Crate xsf 

This crate provides equivalent Rust implementations of scipy.special functions, and bindings to the bundled xsf C++ library that powers scipy.special.

Most of the scipy.special functions are available. See the scipy.special documentation for additional information.

Airy functions

Function	Description
airy	Airy functions and derivatives
airy_scaled	Exponentially scaled Airy functions and derivatives
airy_ai_zeros	Zeros and values of the Airy function Ai and its derivative
airy_bi_zeros	Zeros and values of the Airy function Bi and its derivative
airy_integrals	Integrals of Airy functions

Elliptic functions and integrals

Function	Description
ellipj	Jacobian elliptic functions
ellipk	Complete elliptic integral of the first kind
ellipkm1	Complete elliptic integral of the first kind around $m = 1$
ellipkinc	Incomplete elliptic integral of the first kind
ellipe	Complete elliptic integral of the second kind
ellipeinc	Incomplete elliptic integral of the second kind

Bessel functions

Function	Description
bessel_j	Bessel function of the first kind, $J_v(z)$
bessel_je	Exponentially scaled Bessel function of the first kind
bessel_y	Bessel function of the second kind, $Y_v(z)$
bessel_ye	Exponentially scaled Bessel function of the second kind
bessel_i	Modified Bessel function of the first kind, $I_v(z)$
bessel_ie	Exponentially scaled modified Bessel function of the first kind
bessel_k	Modified Bessel function of the second kind, $K_v(z)$
bessel_ke	Exponentially scaled modified Bessel function of the second kind
hankel_1	Hankel function of the first kind, $H_v^{(1)}(z)$
hankel_1e	Exponentially scaled Hankel function of the first kind
hankel_2	Hankel function of the second kind, $H_v^{(2)}(z)$
hankel_2e	Exponentially scaled Hankel function of the second kind
wright_bessel	Wright’s generalized Bessel function
log_wright_bessel	Natural logarithm of Wright’s generalized Bessel function
jahnke_emden_lambda	Jahnke-Emden Lambda function $\Lambda_{\nu}(x)$ and derivatives

Zeros of Bessel functions

Function	Description
bessel_zeros	Zeros of Bessel functions $J_v(x)$, $J_v'(x)$, $Y_v(x)$, and $Y_v'(x)$

Faster versions of common Bessel functions

Function	Description
bessel_j0	Bessel function of the first kind of order 0, $J_0(x)$
bessel_j1	Bessel function of the first kind of order 1, $J_1(x)$
bessel_y0	Bessel function of the second kind of order 0, $Y_0(x)$
bessel_y1	Bessel function of the second kind of order 1, $Y_1(x)$
bessel_i0	Modified Bessel function of the first kind of order 0, $I_0(x)$
bessel_i0e	Exponentially scaled modified Bessel function of the first kind of order 0
bessel_i1	Modified Bessel function of the first kind of order 1, $I_1(x)$
bessel_i1e	Exponentially scaled modified Bessel function of the first kind of order 1
bessel_k0	Modified Bessel function of the second kind of order 0, $K_0(x)$
bessel_k0e	Exponentially scaled modified Bessel function of the second kind of order 0
bessel_k1	Modified Bessel function of the second kind of order 1, $K_1(x)$
bessel_k1e	Exponentially scaled modified Bessel function of the second kind of order 1

Integrals of Bessel functions

Function	Description
it1j0y0	Integral of Bessel functions of the first kind of order 0
it2j0y0	Integral related to Bessel functions of the first kind of order 0
it1i0k0	Integral of modified Bessel functions of the second kind of order 0
it2i0k0	Integral related to modified Bessel functions of the second kind of order 0
besselpoly	Weighted integral of the Bessel function of the first kind

Derivatives of Bessel functions

Function	Description
bessel_j_prime	$n$-th derivative of bessel_j
bessel_y_prime	$n$-th derivative of bessel_y
bessel_i_prime	$n$-th derivative of bessel_i
bessel_k_prime	$n$-th derivative of bessel_k
hankel_1_prime	$n$-th derivative of hankel_1
hankel_2_prime	$n$-th derivative of hankel_2

Spherical Bessel functions

Function	Description
sph_bessel_j	Spherical Bessel function of the first kind, $j_n(z)$
sph_bessel_j_prime	Derivative of sph_bessel_j, $j_n'(z)$
sph_bessel_y	Spherical Bessel function of the second kind, $y_n(z)$
sph_bessel_y_prime	Derivative of sph_bessel_y, $y_n'(z)$
sph_bessel_i	Modified Spherical Bessel function of the first kind, $i_n(z)$
sph_bessel_i_prime	Derivative of sph_bessel_i, $i_n'(z)$
sph_bessel_k	Modified Spherical Bessel function of the second kind, $k_n(z)$
sph_bessel_k_prime	Derivative of sph_bessel_k, $k_n'(z)$

