scirs2-special

Special mathematical functions for the SciRS2 scientific computing library (v0.4.0).

scirs2-special provides a comprehensive collection of special mathematical functions modeled after SciPy's special module. Version 0.4.0 extends the classical function set with advanced additions: Mathieu functions, Coulomb wave functions, spherical harmonics with Wigner/Gaunt symbols, Jacobi theta functions, Weierstrass elliptic functions, parabolic cylinder functions, Fox H-functions, Appell hypergeometric functions, Meixner-Pollaczek polynomials, Heun functions, polylogarithm, q-analogs, number-theoretic functions, and information-theoretic functions — all as pure Rust.

Features (v0.4.0)

Classical Special Functions

Gamma and Related

• Gamma function Γ(z), log-gamma ln Γ(z)
• Digamma ψ(z), trigamma ψ'(z), polygamma ψ^(n)(z)
• Beta function B(a,b), log-beta
• Incomplete gamma: lower γ(a,x) and upper Γ(a,x)
• Regularized incomplete gamma P(a,x), Q(a,x)
• Incomplete beta I_x(a,b) and its inverse
• Factorial n!, log-factorial, binomial coefficient C(n,k)
• Pochhammer symbol (rising factorial)

Error Functions

• Error function erf(x), complementary erfc(x)
• Scaled complementary erfcx(x) (stable for large x)
• Imaginary error function erfi(x)
• Dawson integral F(x)
• Inverse error function erfinv(x), inverse complementary erfcinv(x)

Bessel Functions

• First kind J_n(x): orders 0, 1, n (integer and real)
• Second kind Y_n(x): orders 0, 1, n
• Modified first kind I_n(x): exponentially scaled variant
• Modified second kind K_n(x): exponentially scaled variant
• Spherical Bessel: j_n(x), y_n(x)
• Hankel functions H_n^(1)(x), H_n^(2)(x) (complex)
• Bessel function zeros

Elliptic Integrals and Functions

• Complete integrals: K(k), E(k), Pi(n,k)
• Incomplete integrals: F(phi,k), E(phi,k), Pi(phi,n,k)
• Jacobi elliptic functions: sn, cn, dn and 12 reciprocal/quotient variants
• Carlson symmetric forms: R_F, R_D, R_J, R_C

Orthogonal Polynomials

• Legendre: P_n(x), associated P_n^m(x)
• Chebyshev first kind T_n(x) and second kind U_n(x)
• Hermite: physicist's H_n(x) and probabilist's He_n(x)
• Laguerre: L_n(x) and associated L_n^alpha(x)
• Gegenbauer (ultraspherical): C_n^lambda(x)
• Jacobi: P_n^(alpha,beta)(x)
• Zernike radial polynomials

Airy Functions

• Ai(x), Bi(x) and their derivatives Ai'(x), Bi'(x)
• Exponentially scaled variants for large |x|
• Complex argument support

Hypergeometric Functions

• Confluent hypergeometric limit _0F_1(b; z)
• Kummer's confluent _1F_1(a; b; z) and Tricomi's U(a,b,z)
• Gauss hypergeometric _2F_1(a,b; c; z): analytic continuation for |z| >= 1
• Generalized hypergeometric _pF_q

Zeta and Related

• Riemann zeta ζ(s), Hurwitz zeta ζ(s,a), Dirichlet eta η(s)
• Lerch transcendent Φ(z,s,a)
• Lambert W function: real branches W_0 and W_{-1}

Other Classical Functions

• Struve functions H_n(x) and L_n(x) with asymptotic expansions
• Kelvin functions: ber, bei, ker, kei and derivatives
• Fresnel integrals S(x) and C(x) with modulus and phase
• Parabolic cylinder functions: Weber D_n(x), U(a,x), V(a,x)
• Spheroidal wave functions: prolate and oblate, angular and radial
• Wright functions: Wright Omega omega(z), Wright Bessel generalizations
• Coulomb wave functions: regular F_l(eta,rho), irregular G_l(eta,rho), H_l^+/-
• Logarithmic integral li(x), offset logarithmic integral Li(x)
• Exponential integrals Ei(x), E_n(x), E_1(x)

