Functions¶

 entr(x) Elementwise function for computing entropy. rel_entr(x, y) Elementwise function for computing relative entropy. kl_div(x, y) Elementwise function for computing Kullback-Leibler divergence. huber(delta, r) Huber loss function. pseudo_huber(delta, r) Pseudo-Huber loss function.

### Error Function and Fresnel Integrals¶

 erf(z) Returns the error function of complex argument. erfc(x) Complementary error function, 1 - erf(x). erfcx(x) Scaled complementary error function, exp(x^2) erfc(x). erfi(z) Imaginary error function, -i erf(i z). erfinv(y) Inverse function for erf. erfcinv(y) Inverse function for erfc. wofz(z) Faddeeva function Returns the value of the Faddeeva function for complex argument:: exp(-z**2)*erfc(-i*z) .. dawsn(x) Dawson’s integral. fresnel(z) Fresnel sin and cos integrals fresnel_zeros(nt) Compute nt complex zeros of sine and cosine Fresnel integrals S(z) and C(z). modfresnelp(x) Modified Fresnel positive integrals modfresnelm(x) Modified Fresnel negative integrals

These are not universal functions:

 erf_zeros(nt) Compute nt complex zeros of error function erf(z). fresnelc_zeros(nt) Compute nt complex zeros of cosine Fresnel integral C(z). fresnels_zeros(nt) Compute nt complex zeros of sine Fresnel integral S(z).

### Legendre Functions¶

 lpmv(m, v, x) Associated legendre function of integer order. sph_harm(m, n, theta, phi) Compute spherical harmonics.

These are not universal functions:

 clpmn(m, n, z[, type]) Associated Legendre function of the first kind, Pmn(z). lpn(n, z) Legendre functions of the first kind, Pn(z). lqn(n, z) Legendre functions of the second kind, Qn(z). lpmn(m, n, z) Associated Legendre function of the first kind, Pmn(z). lqmn(m, n, z) Associated Legendre function of the second kind, Qmn(z).

### Ellipsoidal Harmonics¶

 ellip_harm(h2, k2, n, p, s[, signm, signn]) Ellipsoidal harmonic functions E^p_n(l) These are also known as Lame functions of the first kind, and are solutions to the Lame equation: .. ellip_harm_2(h2, k2, n, p, s) Ellipsoidal harmonic functions F^p_n(l) These are also known as Lame functions of the second kind, and are solutions to the Lame equation: .. ellip_normal(h2, k2, n, p) Ellipsoidal harmonic normalization constants gamma^p_n The normalization constant is defined as ..

### Orthogonal polynomials¶

The following functions evaluate values of orthogonal polynomials:

 assoc_laguerre(x, n[, k]) Compute nth-order generalized (associated) Laguerre polynomial. eval_legendre(n, x[, out]) Evaluate Legendre polynomial at a point. eval_chebyt(n, x[, out]) Evaluate Chebyshev T polynomial at a point. eval_chebyu(n, x[, out]) Evaluate Chebyshev U polynomial at a point. eval_chebyc(n, x[, out]) Evaluate Chebyshev C polynomial at a point. eval_chebys(n, x[, out]) Evaluate Chebyshev S polynomial at a point. eval_jacobi(n, alpha, beta, x[, out]) Evaluate Jacobi polynomial at a point. eval_laguerre(n, x[, out]) Evaluate Laguerre polynomial at a point. eval_genlaguerre(n, alpha, x[, out]) Evaluate generalized Laguerre polynomial at a point. eval_hermite(n, x[, out]) Evaluate Hermite polynomial at a point. eval_hermitenorm(n, x[, out]) Evaluate normalized Hermite polynomial at a point. eval_gegenbauer(n, alpha, x[, out]) Evaluate Gegenbauer polynomial at a point. eval_sh_legendre(n, x[, out]) Evaluate shifted Legendre polynomial at a point. eval_sh_chebyt(n, x[, out]) Evaluate shifted Chebyshev T polynomial at a point. eval_sh_chebyu(n, x[, out]) Evaluate shifted Chebyshev U polynomial at a point. eval_sh_jacobi(n, p, q, x[, out]) Evaluate shifted Jacobi polynomial at a point.