Riccati-Bessel functions

Function	Description
riccati_j	Riccati-Bessel function of the first kind and its derivative
riccati_y	Riccati-Bessel function of the second kind and its derivative

Struve functions

Function	Description
struve_h	Struve function $H_{\nu}(x)$
struve_l	Modified Struve function $L_{\nu}(x)$
itstruve0	Integral of the Struve function of order 0, $H_0(x)$
it2struve0	Integral related to the Struve function of order 0
itmodstruve0	Integral of the modified Struve function of order 0, $L_0(x)$

Raw statistical functions

Binomial distribution

Function	Description
bdtr	Cumulative distribution function
bdtrc	Complement of bdtr
bdtri	Inverse of bdtr

F distribution

Function	Description
fdtr	Cumulative distribution function
fdtrc	Complement of fdtr
fdtri	Inverse of fdtr

Gamma distribution

Function	Description
gdtr	Cumulative distribution function
gdtrc	Complement of gdtr
gdtrib	Inverse of gdtr(a, b, x) with respect to b

Negative binomial distribution

Function	Description
nbdtr	Cumulative distribution function
nbdtrc	Complement of nbdtr
nbdtri	Inverse of nbdtr

Normal distribution

Function	Description
ndtr	Cumulative distribution function
ndtri	Inverse of ndtr
log_ndtr	Logarithm of ndtr
ndtri_exp	Inverse of log_ndtr

Poisson distribution

Function	Description
pdtr	Cumulative distribution function
pdtrc	Complement of pdtr
pdtri	Inverse of pdtr

Student’s t distribution

Function	Description
stdtr	Cumulative distribution function
stdtri	Inverse of stdtr

Chi square distribution

Function	Description
chdtr	Cumulative distribution function
chdtrc	Complement of chdtr
chdtri	Inverse of chdtr

Kolmogorov distribution

Function	Description
kolmogorov	Survival function
kolmogp	Derivative of kolmogorov
kolmogi	Inverse of kolmogorov
kolmogc	Complement of kolmogorov
kolmogci	Inverse of kolmogc

Kolmogorov-Smirnov distribution

Function	Description
smirnov	Survival function
smirnovp	Derivative of smirnov
smirnovi	Inverse of smirnov
smirnovc	Complement of smirnov
smirnovci	Inverse of smirnovc

Box-Cox transformation

Function	Description
boxcox	Box-Cox transformation of $x$
boxcox1p	Box-Cox transformation of $1 + x$
inv_boxcox	Inverse of boxcox
inv_boxcox1p	Inverse of boxcox1p

Sigmoidal functions

Function	Description
logit	Logit function, $\ln ( \frac{x}{1-x} )$
expit	Expit function, $\frac{1}{1 + \exp(-x)}$
log_expit	Logarithm of expit

Miscellaneous

Function	Description
tukeylambdacdf	Tukey-Lambda cumulative distribution function
owens_t	Owen’s T function

Information Theory functions

Function	Description
entr	Elementwise function for computing entropy, $H[X]$
rel_entr	Elementwise function for computing relative entropy, $H[X \rvert Y]$
kl_div	Elementwise function for computing Kullback-Leibler divergence
huber	Huber loss function, $L_\delta(r)$
pseudo_huber	Pseudo-Huber loss function, $\widetilde{L}_\delta(r)$

Gamma and related functions

Function	Description
gamma	Gamma function, $\Gamma(z)$
gammaln	Log-gamma function, $\ln\abs{\Gamma(z)}$
loggamma	Principal branch of $\ln \Gamma(z)$
gammasgn	Sign of gamma, $\sgn(\Gamma(z))$
gammainc	Regularized lower incomplete gamma function $P(a,x) = 1 - Q(a,x)$
gammaincinv	Inverse of gammainc, $P^{-1}(a,y)$
gammaincc	Regularized upper incomplete gamma function $Q(a,x) = 1 - P(a,x)$
gammainccinv	Inverse of gammaincc, $Q^{-1}(a,y)$
beta	Beta function, $\B(a,b) = {\Gamma(a)\Gamma(b) \over \Gamma(a+b)}$
betaln	Log-Beta function, $\ln\abs{\B(a,b)}$
betainc	Regularized incomplete beta function, $\I_x(a,b)$
betaincinv	Inverse of betainc, $\I_y^{-1}(a,b)$
digamma	The digamma function, $\psi(z)$
polygamma	The polygamma function, $\psi^{(n)}(x)$
multigammaln	Log of multivariate gamma, sometimes called the generalized gamma
rgamma	Reciprocal of the gamma function, $\frac{1}{\Gamma(z)}$
pow_rising	Rising factorial $\rpow x m = {\Gamma(x+m) \over \Gamma(x)}$
pow_falling	Falling factorial $\fpow x m = {\Gamma(x+1) \over \Gamma(x+1-m)}$

