Special functions (scipy.special)#

Almost all of the functions below accept NumPy arrays as input arguments as well as single numbers. This means they follow broadcasting and automatic array-looping rules. Technically, they are NumPy universal functions. Functions which do not accept NumPy arrays are marked by a warning in the section description.

See also

scipy.special.cython_special – Typed Cython versions of special functions

Error handling#

Errors are handled by returning NaNs or other appropriate values. Some of the special function routines can emit warnings or raise exceptions when an error occurs. By default this is disabled, except for memory allocation errors, which result in an exception being raised. To query and control the current error handling state the following functions are provided.

geterr()

Get the current way of handling special-function errors.

seterr(**kwargs)

Set how special-function errors are handled.

errstate(**kwargs)

Context manager for special-function error handling.

SpecialFunctionWarning

Warning that can be emitted by special functions.

SpecialFunctionError

Exception that can be raised by special functions.

Available functions#

Airy functions#

airy

airye

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

Elliptic functions and integrals#

ellipj

ellipk

ellipkm1

ellipkinc

ellipe

ellipeinc

elliprc

elliprd

elliprf

elliprg

elliprj

Bessel functions#

jv

jve

yn

yv

yve

iv

ive

kn

kv

kve

hankel1

hankel1e

hankel2

hankel2e

wright_bessel

log_wright_bessel

The following function does not accept NumPy arrays (it is not a universal function):

lmbda(v, x)

Jahnke-Emden Lambda function, Lambdav(x).

Zeros of Bessel functions#

The following functions do not accept NumPy arrays (they 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 functions Jn.

jnp_zeros(n, nt)

Compute zeros of integer-order Bessel function derivatives Jn'.

yn_zeros(n, nt)

Compute zeros of integer-order Bessel function Yn(x).

ynp_zeros(n, nt)

Compute zeros of integer-order Bessel function derivatives 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

j1

y0

y1

i0

i0e

i1

i1e

k0

k0e

k1

k1e

Integrals of Bessel functions#

itj0y0

it2j0y0

iti0k0

it2i0k0

besselpoly

Derivatives of Bessel functions#

jvp(v, z[, n])

Compute derivatives of Bessel functions of the first kind.

yvp(v, z[, n])

Compute derivatives of Bessel functions of the second kind.

ivp(v, z[, n])

Compute derivatives of modified Bessel functions of the first kind.

kvp(v, z[, n])

Compute derivatives of real-order modified Bessel function Kv(z)

h1vp(v, z[, n])

Compute derivatives of Hankel function H1v(z) with respect to z.

h2vp(v, z[, n])

Compute derivatives of Hankel function H2v(z) with respect to z.

Spherical Bessel functions#

spherical_jn(n, z[, derivative])

Spherical Bessel function of the first kind or its derivative.

spherical_yn(n, z[, derivative])

Spherical Bessel function of the second kind or its derivative.

spherical_in(n, z[, derivative])

Modified spherical Bessel function of the first kind or its derivative.

spherical_kn(n, z[, derivative])

Modified spherical Bessel function of the second kind or its derivative.

Riccati-Bessel functions#

The following functions do not accept NumPy arrays (they are not universal functions):

riccati_jn(n, x)

Compute Riccati-Bessel function of the first kind and its derivative.

riccati_yn(n, x)

Compute Riccati-Bessel function of the second kind and its derivative.

Struve functions#

struve

modstruve

itstruve0

it2struve0

itmodstruve0

Raw statistical functions#

See also

scipy.stats: Friendly versions of these functions.

Binomial distribution#

bdtr

bdtrc

bdtri

bdtrik

bdtrin

Beta distribution#

btdtria

btdtrib

F distribution#

fdtr

fdtrc

fdtri

fdtridfd

Gamma distribution#

gdtr

gdtrc

gdtria

gdtrib

gdtrix

Negative binomial distribution#

nbdtr

nbdtrc

nbdtri

nbdtrik

nbdtrin

Noncentral F distribution#

ncfdtr

ncfdtridfd

ncfdtridfn

ncfdtri

ncfdtrinc

Noncentral t distribution#

nctdtr

nctdtridf

nctdtrit

nctdtrinc

Normal distribution#

nrdtrimn

nrdtrisd

ndtr

log_ndtr

ndtri

ndtri_exp

Poisson distribution#

pdtr

pdtrc

pdtri

pdtrik

Student t distribution#

stdtr

stdtridf

stdtrit

Chi square distribution#

chdtr

chdtrc

chdtri

chdtriv

Non-central chi square distribution#

chndtr

chndtridf

chndtrinc

chndtrix

Kolmogorov distribution#

smirnov

smirnovi

kolmogorov

kolmogi

Box-Cox transformation#

boxcox

boxcox1p

inv_boxcox

inv_boxcox1p

Sigmoidal functions#

logit

expit

log_expit

Miscellaneous#

tklmbda

owens_t

Information Theory functions#

entr

rel_entr

kl_div

huber

pseudo_huber

Error function and Fresnel integrals#

erf

erfc

erfcx

erfi

erfinv

erfcinv

wofz

dawsn

fresnel

fresnel_zeros(nt)

Compute nt complex zeros of sine and cosine Fresnel integrals S(z) and C(z).

modfresnelp

modfresnelm

voigt_profile

The following functions do not accept NumPy arrays (they are not universal functions):

erf_zeros(nt)

Compute the first nt zero in the first quadrant, ordered by absolute value.

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#

legendre_p

legendre_p_all

assoc_legendre_p

assoc_legendre_p_all

sph_legendre_p

sph_legendre_p_all

sph_harm_y

sph_harm_y_all

The following functions are in the process of being deprecated in favor of the above, which provide a more flexible and consistent interface.

lpmv

lqn(n, z)

Legendre function of the second kind.

lqmn(m, n, z)

Sequence of associated Legendre functions of the second kind.