The functions below, in turn, return the polynomial coefficients in orthopoly1d objects, which function similarly as numpy.poly1d. The orthopoly1d class also has an attribute weights which returns the roots, weights, and total weights for the appropriate form of Gaussian quadrature. These are returned in an n x 3 array with roots in the first column, weights in the second column, and total weights in the final column. Note that orthopoly1d objects are converted to poly1d when doing arithmetic, and lose information of the original orthogonal polynomial.

 legendre(n[, monic]) Legendre polynomial coefficients Returns the nth-order Legendre polynomial, P_n(x), orthogonal over [-1, 1] with weight function 1. chebyt(n[, monic]) Return nth order Chebyshev polynomial of first kind, Tn(x). chebyu(n[, monic]) Return nth order Chebyshev polynomial of second kind, Un(x). chebyc(n[, monic]) Return n-th order Chebyshev polynomial of first kind, $$C_n(x)$$. chebys(n[, monic]) Return nth order Chebyshev polynomial of second kind, $$S_n(x)$$. jacobi(n, alpha, beta[, monic]) Returns the nth order Jacobi polynomial, P^(alpha,beta)_n(x) orthogonal over [-1,1] with weighting function (1-x)**alpha (1+x)**beta with alpha,beta > -1. laguerre(n[, monic]) Return the nth order Laguerre polynoimal, L_n(x), orthogonal over genlaguerre(n, alpha[, monic]) Returns the nth order generalized (associated) Laguerre polynomial, hermite(n[, monic]) Return the nth order Hermite polynomial, H_n(x), orthogonal over hermitenorm(n[, monic]) Return the nth order normalized Hermite polynomial, He_n(x), orthogonal gegenbauer(n, alpha[, monic]) Return the nth order Gegenbauer (ultraspherical) polynomial, sh_legendre(n[, monic]) Returns the nth order shifted Legendre polynomial, P^*_n(x), orthogonal over [0,1] with weighting function 1. sh_chebyt(n[, monic]) Return nth order shifted Chebyshev polynomial of first kind, Tn(x). sh_chebyu(n[, monic]) Return nth order shifted Chebyshev polynomial of second kind, Un(x). sh_jacobi(n, p, q[, monic]) Returns the nth order Jacobi polynomial, G_n(p,q,x) orthogonal over [0,1] with weighting function (1-x)**(p-q) (x)**(q-1) with p>q-1 and q > 0.

Warning

Computing values of high-order polynomials (around order > 20) using polynomial coefficients is numerically unstable. To evaluate polynomial values, the eval_* functions should be used instead.

Roots and weights for orthogonal polynomials

 c_roots(n[, mu]) Gauss-Chebyshev (first kind) quadrature. cg_roots(n, alpha[, mu]) Gauss-Gegenbauer quadrature. h_roots(n[, mu]) Gauss-Hermite (physicst’s) quadrature. he_roots(n[, mu]) Gauss-Hermite (statistician’s) quadrature. j_roots(n, alpha, beta[, mu]) Gauss-Jacobi quadrature. js_roots(n, p1, q1[, mu]) Gauss-Jacobi (shifted) quadrature. l_roots(n[, mu]) Gauss-Laguerre quadrature. la_roots(n, alpha[, mu]) Gauss-generalized Laguerre quadrature. p_roots(n[, mu]) Gauss-Legendre quadrature. ps_roots(n[, mu]) Gauss-Legendre (shifted) quadrature. s_roots(n[, mu]) Gauss-Chebyshev (second kind) quadrature. t_roots(n[, mu]) Gauss-Chebyshev (first kind) quadrature. ts_roots(n[, mu]) Gauss-Chebyshev (first kind, shifted) quadrature. u_roots(n[, mu]) Gauss-Chebyshev (second kind) quadrature. us_roots(n[, mu]) Gauss-Chebyshev (second kind, shifted) quadrature.

### Hypergeometric Functions¶

 hyp2f1(a, b, c, z) Gauss hypergeometric function 2F1(a, b; c; z). hyp1f1(a, b, x) Confluent hypergeometric function 1F1(a, b; x) hyperu(a, b, x) Confluent hypergeometric function U(a, b, x) of the second kind hyp0f1(v, z) Confluent hypergeometric limit function 0F1. hyp2f0(a, b, x, type) Hypergeometric function 2F0 in y and an error estimate The parameter type determines a convergence factor and can be either 1 or 2. hyp1f2(a, b, c, x) Hypergeometric function 1F2 and error estimate hyp3f0(a, b, c, x) Hypergeometric function 3F0 in y and an error estimate

### Parabolic Cylinder Functions¶

 pbdv(v, x) Parabolic cylinder function D Returns (d, dp) the parabolic cylinder function Dv(x) in d and the derivative, Dv’(x) in dp. pbvv(v, x) Parabolic cylinder function V Returns the parabolic cylinder function Vv(x) in v and the derivative, Vv’(x) in vp. pbwa(a, x) Parabolic cylinder function W Returns the parabolic cylinder function W(a, x) in w and the derivative, W’(a, x) in wp.