Error function and Fresnel integrals

Function	Description
erf	Error function, $\erf(z)$
erfc	Complementary error function, $\erfc(z) = 1 - \erf(z)$
erfcx	Scaled complementary error function, $e^{z^2} \erfc(z)$
erfi	Imaginary error function $\erfi(z) = -i \erf(i z)$
erfinv	Inverse of erf, $\erf^{-1}(z)$
erfcinv	Inverse of erfc, $\erfc^{-1}(z) = \erf^{-1}(1 - z)$
erf_zeros	Zeros (roots) of erf
wofz	Faddeeva function, $w(z) = e^{-z^2} \erfc(-iz)$
dawsn	Dawson function $D(z) = \frac{\sqrt{\pi}}{2} e^{-z^2} \erfi(z)$
fresnel	Fresnel integrals $S(z)$ and $C(z)$
fresnel_zeros	Zeros (roots) of Fresnel integrals $S(z)$ and $C(z)$
modified_fresnel_plus	Modified Fresnel positive integrals
modified_fresnel_minus	Modified Fresnel negative integrals
voigt_profile	Voigt profile

Legendre functions

Function	Description
legendre_p	Legendre polynomial of the first kind, $P_n(z)$
legendre_p_all	All Legendre polynomials of the first kind
assoc_legendre_p	Associated Legendre polynomial of the 1st kind, $P_n^m(z)$
assoc_legendre_p_all	All associated Legendre polynomials of the 1st kind
assoc_legendre_p_norm	Normalized associated Legendre polynomial
assoc_legendre_p_norm_all	All normalized associated Legendre polynomials
sph_legendre_p	Spherical Legendre polynomial of the first kind
sph_legendre_p_all	All spherical Legendre polynomials of the first kind
sph_harm_y	Spherical harmonics, $Y_n^m(\theta,\phi)$
sph_harm_y_all	All spherical harmonics
legendre_q_all	All Legendre functions of the 2nd kind and derivatives
assoc_legendre_q_all	All associated Legendre functions of the 2nd kind and derivatives

Orthogonal polynomials

The following functions evaluate values of orthogonal polynomials:

Function	Name	Notation
eval_jacobi	Jacobi	$P_n^{(\alpha,\beta)}(z)$
eval_legendre	Legendre	$P_n(z)$
eval_chebyshev_t	Chebyshev (first kind)	$T_n(z)$
eval_chebyshev_u	Chebyshev (second kind)	$U_n(z)$
eval_gegenbauer	Gegenbauer / Ultraspherical	$C_n^{(\alpha)}(z)$
eval_genlaguerre	Generalized Laguerre	$L_n^{(\alpha)}(z)$
eval_laguerre	Laguerre	$L_n(z)$
eval_hermite_h	Hermite (physicist’s)	$H_n(x)$
eval_hermite_he	Hermite (probabilist’s)	$He_n(x)$

Hypergeometric functions

Function	Description	Notation
hyp0f0	Generalized hypergeometric function	$_0F_0\left[ \middle| z\right]$
hyp1f0	Generalized hypergeometric function	$_1F_0\left[a\middle| z\right]$
hyp0f1	Confluent hypergeometric limit function	$_0F_1\left[b\middle| z\right]$
hyp1f1	Confluent hypergeometric function	$\hyp 1 1 a b z$
hyp2f1	Gauss’ hypergeometric function	$\hyp 2 1 {a_1\enspace a_2} b z$
hypu	Confluent hypergeometric function	$U(a_1,a_2,x)$

Parabolic cylinder functions

Function	Description
pbdv	Parabolic cylinder function $D_v(x)$ and its derivative $D_v'(x)$
pbvv	Parabolic cylinder function $V_v(x)$ and its derivative $V_v'(x)$
pbwa	Parabolic cylinder function $W_a(x)$ and its derivative $W_a'(x)$

Mathieu and related functions

Even	Odd	Description
mathieu_a	mathieu_b	Characteristic value of the Mathieu functions
mathieu_cem	mathieu_sem	Mathieu functions
mathieu_modcem1	mathieu_modsem1	Modified Mathieu functions of the first kind
mathieu_modcem2	mathieu_modsem2	Modified Mathieu functions of the second kind
mathieu_even_coef	mathieu_odd_coef	Fourier coefficients for Mathieu functions

