# 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 the zeros of Airy Functions Ai(x) and Ai’(x), a and a’ respectively, and the associated values of Ai(a’) and Ai’(a). bi_zeros(nt) Compute the zeros of Airy Functions Bi(x) and Bi’(x), b and b’ respectively, and the associated values of Ai(b’) and Ai’(b).

### Elliptic Functions and Integrals¶

 ellipj(u, m) Jacobian elliptic functions ellipk(m) Complete elliptic integral of the first kind ellipkm1(p) Complete elliptic integral of the first kind around m = 1 ellipkinc(phi, m) Incomplete elliptic integral of the first kind ellipe(m) Complete elliptic integral of the second kind ellipeinc(phi, m) Incomplete elliptic integral of the second kind

### Bessel Functions¶

 jv(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 yv(v,z) Bessel function of the second kind of real order 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 kv(v,z) Modified Bessel function of the second kind of real order v 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 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) Compute sequence of lambda functions with arbitrary order v and their derivatives.

#### Zeros of Bessel Functions¶

These are not universal functions:

 jnjnp_zeros(nt) Compute nt (<=1200) zeros of the Bessel functions Jn and Jn’ and arange them in order of their magnitudes. jnyn_zeros(n, nt) Compute nt zeros of the Bessel functions Jn(x), Jn’(x), Yn(x), and Yn’(x), respectively. jn_zeros(n, nt) Compute nt zeros of the Bessel function Jn(x). jnp_zeros(n, nt) Compute nt zeros of the Bessel function Jn’(x). yn_zeros(n, nt) Compute nt zeros of the Bessel function Yn(x). ynp_zeros(n, nt) Compute nt zeros of the Bessel function Yn’(x). y0_zeros(nt[, complex]) Returns nt (complex or real) zeros of Y0(z), z0, and the value of Y0’(z0) = -Y1(z0) at each zero. y1_zeros(nt[, complex]) Returns nt (complex or real) zeros of Y1(z), z1, and the value of Y1’(z1) = Y0(z1) at each zero. y1p_zeros(nt[, complex]) Returns nt (complex or real) zeros of Y1’(z), z1’, and the value of Y1(z1’) 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 y1(x) Bessel function of the second kind of order 1 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 0. k0(x) Modified Bessel function K of order 0 k0e(x) Exponentially scaled modified Bessel function K of order 0 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 it2j0y0(x) Integrals related to Bessel functions of order 0 iti0k0(x) Integrals of modified Bessel functions of order 0 it2i0k0(x) Integrals related to modified Bessel functions of order 0 besselpoly(a, lmb, nu) Weighed integral of a Bessel function.

#### Derivatives of Bessel Functions¶

 jvp(v, z[, n]) Return the nth derivative of Jv(z) with respect to z. yvp(v, z[, n]) Return the nth derivative of Yv(z) with respect to z. kvp(v, z[, n]) Return the nth derivative of Kv(z) with respect to z. ivp(v, z[, n]) Return the nth derivative of Iv(z) with respect to z. h1vp(v, z[, n]) Return the nth derivative of H1v(z) with respect to z. h2vp(v, z[, n]) Return the nth derivative of H2v(z) with respect to z.

#### Spherical Bessel Functions¶

These are not universal functions:

 sph_jn(n, z) Compute the spherical Bessel function jn(z) and its derivative for all orders up to and including n. sph_yn(n, z) Compute the spherical Bessel function yn(z) and its derivative for all orders up to and including n. sph_jnyn(n, z) Compute the spherical Bessel functions, jn(z) and yn(z) and their derivatives for all orders up to and including n. sph_in(n, z) Compute the spherical Bessel function in(z) and its derivative for all orders up to and including n. sph_kn(n, z) Compute the spherical Bessel function kn(z) and its derivative for all orders up to and including n. sph_inkn(n, z) Compute the spherical Bessel functions, in(z) and kn(z) and their derivatives for all orders up to and including n.

#### Riccati-Bessel Functions¶

These are not universal functions:

 riccati_jn(n, x) Compute the Ricatti-Bessel function of the first kind and its derivative for all orders up to and including n. riccati_yn(n, x) Compute the Ricatti-Bessel function of the second kind and its derivative for all orders up to and including n.