These are not universal functions:

 pbdv_seq(v, x) Parabolic cylinder functions Dv(x) and derivatives. pbvv_seq(v, x) Parabolic cylinder functions Vv(x) and derivatives. pbdn_seq(n, z) Parabolic cylinder functions Dn(z) and derivatives.

### Spheroidal Wave Functions¶

 pro_ang1(m, n, c, x) Prolate spheroidal angular function of the first kind and its derivative Computes the prolate spheroidal angular function of the first kind and its derivative (with respect to x) for mode parameters m>=0 and n>=m, spheroidal parameter c and |x| < 1.0. pro_rad1(m, n, c, x) Prolate spheroidal radial function of the first kind and its derivative Computes the prolate spheroidal radial function of the first kind and its derivative (with respect to x) for mode parameters m>=0 and n>=m, spheroidal parameter c and |x| < 1.0. pro_rad2(m, n, c, x) Prolate spheroidal radial function of the secon kind and its derivative Computes the prolate spheroidal radial function of the second kind and its derivative (with respect to x) for mode parameters m>=0 and n>=m, spheroidal parameter c and |x| < 1.0. obl_ang1(m, n, c, x) Oblate spheroidal angular function of the first kind and its derivative Computes the oblate spheroidal angular function of the first kind and its derivative (with respect to x) for mode parameters m>=0 and n>=m, spheroidal parameter c and |x| < 1.0. obl_rad1(m, n, c, x) Oblate spheroidal radial function of the first kind and its derivative Computes the oblate spheroidal radial function of the first kind and its derivative (with respect to x) for mode parameters m>=0 and n>=m, spheroidal parameter c and |x| < 1.0. obl_rad2(m, n, c, x) Oblate spheroidal radial function of the second kind and its derivative. pro_cv(m, n, c) Characteristic value of prolate spheroidal function Computes the characteristic value of prolate spheroidal wave functions of order m, n (n>=m) and spheroidal parameter c. obl_cv(m, n, c) Characteristic value of oblate spheroidal function Computes the characteristic value of oblate spheroidal wave functions of order m, n (n>=m) and spheroidal parameter c. pro_cv_seq(m, n, c) Characteristic values for prolate spheroidal wave functions. obl_cv_seq(m, n, c) Characteristic values for oblate spheroidal wave functions.

The following functions require pre-computed characteristic value:

 pro_ang1_cv(m, n, c, cv, x) Prolate spheroidal angular function pro_ang1 for precomputed characteristic value Computes the prolate spheroidal angular function of the first kind and its derivative (with respect to x) for mode parameters m>=0 and n>=m, spheroidal parameter c and |x| < 1.0. pro_rad1_cv(m, n, c, cv, x) Prolate spheroidal radial function pro_rad1 for precomputed characteristic value Computes the prolate spheroidal radial function of the first kind and its derivative (with respect to x) for mode parameters m>=0 and n>=m, spheroidal parameter c and |x| < 1.0. pro_rad2_cv(m, n, c, cv, x) Prolate spheroidal radial function pro_rad2 for precomputed characteristic value Computes the prolate spheroidal radial function of the second kind and its derivative (with respect to x) for mode parameters m>=0 and n>=m, spheroidal parameter c and |x| < 1.0. obl_ang1_cv(m, n, c, cv, x) Oblate spheroidal angular function obl_ang1 for precomputed characteristic value Computes the oblate spheroidal angular function of the first kind and its derivative (with respect to x) for mode parameters m>=0 and n>=m, spheroidal parameter c and |x| < 1.0. obl_rad1_cv(m, n, c, cv, x) Oblate spheroidal radial function obl_rad1 for precomputed characteristic value Computes the oblate spheroidal radial function of the first kind and its derivative (with respect to x) for mode parameters m>=0 and n>=m, spheroidal parameter c and |x| < 1.0. obl_rad2_cv(m, n, c, cv, x) Oblate spheroidal radial function obl_rad2 for precomputed characteristic value Computes the oblate spheroidal radial function of the second kind and its derivative (with respect to x) for mode parameters m>=0 and n>=m, spheroidal parameter c and |x| < 1.0.