Spheroidal wave functions

Function	Description
prolate_aswfa_nocv	Prolate spheroidal angular function of the first kind
prolate_radial1_nocv	Prolate spheroidal radial function of the first kind
prolate_radial2_nocv	Prolate spheroidal radial function of the second kind
oblate_aswfa_nocv	Oblate spheroidal angular function of the first kind
oblate_radial1_nocv	Oblate spheroidal radial function of the first kind
oblate_radial2_nocv	Oblate spheroidal radial function of the second kind
prolate_segv	Characteristic value of prolate spheroidal function
oblate_segv	Characteristic value of oblate spheroidal function

The following functions require pre-computed characteristic value:

Function	Description
prolate_aswfa	Prolate spheroidal angular function of the first kind
prolate_radial1	Prolate spheroidal radial function of the first kind
prolate_radial2	Prolate spheroidal radial function of the second kind
oblate_aswfa	Oblate spheroidal angular function of the first kind
oblate_radial1	Oblate spheroidal radial function of the first kind
oblate_radial2	Oblate spheroidal radial function of the second kind

Kelvin functions

Function	Zeros	Description
kelvin	kelvin_zeros	Kelvin functions as complex numbers
ber	ber_zeros	Kelvin function $\ber(x)$
berp	berp_zeros	Derivative of ber, $\ber'(x)$
bei	bei_zeros	Kelvin function $\bei(x)$
beip	beip_zeros	Derivative of bei, $\bei'(x)$
ker	ker_zeros	Kelvin function $\ker(x)$
kerp	kerp_zeros	Derivative of ker, $\ker'(x)$
kei	kei_zeros	Kelvin function $\kei(x)$
keip	keip_zeros	Derivative of kei, $\kei'(x)$

Combinatorics

Function	Description
comb	$k$-combinations of $n$ things, $_nC_k = {n \choose k}$
comb_rep	$k$-combinations with replacement, $\big(\!\!{n \choose k}\!\!\big)$
perm	$k$-permutations of $n$ things, $_nP_k = {n! \over (n-k)!}$
stirling2	Stirling number of the second kind $S(n,k)$

Factorials

Function	Description
factorial	Factorial $n!$
factorial_checked	factorial with overflow checking
multifactorial	Multifactorial $n!_{(k)}$
multifactorial_checked	multifactorial with overflow checking

Exponential integrals

Function	Description
expn	Generalized exponential integral $E_n(x)$
expi	Exponential integral $Ei(x)$
exp1	Exponential integral $E_1(x)$
scaled_exp1	Scaled exponential integral $x e^x E_1(x)$

Zeta functions

Function	Description
zeta	Hurwitz zeta function $\zeta(z,q)$ for real or complex $z$
riemann_zeta	Riemann zeta function $\zeta(z)$ for real or complex $z$
zetac	$\zeta(x) - 1$ for real $x$

Lambert W and related functions

Function	Description
lambertw	Lambert W function
wrightomega	Wright Omega function

Other special functions

Function	Description
agm	Arithmetic-geometric mean of two scalars
bernoulli	Bernoulli numbers $B_0,\dotsc,B_{N-1}$
binom	Binomial coefficient $\binom{n}{k}$ for real input
diric	Periodic sinc function, also called the Dirichlet kernel
euler	Euler numbers $E_0,\dotsc,E_{N-1}$
sici	Sine and cosine integrals $\Si(z)$ and $\Ci(z)$
shichi	Hyperbolic sine and cosine integrals $\Shi(z)$ and $\Chi(z)$
softmax	Softmax function
log_softmax	Logarithm of the softmax function
spence	Spence’s function, also known as the dilogarithm
softplus	$\ln(1 + e^x)$
log1mexp	$\ln(1 - e^x)$

Convenience functions

Function	Description
cbrt	$\sqrt[3]{x}$
exp10	$10^x$
exp2	$2^x$
radian	Convert from degrees to radians
cosdg	Cosine of an angle in degrees
sindg	Sine of an angle in degrees
tandg	Tangent of an angle in degrees
cotdg	Cotangent of an angle in degrees
expm1	$e^x - 1$
cosm1	$\cos(x) - 1$
round	Round to nearest or even integer-valued float
xlogy	$x \ln(y)$ or $0$ if $x = 0$
xlog1py	$x \ln(1+y)$ or $0$ if $x = 0$
logaddexp	$\ln(e^x + e^y)$
logaddexp2	$\log_2(2^x + 2^y)$
logsumexp	$\ln \sum e^{x_i}$
exprel	Relative error exponential, $e^x - 1 \over x$
sinc	Normalized sinc function, $\sin(\pi x) \over \pi x$