Advanced Special Functions

Mathieu Functions

• Characteristic values of even functions a_r(q) and odd functions b_r(q)
• Even (cosine-elliptic) functions ce_r(q, x) with Fourier coefficients
• Odd (sine-elliptic) functions se_r(q, x) with Fourier coefficients
• Radial Mathieu functions of the first kind Mc_r(q, x) and second kind Ms_r(q, x)
• Asymptotic expansions for large |q|

Spherical Harmonics and Angular Momentum

• Real spherical harmonics Y_l^m(theta, phi) for arbitrary l, m
• Complex spherical harmonics with Condon-Shortley phase
• Gaunt coefficients: integrals of products of three spherical harmonics
• Wigner 3-j symbols by Racah formula
• Wigner 6-j symbols (Racah W-coefficients)
• Wigner 9-j symbols for compound angular momentum coupling
• Clebsch-Gordan coefficients

Jacobi Theta Functions

• theta_1(z, q), theta_2(z, q), theta_3(z, q), theta_4(z, q)
• Logarithmic derivatives (theta functions of the second kind)
• Quasi-periodicity relations and heat equation connections
• Modular transformations

Weierstrass Elliptic Functions

• Weierstrass P-function p(z; g2, g3)
• Weierstrass zeta function zeta(z; g2, g3)
• Weierstrass sigma function sigma(z; g2, g3)
• Elliptic invariants g2, g3 from half-periods omega1, omega2
• Discriminant Delta and Klein j-invariant

Parabolic Cylinder Functions (Extended)

• D_n(x) for non-integer n via connection to Whittaker functions
• U(a, x) and V(a, x): standard parabolic cylinder pair
• Asymptotic expansions for large |x| and large |a|

Fox H-Function

• Generalized Fox H-function H_{p,q}^{m,n}[z | (a,alpha)_p; (b,beta)_q]
• Series and integral representations
• Special cases: Meijer G-function, Wright function, stable distributions

Appell Hypergeometric Functions

• F_1(a; b1, b2; c; x, y): Appell first hypergeometric function
• F_2(a; b1, b2; c1, c2; x, y): Appell second
• F_3(a1, a2; b1, b2; c; x, y): Appell third
• F_4(a; b; c1, c2; x, y): Appell fourth

Meixner-Pollaczek Polynomials

• P_n^lambda(x; phi): Meixner-Pollaczek polynomials
• Recurrence relations and generating functions
• Orthogonality weight and inner product

Heun Functions

• General Heun function Hl(a, q; alpha, beta, gamma, delta; z)
• Confluent Heun function HeunC
• Double-confluent Heun function HeunD
• Biconfluent Heun function HeunB
• Triconfluent Heun function HeunT

Polylogarithm

• Li_s(z) for complex s and z
• Fermi-Dirac integrals F_j(x)
• Bose-Einstein integrals G_j(x)
• Clausen function Cl_2(theta)

Q-Analogs

• Q-Gamma function Gamma_q(x)
• Q-Pochhammer symbol (a; q)_n and infinite product (a; q)_infty
• Q-Binomial coefficient (Gaussian binomial)
• Q-Exponential functions e_q(z) and E_q(z)
• Q-Bessel functions of the first and second kind
• Q-orthogonal polynomials: big and little q-Jacobi, q-Laguerre, q-Hermite

Number Theory Functions

• Ramanujan tau function tau(n)
• Euler's totient function phi(n) and Jordan's totient
• Liouville function lambda(n), von Mangoldt function Lambda(n)
• Mobius function mu(n) and Mertens function M(n)
• Number of divisors d(n), sum of divisors sigma_k(n)
• Partition function p(n) via pentagonal recurrence
• Bell polynomials (complete and partial)
• Bernoulli numbers and polynomials; Euler numbers and polynomials
• Stirling numbers of the first and second kind; Lah numbers

Information-Theoretic Functions

• KL divergence D_KL(P || Q) for discrete and Gaussian distributions
• Jensen-Shannon divergence
• Shannon entropy H(p), Renyi entropy H_alpha(p)
• Mutual information and conditional entropy
• Cross-entropy and binary cross-entropy
• Logistic function, softmax, logsumexp (numerically stable)
• Sinc function

Combinatorics Extensions

• Bell polynomials Y_n(x_1,...,x_n) complete and partial
• Falling factorial, rising factorial (Pochhammer)
• Catalan numbers, Narayana numbers, Motzkin numbers
• Derangement count, subfactorial
• Restricted growth strings and set partition enumeration