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])

Compute the generalized (associated) Laguerre polynomial of degree n and order k.

eval_legendre

eval_chebyt

eval_chebyu

eval_chebyc

eval_chebys

eval_jacobi

eval_laguerre

eval_genlaguerre

eval_hermite

eval_hermitenorm

eval_gegenbauer

eval_sh_legendre

eval_sh_chebyt

eval_sh_chebyu

eval_sh_jacobi

The following functions compute roots and quadrature weights for orthogonal polynomials:

roots_legendre(n[, mu])

Gauss-Legendre quadrature.

roots_chebyt(n[, mu])

Gauss-Chebyshev (first kind) quadrature.

roots_chebyu(n[, mu])

Gauss-Chebyshev (second kind) quadrature.

roots_chebyc(n[, mu])

Gauss-Chebyshev (first kind) quadrature.

roots_chebys(n[, mu])

Gauss-Chebyshev (second kind) quadrature.

roots_jacobi(n, alpha, beta[, mu])

Gauss-Jacobi quadrature.

roots_laguerre(n[, mu])

Gauss-Laguerre quadrature.

roots_genlaguerre(n, alpha[, mu])

Gauss-generalized Laguerre quadrature.

roots_hermite(n[, mu])

Gauss-Hermite (physicist's) quadrature.

roots_hermitenorm(n[, mu])

Gauss-Hermite (statistician's) quadrature.

roots_gegenbauer(n, alpha[, mu])

Gauss-Gegenbauer quadrature.

roots_sh_legendre(n[, mu])

Gauss-Legendre (shifted) quadrature.

roots_sh_chebyt(n[, mu])

Gauss-Chebyshev (first kind, shifted) quadrature.

roots_sh_chebyu(n[, mu])

Gauss-Chebyshev (second kind, shifted) quadrature.

roots_sh_jacobi(n, p1, q1[, mu])

Gauss-Jacobi (shifted) quadrature.

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.

chebyt(n[, monic])

Chebyshev polynomial of the first kind.

chebyu(n[, monic])

Chebyshev polynomial of the second kind.

chebyc(n[, monic])

Chebyshev polynomial of the first kind on \([-2, 2]\).

chebys(n[, monic])

Chebyshev polynomial of the second kind on \([-2, 2]\).

jacobi(n, alpha, beta[, monic])

Jacobi polynomial.

laguerre(n[, monic])

Laguerre polynomial.

genlaguerre(n, alpha[, monic])

Generalized (associated) Laguerre polynomial.

hermite(n[, monic])

Physicist's Hermite polynomial.

hermitenorm(n[, monic])

Normalized (probabilist's) Hermite polynomial.

gegenbauer(n, alpha[, monic])

Gegenbauer (ultraspherical) polynomial.

sh_legendre(n[, monic])

Shifted Legendre polynomial.

sh_chebyt(n[, monic])

Shifted Chebyshev polynomial of the first kind.

sh_chebyu(n[, monic])

Shifted Chebyshev polynomial of the second kind.

sh_jacobi(n, p, q[, monic])

Shifted Jacobi polynomial.

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

hyp1f1

hyperu

hyp0f1

Parabolic cylinder functions#

pbdv

pbvv

pbwa

The following functions do not accept NumPy arrays (they 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

pro_rad1

pro_rad2

obl_ang1

obl_rad1

obl_rad2

pro_cv

obl_cv

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

pro_rad1_cv

pro_rad2_cv

obl_ang1_cv

obl_rad1_cv

obl_rad2_cv

Kelvin functions#

kelvin

kelvin_zeros(nt)

Compute nt zeros of all Kelvin functions.

ber

bei

berp

beip

ker

kei

kerp

keip

The following functions do not accept NumPy arrays (they are not universal functions):

ber_zeros(nt)

Compute nt zeros of the Kelvin function ber.

bei_zeros(nt)

Compute nt zeros of the Kelvin function bei.

berp_zeros(nt)

Compute nt zeros of the derivative of the Kelvin function ber.

beip_zeros(nt)

Compute nt zeros of the derivative of the Kelvin function bei.

ker_zeros(nt)

Compute nt zeros of the Kelvin function ker.

kei_zeros(nt)

Compute nt zeros of the Kelvin function kei.

kerp_zeros(nt)

Compute nt zeros of the derivative of the Kelvin function ker.

keip_zeros(nt)

Compute nt zeros of the derivative of the Kelvin function kei.

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.

stirling2(N, K, *[, exact])

Generate Stirling number(s) of the second kind.

Other special functions#

agm

bernoulli(n)

Bernoulli numbers B0..Bn (inclusive).

binom

diric(x, n)

Periodic sinc function, also called the Dirichlet kernel.

euler(n)

Euler numbers E(0), E(1), ..., E(n).

expn

exp1

expi

factorial(n[, exact, extend])

The factorial of a number or array of numbers.

factorial2(n[, exact, extend])

Double factorial.

factorialk(n, k[, exact, extend])

Multifactorial of n of order k, n(!!...!).

shichi

sici

softmax(x[, axis])

Compute the softmax function.

log_softmax(x[, axis])

Compute the logarithm of the softmax function.

spence

zeta(x[, q, out])

Riemann or Hurwitz zeta function.

zetac

softplus(x, **kwargs)

Compute the softplus function element-wise.

Convenience functions#

cbrt

exp10

exp2

radian

cosdg

sindg

tandg

cotdg

log1p

expm1

cosm1

powm1

round

xlogy

xlog1py

logsumexp(a[, axis, b, keepdims, return_sign])

Compute the log of the sum of exponentials of input elements.

exprel

sinc(x)

Return the normalized sinc function.