Functions

agm

Arithmetic-geometric mean

airy

Airy functions and derivatives

airy_ai_zeros

Zeros and values of the Airy function Ai and its derivative

airy_bi_zeros

Zeros and values of the Airy function Bi and its derivative

airy_integrals

Integrals of Airy functions

airy_scaled

Exponentially scaled Airy functions and derivatives

assoc_legendre_p

Associated Legendre polynomial of the 1st kind

assoc_legendre_p_all

All associated Legendre polynomials of the 1st kind

assoc_legendre_p_norm

Normalized associated Legendre polynomial of the 1st kind

assoc_legendre_p_norm_all

All normalized associated Legendre polynomials of the 1st kind

assoc_legendre_q_all

All associated Legendre polynomials of the 2nd kind and their derivatives

bdtr

Binomial distribution function

bdtrc

Binomial survival function

bdtri

Binomial quantile function

bei

Kelvin function bei

bei_zeros

First nt zeros of Kelvin function bei

beip

Derivative of the Kelvin function bei

beip_zeros

First nt zeros of Kelvin function derivative beip

ber

Kelvin function ber

ber_zeros

First nt zeros of Kelvin function ber

bernoulli

Bernoulli numbers $B_0, \ldots, B_{N-1}$

berp

Derivative of the Kelvin function ber

berp_zeros

First nt zeros of Kelvin function derivative berp

bessel_i

Modified Bessel function of the first kind, $I_v(z)$

bessel_i0

Modified Bessel function of the first kind of order 0, $I_0(x)$

bessel_i0e

Exponentially scaled modified Bessel function of the first kind of order 0, $e^{-|x|} I_0(x)$

bessel_i1

Modified Bessel function of the first kind of order 1, $I_1(x)$

bessel_i1e

Exponentially scaled modified Bessel function of the first kind of order 1, $e^{-|x|} I_1(x)$

bessel_i_prime

Compute the $n$th derivative of bessel_i(v, z) w.r.t. z

bessel_ie

Exponentially scaled modified Bessel function of the first kind

bessel_j

Bessel function of the first kind, $J_v(z)$

bessel_j0

Bessel function of the first kind of order 0, $J_0(x)$

bessel_j1

Bessel function of the first kind of order 1, $J_1(x)$

bessel_j_prime

Compute the $n$th derivative of bessel_j(v, z) w.r.t. z

bessel_je

Exponentially scaled Bessel function of the first kind

bessel_k

Modified Bessel function of the second kind, $K_v(z)$

bessel_k0

Modified Bessel function of the second kind of order 0, $K_0(x)$

bessel_k0e

Exponentially scaled modified Bessel function of the second kind of order 0, $e^x K_0(x)$

bessel_k1

Modified Bessel function of the second kind of order 1, $K_1(x)$

bessel_k1e

Exponentially scaled modified Bessel function of the second kind of order 1, $e^x K_1(x)$

bessel_k_prime

Compute the $n$th derivative of bessel_k(v, z) w.r.t. z

bessel_ke

Exponentially scaled modified Bessel function of the second kind

bessel_y

Bessel function of the second kind, $Y_v(z)$

bessel_y0

Bessel function of the second kind of order 0, $Y_0(x)$

bessel_y1

Bessel function of the second kind of order 1, $Y_1(x)$

bessel_y_prime

Compute the $n$th derivative of bessel_y(v, z) w.r.t. z

bessel_ye

Exponentially scaled Bessel function of the second kind

bessel_zeros

Compute $N$ zeros of Bessel functions $J_v(x)$, $J_v'(x)$, $Y_v(x)$, and $Y_v'(x)$

besselpoly

Weighted integral of the Bessel function of the first kind, $\int_0^1 x^\lambda \mathop{J}_v(2ax) \dd x$

beta

Beta function $\B(a,b)$

betainc

Regularized incomplete Beta function $\I_x(a,b)$

betaincinv

Inverse of the regularized incomplete Beta function, $y = \I_x(a, b)$

betaln

Natural logarithm of the absolute value of beta, $\ln{\abs{\B(a,b)}}$

binom

Binomial coefficient considered as a function of two real variables

boxcox

Box-Cox transformation

boxcox1p

Box-Cox transformation of 1 + x

cbrt

Cube root of $x$, $\sqrt[3]{x}$

cevalpoly

Evaluate polynomials

chdtr

CDF of the Chi-squared distribution

chdtrc

Survival function of the Chi-squared distribution

chdtri

Quantile function of the Chi-squared distribution

comb

k-combinations of n things, _nC_k

comb_rep

k-combinations of n things with replacement

cosdg

Cosine of angle in degrees

cosm1

$\cos(x) - 1$ for use when $x$ is near zero

cospi

$\cos(\pi z)$ for real or complex $z$

cotdg

Cotangent of the angle x given in degrees

dawsn

Dawson function sqrt(pi)/2 * exp(-z^2) * erfi(z) for real or complex input

digamma

Digamma function for real or complex input

diric

Dirichlet kernel, also known as the periodic sinc function.