### Kelvin Functions¶

 kelvin(x) Kelvin functions as complex numbers :Returns: Be, Ke, Bep, Kep The tuple (Be, Ke, Bep, Kep) contains complex numbers representing the real and imaginary Kelvin functions and their derivatives evaluated at x. kelvin_zeros(nt) Compute nt zeros of all Kelvin functions. ber(x) Kelvin function ber. bei(x) Kelvin function bei berp(x) Derivative of the Kelvin function ber beip(x) Derivative of the Kelvin function bei ker(x) Kelvin function ker kei(x) Kelvin function ker kerp(x) Derivative of the Kelvin function ker keip(x) Derivative of the Kelvin function kei

These are not universal functions:

 ber_zeros(nt) Compute nt zeros of the Kelvin function ber(x). bei_zeros(nt) Compute nt zeros of the Kelvin function bei(x). berp_zeros(nt) Compute nt zeros of the Kelvin function ber’(x). beip_zeros(nt) Compute nt zeros of the Kelvin function bei’(x). ker_zeros(nt) Compute nt zeros of the Kelvin function ker(x). kei_zeros(nt) Compute nt zeros of the Kelvin function kei(x). kerp_zeros(nt) Compute nt zeros of the Kelvin function ker’(x). keip_zeros(nt) Compute nt zeros of the Kelvin function kei’(x).

### Combinatorics¶

 comb(N, k[, exact, repetition]) The number of combinations of N things taken k at a time. perm(N, k[, exact]) Permutations of N things taken k at a time, i.e., k-permutations of N.

### Other Special Functions¶

 agm(a, b) Arithmetic, Geometric Mean. bernoulli(n) Bernoulli numbers B0..Bn (inclusive). binom(n, k) Binomial coefficient diric(x, n) Periodic sinc function, also called the Dirichlet function. euler(n) Euler numbers E0..En (inclusive). expn(n, x) Exponential integral E_n Returns the exponential integral for integer n and non-negative x and n:: integral(exp(-x*t) / t**n, t=1..inf). exp1(z) Exponential integral E_1 of complex argument z :: integral(exp(-z*t)/t, t=1..inf). expi(x) Exponential integral Ei Defined as:: integral(exp(t)/t, t=-inf..x) See expn for a different exponential integral. factorial(n[, exact]) The factorial function, n! = special.gamma(n+1). factorial2(n[, exact]) Double factorial. factorialk(n, k[, exact]) Multifactorial of n of order k, n(!!...!). shichi(x) Hyperbolic sine and cosine integrals :Returns: shi integral(sinh(t)/t, t=0..x) chi eul + ln x + integral((cosh(t)-1)/t, t=0..x) where eul is Euler’s constant. sici(x) Sine and cosine integrals :Returns: si integral(sin(t)/t, t=0..x) ci eul + ln x + integral((cos(t) - 1)/t, t=0..x) where eul is Euler’s constant. spence(x) Dilogarithm integral lambertw(z[, k, tol]) Lambert W function [R744]. zeta(x, q) Hurwitz zeta function The Riemann zeta function of two arguments (also known as the Hurwitz zeta function). zetac(x) Riemann zeta function minus 1.

### Convenience Functions¶

 cbrt(x) Cube root of x exp10(x) 10**x exp2(x) 2**x radian(d, m, s) Convert from degrees to radians Returns the angle given in (d)egrees, (m)inutes, and (s)econds in radians. cosdg(x) Cosine of the angle x given in degrees. sindg(x) Sine of angle given in degrees tandg(x) Tangent of angle x given in degrees. cotdg(x) Cotangent of the angle x given in degrees. log1p(x) Calculates log(1+x) for use when x is near zero expm1(x) exp(x) - 1 for use when x is near zero. cosm1(x) cos(x) - 1 for use when x is near zero. round(x) Round to nearest integer Returns the nearest integer to x as a double precision floating point result. xlogy(x, y) Compute x*log(y) so that the result is 0 if x = 0. xlog1py(x, y) Compute x*log1p(y) so that the result is 0 if x = 0. exprel(x) Relative error exponential, (exp(x)-1)/x, for use when x is near zero. sinc(x) Return the sinc function.