### Struve Functions¶

 struve(v,x) Struve function modstruve(v, x) Modified Struve function 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¶

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. btdtr(a,b,x) Cumulative beta distribution. btdtri(a,b,p) p-th quantile of the beta distribution. fdtr(dfn, dfd, x) F cumulative distribution function fdtrc(dfn, dfd, x) F survival function fdtri(dfn, dfd, p) Inverse to fdtr vs x 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 nbdtrc(k,n,p) Negative binomial survival function nbdtri(k, n, y) Inverse of nbdtr vs p 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 pdtrc(k, m) Poisson survival function pdtri(k,y) Inverse to pdtr vs m stdtr(df,t) Student t distribution cumulative density function stdtridf(p,t) Inverse of stdtr vs df stdtrit(df,p) Inverse of stdtr vs t chdtr(v, x) Chi square cumulative distribution function chdtrc(v,x) Chi square survival function chdtri(v,p) Inverse to chdtrc ndtr(x) Gaussian cumulative distribution function ndtri(y) Inverse of ndtr vs x smirnov(n, e) Kolmogorov-Smirnov complementary cumulative distribution function smirnovi(n, y) Inverse to smirnov kolmogorov(y) Complementary cumulative distribution function of Kolmogorov distribution kolmogi(p) Inverse function to kolmogorov 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.

### 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 dawsn(x) Dawson’s integral. fresnel(z) Fresnel sin and cos integrals fresnel_zeros(nt) Compute nt complex zeros of the 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 the error function erf(z). fresnelc_zeros(nt) Compute nt complex zeros of the cosine Fresnel integral C(z). fresnels_zeros(nt) Compute nt complex zeros of the 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) Compute sequence of Legendre functions of the first kind (polynomials), Pn(z) and derivatives for all degrees from 0 to n (inclusive). lqn(n, z) Compute sequence of Legendre functions of the second kind, Qn(z) and derivatives for all degrees from 0 to n (inclusive). lpmn(m, n, z) Associated Legendre function of the first kind, Pmn(z) lqmn(m, n, z) Associated Legendre functions of the second kind, Qmn(z) and its derivative, Qmn'(z) of order m and degree n.

### Ellipsoidal Harmonics¶

 ellip_harm(h2, k2, n, p, s[, signm, signn]) Ellipsoidal harmonic functions E^p_n(l) ellip_harm_2(h2, k2, n, p, s) Ellipsoidal harmonic functions F^p_n(l) ellip_normal(h2, k2, n, p) Ellipsoidal harmonic normalization constants gamma^p_n

### Orthogonal polynomials¶

The following functions evaluate values of orthogonal polynomials:

 assoc_laguerre(x, n[, k]) Returns the n-th 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 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 nth order Chebyshev polynomial of first kind, Cn(x). chebys(n[, monic]) Return nth order Chebyshev polynomial of second kind, Sn(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.

### 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 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 pbvv(v,x) Parabolic cylinder function V pbwa(a,x) Parabolic cylinder function W

These are not universal functions:

 pbdv_seq(v, x) Compute sequence of parabolic cylinder functions Dv(x) and their derivatives for Dv0(x)..Dv(x) with v0=v-int(v). pbvv_seq(v, x) Compute sequence of parabolic cylinder functions Dv(x) and their derivatives for Dv0(x)..Dv(x) with v0=v-int(v). pbdn_seq(n, z) Compute sequence of parabolic cylinder functions Dn(z) and their derivatives for D0(z)..Dn(z).

### Spheroidal Wave Functions¶

 pro_ang1(m,n,c,x) Prolate spheroidal angular function of the first kind and its derivative pro_rad1(m,n,c,x) Prolate spheroidal radial function of the first kind and its derivative pro_rad2(m,n,c,x) Prolate spheroidal radial function of the secon kind and its derivative obl_ang1(m, n, c, x) Oblate spheroidal angular function of the first kind and its derivative obl_rad1(m,n,c,x) Oblate spheroidal radial function of the first kind and its derivative 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 obl_cv(m, n, c) Characteristic value of oblate spheroidal function pro_cv_seq(m, n, c) Compute a sequence of characteristic values for the prolate spheroidal wave functions for mode m and n’=m..n and spheroidal parameter c. obl_cv_seq(m, n, c) Compute a sequence of characteristic values for the oblate spheroidal wave functions for mode m and n’=m..n and spheroidal parameter c.

The following functions require pre-computed characteristic value:

### Kelvin Functions¶

 kelvin(x) Kelvin functions as complex numbers kelvin_zeros(nt) Compute nt zeros of all the Kelvin functions returned in a length 8 tuple of arrays of length nt. 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) Return an array of the Bernoulli numbers B0..Bn binom(n, k) Binomial coefficient diric(x, n) Return the periodic sinc function, also called the Dirichlet function. euler(n) Return an array of the Euler numbers E0..En (inclusive) expn(n, x) Exponential integral E_n exp1(z) Exponential integral E_1 of complex argument z expi(x) Exponential integral Ei factorial(n[, exact]) The factorial function, n! = special.gamma(n+1). factorial2(n[, exact]) Double factorial. factorialk(n, k[, exact]) n(!!...!) = multifactorial of order k shichi(x) Hyperbolic sine and cosine integrals sici(x) Sine and cosine integrals spence(x) Dilogarithm integral lambertw(z[, k, tol]) Lambert W function [R566]. zeta(x, q) 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 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 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.