ellipe

Complete elliptic integral of the second kind

ellipeinc

Incomplete elliptic integral of the second kind

ellipj

Jacobi elliptic functions

ellipk

Complete elliptic integral of the first kind

ellipkinc

Incomplete elliptic integral of the first kind

ellipkm1

Complete elliptic integral of the first kind around $m = 1$

entr

Elementwise function for computing the entropy

erf

Error function $\erf(z)$ for real or complex input

erf_zeros

Zeros of the error function erf(z) in the first quadrant of the complex plane

erfc

Complementary error function $1 - \erf(z)$ for real or complex input

erfcinv

Inverse of the complementary error function erfc(x)

erfcx

Scaled complementary error function $e^{z^2} \erfc(z)$ for real or complex input

erfi

Imaginary error function -i erf(i z) for real or complex input

erfinv

Inverse of the error function erf(x)

euler

Euler numbers $E_0, \ldots, E_{N-1}$

eval_chebyshev_t

Evaluate Chebyshev polynomial of the first kind $T_n$ at a point.

eval_chebyshev_u

Evaluate Chebyshev polynomial of the second kind $U_n$ at a point.

eval_gegenbauer

Evaluate Gegenbauer polynomial $C_n^{(\alpha)}$ at a point.

eval_genlaguerre

Evaluate generalized Laguerre polynomial $L_n^{(\alpha)}$ at a point.

eval_hermite_h

Evaluate physicists’ Hermite polynomial $H_n(x)$ at a point

eval_hermite_he

Evaluate probabilists’ (normalized) Hermite polynomial $He_n(x)$ at a point

eval_jacobi

Evaluate Jacobi polynomial $P_n^{(\alpha, \beta)}$ at a point.

eval_laguerre

Evaluate Laguerre polynomial $L_n$ at a point

eval_legendre

Evaluate Legendre polynomial $P_n$ at a point.

exp1

Exponential integral $E_1$ for real or complex input

exp2

$2^x$

exp10

$10^x$

expi

Exponential integral $Ei$ for real or complex input

expit

Expit (a.k.a. logistic sigmoid) function, $1 / (1 + e^{-x})$

expm1

$e^x - 1$ for real or complex input

expn

Generalized exponential integral $E_n(x)$

exprel

Relative error exponential, $(e^x - 1) / x$

extended_absolute_error

Extended absolute error metric between two f64 or Complex<f64> values

extended_relative_error

Extended relative error metric between two f64 or Complex<f64> values

factorial

Factorial n!

factorial_checked

Factorial n! with overflow checking

fdtr

CDF of the F distribution

fdtrc

Survival function of the F distribution

fdtri

Quantile function of the F distribution

fresnel

Fresnel integrals S(z) and C(z) for real or complex argument

fresnel_zeros

Zeros of Fresnel integrals S(z) and C(z)

gamma

Gamma function $\Gamma(z)$ for real or complex argument.

gammainc

Regularized lower incomplete gamma function $P(a,x)$

gammaincc

Regularized upper incomplete gamma function $Q(a,x)$

gammainccinv

Inverse of gammaincc

gammaincinv

Inverse of gammainc

gammaln

Logarithm of the absolute value of gamma, $\ln{\abs{\Gamma(x)}}$

gammasgn

Sign of the Gamma function, $\sgn \Gamma(x)$

gdtr

CDF of the Gamma distribution

gdtrc

Survival function of the Gamma distribution

gdtrib

Inverse of gdtr with respect to b

hankel_1

Hankel function of the first kind, $H_v^{(1)}(z)$

hankel_2

Hankel function of the second kind, $H_v^{(2)}(z)$

hankel_1_prime

Compute the $n$th derivative of hankel_1(v, z) w.r.t. z

hankel_1e

Exponentially scaled Hankel function of the first kind

hankel_2_prime

Compute the $n$th derivative of hankel_2(v, z) w.r.t. z

hankel_2e

Exponentially scaled Hankel function of the second kind

huber

Huber Loss function

hyp0f0

Hypergeometric function $_0F_0\left[\middle| z\right]$ for real or complex $z$

hyp0f1

Confluent hypergeometric limit function $_0F_1\left[b\middle| z\right]$ for real or complex $z$

hyp1f0

Hypergeometric function $_1F_0\left[a\middle| z\right]$ for real or complex $z$

hyp1f1

Kummer’s Confluent hypergeometric function $_1F_1$

hyp2f1

Gauss’ hypergeometric function $_2F_1$

hypu

Tricomi’s confluent hypergeometric function $U(a,b,x)$

inv_boxcox

Inverse of the Box-Cox transformation

inv_boxcox1p

Inverse of the Box-Cox transformation of 1 + x

it1i0k0

Integrals of modified Bessel functions of order 0

it1j0y0

Integrals of Bessel functions of order 0

it2i0k0

Integrals related to modified Bessel functions of order 0.

