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
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. |
| 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. |
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:
| pro_ang1_cv(m,n,c,cv,x) |
Prolate spheroidal angular function pro_ang1 for precomputed characteristic value |
| pro_rad1_cv(m,n,c,cv,x) |
Prolate spheroidal radial function pro_rad1 for precomputed characteristic value |
| pro_rad2_cv(m,n,c,cv,x) |
Prolate spheroidal radial function pro_rad2 for precomputed characteristic value |
| obl_ang1_cv(m, n, c, cv, x) |
Oblate spheroidal angular function obl_ang1 for precomputed characteristic value |
| obl_rad1_cv(m,n,c,cv,x) |
Oblate spheroidal radial function obl_rad1 for precomputed characteristic value |
| obl_rad2_cv(m,n,c,cv,x) |
Oblate spheroidal radial function obl_rad2 for precomputed 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) |
Returns 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 [R497]. |
| 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. |