Orthogonal Polynomial Extensions

• Wilson polynomials and Racah polynomials
• Askey-Wilson polynomials (q-extensions)
• Dual Hahn polynomials, continuous dual Hahn
• Krawtchouk, Meixner, Charlier discrete orthogonal polynomials

Quick Start

Add to your Cargo.toml:

[dependencies]
scirs2-special = "0.4.0"

With parallel processing:

[dependencies]
scirs2-special = { version = "0.4.0", features = ["parallel"] }

Classical functions

use scirs2_special::{gamma, bessel, erf, elliptic, orthogonal};

// Gamma(5) = 4! = 24
let g = gamma(5.0f64);
assert!((g - 24.0).abs() < 1e-10);

// Bessel J_0(2.0)
let j0 = bessel::j0(2.0f64);
println!("J_0(2) = {}", j0);

// erf(1.0)
let e = erf::erf(1.0f64);
println!("erf(1) = {}", e);

// Complete elliptic integral K(0.5)
let k = elliptic::ellipk(0.5f64)?;
println!("K(0.5) = {}", k);

// Legendre polynomial P_3(0.5)
let p3 = orthogonal::legendre(3, 0.5f64)?;
println!("P_3(0.5) = {}", p3);

Mathieu functions

use scirs2_special::mathieu::{mathieu_a, mathieu_ce};

// Characteristic value a_0(q=1.0)
let a0 = mathieu_a(0, 1.0f64)?;
println!("a_0(1.0) = {}", a0);

// Even Mathieu function ce_0(q=1.0, x=0.5)
let ce0 = mathieu_ce(0, 1.0f64, 0.5f64)?;
println!("ce_0(1.0, 0.5) = {}", ce0);

Spherical harmonics and Wigner symbols

use scirs2_special::spherical_harmonics::{sph_harm, wigner_3j, gaunt};

// Y_2^1(theta=0.5, phi=1.0)
let y21 = sph_harm(2, 1, 0.5f64, 1.0f64)?;
println!("Y_2^1 = {:?}", y21); // complex value

// Wigner 3-j symbol (j1=1, j2=1, j3=2, m1=-1, m2=0, m3=1)
let w3j = wigner_3j(1, 1, 2, -1, 0, 1)?;
println!("Wigner 3-j = {}", w3j);

// Gaunt coefficient integral of Y_2^1 * Y_2^0 * Y_2^(-1)
let g = gaunt(2, 1, 2, 0, 2, -1)?;
println!("Gaunt = {}", g);

Jacobi theta functions

use scirs2_special::theta_functions::{theta_1, theta_3};

// theta_1(z=0.5, q=0.3)
let t1 = theta_1(0.5f64, 0.3f64)?;
println!("theta_1(0.5, 0.3) = {}", t1);

// theta_3(z=0.0, q=0.5) -- the Jacobi nome series
let t3 = theta_3(0.0f64, 0.5f64)?;
println!("theta_3(0, 0.5) = {}", t3);

Polylogarithm

use scirs2_special::polylogarithm::polylog;
use num_complex::Complex64;

// Li_2(0.5) = pi^2/12 - (ln 0.5)^2 / 2
let li2 = polylog(2.0f64, Complex64::new(0.5, 0.0))?;
println!("Li_2(0.5) = {}", li2.re);

Q-analogs

use scirs2_special::q_analogs::{q_gamma, q_pochhammer};

// Q-Gamma(4, q=0.5) -> should approach 3! = 6 as q -> 1
let qg = q_gamma(4.0f64, 0.5f64)?;
println!("Gamma_q(4; q=0.5) = {}", qg);

// Q-Pochhammer (0.3; 0.5)_5
let qp = q_pochhammer(0.3f64, 0.5f64, 5)?;
println!("(0.3; 0.5)_5 = {}", qp);

Information-theoretic functions

use scirs2_special::information_theoretic::{kl_divergence, shannon_entropy, mutual_information};

let p = vec![0.25, 0.25, 0.25, 0.25f64];
let q = vec![0.5, 0.25, 0.125, 0.125f64];

let kl = kl_divergence(&p, &q)?;
let h = shannon_entropy(&p)?;
println!("KL(P||Q) = {}, H(P) = {}", kl, h);

API Overview

Feature Flags

Documentation

Full API documentation is available at docs.rs/scirs2-special.

License

Licensed under the Apache License 2.0. See LICENSE for details.