it2j0y0

Integrals related to Bessel functions of the first kind of order 0

it2struve0

Integral related to the Struve $H_0$ function

itmodstruve0

Integral of the modified Struve $L_0$ function

itstruve0

Integral of the Struve $H_0$ function

iv_ratio

Compute iv(v,x)/iv(v-1,x) of the modified Bessel function of the first kind

iv_ratio_c

Compute iv(v,x)/iv(v-1,x) of the modified Bessel function of the first kind

jahnke_emden_lambda

Evaluate the Jahnke-Emden Lambda function $\Lambda_v(x)$ and its derivatives

kei

Kelvin function kei

kei_zeros

First nt zeros of Kelvin function kei

keip

Derivative of the Kelvin function kei

keip_zeros

First nt zeros of Kelvin function derivative keip

kelvin

Kelvin functions as complex numbers

kelvin_zeros

First nt zeros of all Kelvin functions and their derivatives

ker

Kelvin function ker

ker_zeros

First nt zeros of Kelvin function ker

kerp

Derivative of the Kelvin function ker

kerp_zeros

First nt zeros of Kelvin function derivative kerp

kl_div

Elementwise function for computing the Kullback-Leibler divergence

kolmogc

CDF of the Kolmogorov distribution

kolmogci

Inverse of kolmogc, the quantile function of the Kolmogorov distribution

kolmogi

Inverse of kolmogorov

kolmogorov

Survival function of the Kolmogorov distribution

kolmogp

Derivative of kolmogorov

lambertw

Lambert W function.

legendre_p

Legendre polynomial of degree n

legendre_p_all

All Legendre polynomials of the 1st kind

legendre_q_all

All Legendre polynomials of the 2nd kind and their derivatives

log1mexp

Compute $\ln(1 - e^x)$

log1p

$ \log(1+z)$ for real or complex input

log1pmx

Compute $\log(1+x)-x$ for real input

log_expit

Natural logarithm of expit

log_ndtr

Logarithm of ndtr, $\ln \Phi(z)$

log_softmax

Compute the logarithm of the softmax function

log_wright_bessel

Natural logarithm of wright_bessel

logaddexp

ln(e^x + e^y)

logaddexp2

log₂(2^x + 2^y)

loggamma

Principal branch of the logarithm of gamma(z)

logit

Logit function, $\ln(x / (1 - x))$

logsumexp

Compute the log of the sum of exponentials of input elements

mathieu_a

Characteristic value of even Mathieu functions

mathieu_b

Characteristic value of odd Mathieu functions

mathieu_cem

Even Mathieu function and its derivative

mathieu_even_coef

Fourier coefficients for even Mathieu and modified Mathieu functions

mathieu_modcem1

Even modified Mathieu function of the first kind and its derivative

mathieu_modcem2

Even modified Mathieu function of the second kind and its derivative

mathieu_modsem1

Odd modified Mathieu function of the first kind and its derivative

mathieu_modsem2

Odd modified Mathieu function of the second kind and its derivative

mathieu_odd_coef

Fourier coefficients for odd Mathieu and modified Mathieu functions

mathieu_sem

Odd Mathieu function and its derivative

modified_fresnel_minus

Modified Fresnel negative integrals

modified_fresnel_plus

Modified Fresnel positive integrals

multifactorial

Multifactorial n!_(k) for positive k

multifactorial_checked

Multifactorial n!_(k) for positive k with overflow checking

multigammaln

Log of multivariate gamma, $\ln \Gamma_d(a)$, sometimes called the generalized gamma

nbdtr

Negative binomial distribution function

nbdtrc

Negative binomial survival function

nbdtri

Negative binomial quantile function

ndtr

CDF of the standard normal distribution, $\Phi(z)$

ndtri

Inverse of ndtr, the probit function $\Phi^{-1}(p)$

ndtri_exp

Inverse of log_ndtr vs x.

oblate_aswfa

Oblate spheroidal angular function for precomputed characteristic value

oblate_aswfa_nocv

Oblate spheroidal angular function of the 1st kind and its derivative

oblate_radial1

Oblate spheroidal radial function of the 1st kind for precomputed characteristic value

oblate_radial2

Oblate spheroidal radial function of the 2nd kind for precomputed characteristic value

oblate_radial1_nocv

Oblate spheroidal radial function of the 1st kind and its derivative

oblate_radial2_nocv

Oblate spheroidal radial function of the 2nd kind and its derivative

oblate_segv

Characteristic value of oblate spheroidal function

owens_t

Owen’s T function, $T(h, a)$

pbdv

Parabolic cylinder function $D$

pbvv

Parabolic cylinder function $V$

pbwa

Parabolic cylinder function $W$

pdtr

Poisson distribution function

pdtrc

Poisson survival function

pdtri

Poisson quantile function

perm

k-permutations of n things, _nP_k

pmv

Associated Legendre function for $|x| \leq 1$

polygamma

Polygamma function ψ⁽ⁿ⁾(x)

pow_falling

Falling factorial $\fpow x m$

pow_rising

Rising factorial $\rpow x m$

prolate_aswfa

Prolate spheroidal angular function for precomputed characteristic value

prolate_aswfa_nocv

Prolate spheroidal angular function of the 1st kind and its derivative

prolate_radial1

Prolate spheroidal radial function of the 1st kind for precomputed characteristic value

prolate_radial2

Prolate spheroidal radial function of the 2nd kind for precomputed characteristic value

prolate_radial1_nocv

Prolate spheroidal radial function of the 1st kind and its derivative

prolate_radial2_nocv

Prolate spheroidal radial function of the 2nd kind and its derivative

prolate_segv

Characteristic value of prolate spheroidal function

pseudo_huber

Pseudo-Huber Loss function

radian

Convert from degrees to radians.

rel_entr

Elementwise function for computing relative entropy

rgamma

Reciprocal Gamma function 1 / gamma(z)

riccati_j

Compute Riccati-Bessel function of the first kind and derivatives for the first $N$ orders

riccati_y

Compute Riccati-Bessel function of the second kind and derivatives for the first $N$ orders

riemann_zeta

Riemann zeta function $\zeta(z)$ for real or complex $z$

round

Round to nearest or even integer-valued float

scaled_exp1

Scaled version of the exponential integral $E_1$ for real input

shichi

Hyperbolic sine and cosine integrals.

sici

Sine and cosine integrals.

sinc

Normalized sinc function

sindg

Sine of the angle x given in degrees.

sinpi

$\sin(\pi z)$ for real or complex $z$

smirnov

Survival function of the Kolmogorov-Smirnov distribution

smirnovc

CDF of the Kolmogorov-Smirnov distribution

smirnovci

Inverse of smirnovc

smirnovi

Inverse of smirnov

smirnovp

Derivative of smirnov

softmax

Compute the softmax function

softplus

ln(1 + e^x)

spence

Spence’s function for real or complex argument, also known as the dilogarithm

sph_bessel_i

Modified spherical Bessel function of the first kind, $i_n(z)$

sph_bessel_i_prime

Derivative of sph_bessel_i, $i_n'(z)$

sph_bessel_j

Spherical Bessel function of the first kind, $j_n(z)$

sph_bessel_j_prime

Derivative of sph_bessel_j, $j_n'(z)$

sph_bessel_k

Modified spherical Bessel function of the second kind, $k_n(z)$

sph_bessel_k_prime

Derivative of sph_bessel_k, $k_n'(z)$

sph_bessel_y

Spherical Bessel function of the second kind, $y_n(z)$

sph_bessel_y_prime

Derivative of sph_bessel_y, $y_n'(z)$

sph_harm_y

Spherical harmonics

sph_harm_y_all

All spherical harmonics up to the specified degree $n$ and order $m$

sph_legendre_p

Spherical Legendre polynomial of degree n and order m

sph_legendre_p_all

All spherical Legendre polynomials of the 1st kind

stdtr

Student’s t distribution cumulative distribution function

stdtri

Inverse of stdtr

stirling2

Stirling number of the second kind S(n,k)

struve_h

Struve $H_v$ function

struve_l

Modified Struve $L_v$ function

tandg

Tangent of angle x given in degrees

tukeylambdacdf

CDF of the Tukey-Lambda distribution

voigt_profile

Voigt profile

wofz

Faddeeva function exp(-z^2) * erfc(-i z)

wright_bessel

Wright’s generalized Bessel function

wrightomega

Wright Omega function, $\omega(z)$

xlog1py

Compute $x \log(1+y)$ for real or complex input

xlogy

Compute $x \log(y)$ for real or complex input

zeta

Hurwitz zeta function $\zeta(z, q)$ for real or complex $z$

zetac

Riemann zeta function minus one, $\zeta(x) - 1$