Crate scirs2_special

Expand description

§SciRS2 Special - Special Mathematical Functions

scirs2-special provides production-ready special mathematical functions modeled after SciPy’s special module, offering gamma, Bessel, error functions, elliptic integrals, hypergeometric functions, and more, with enhanced numerical stability, GPU acceleration, and arbitrary precision support.

§🎯 Key Features

SciPy Compatibility: Drop-in replacement for scipy.special functions
100+ Functions: Gamma, Bessel, error, elliptic, hypergeometric, orthogonal polynomials
Numerical Stability: Carefully implemented algorithms avoiding overflow/underflow
GPU Acceleration: CUDA/ROCm support for array operations
Arbitrary Precision: High-precision computations with rug backend
Memory Efficient: Chunked processing for large arrays
SIMD & Parallel: Vectorized and multi-threaded execution

§📦 Module Overview

SciRS2 Module	SciPy Equivalent	Description
`gamma`	`scipy.special.gamma`	Gamma and related functions
`bessel`	`scipy.special.jv`, `yv`	Bessel functions (J, Y, I, K)
`erf`	`scipy.special.erf`, `erfc`	Error and complementary error functions
`elliptic`	`scipy.special.ellipk`	Elliptic integrals and functions
`hypergeometric`	`scipy.special.hyp2f1`	Hypergeometric functions
`combinatorial`	`scipy.special.factorial`	Factorials, binomial coefficients
`orthogonal`	`scipy.special.eval_legendre`	Orthogonal polynomials (Legendre, Chebyshev, etc.)
`zeta`	`scipy.special.zeta`	Riemann zeta and related functions

§🚀 Quick Start

[dependencies]
scirs2-special = "0.5.0"

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

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

// Bessel function of the first kind: J₀(x)
let j0 = bessel::j0(2.0);

// Error function
let erf_val = erf::erf(1.0);

§🏗️ Architecture

scirs2-special
├── Gamma Functions (gamma, lgamma, digamma, polygamma, beta)
├── Bessel Functions (J, Y, I, K, modified, spherical)
├── Error Functions (erf, erfc, erfcx, erfi, dawson)
├── Elliptic Integrals (complete, incomplete, Jacobi)
├── Hypergeometric (2F1, 1F1, 0F1, generalized)
├── Combinatorial (factorial, binomial, multinomial, Stirling)
├── Orthogonal Polynomials (Legendre, Chebyshev, Hermite, Laguerre)
├── Statistical (logistic, softmax, sinc, logsumexp)
├── Special Integrals (exponential, logarithmic, Fresnel)
├── Zeta & Related (Riemann zeta, Hurwitz zeta, polylog)
├── Lambert W & Wright Omega
├── Airy Functions (Ai, Bi, derivatives)
├── Performance Features
│   ├── GPU acceleration (gamma, Bessel, erf)
│   ├── Chunked processing (memory-efficient)
│   ├── SIMD vectorization
│   └── Parallel execution
└── Arbitrary Precision (via rug, optional)

§📊 Performance

Function	Array Size	CPU	GPU	Speedup
Gamma	10⁶	120ms	6ms	20×
Bessel J0	10⁶	180ms	8ms	22.5×
Erf	10⁶	85ms	4ms	21×

§🔒 Version Information

Version: 0.5.0
Release Date: March 27, 2026
Repository: github.com/cool-japan/scirs

Re-exports§

pub use error::SpecialError;
pub use error::SpecialResult;
pub use advanced_performance_enhancement::benchmark_function;
pub use advanced_performance_enhancement::erf_advancedfast;
pub use advanced_performance_enhancement::gamma_advancedfast;
pub use advanced_performance_enhancement::gamma_array_advancedfast;
pub use advanced_performance_enhancement::j0_advancedfast;
pub use advanced_performance_enhancement::PerformanceConfig;
pub use advanced_performance_enhancement::PerformanceMetrics;
pub use bessel::h1vp;
pub use bessel::h2vp;
pub use bessel::hankel1;
pub use bessel::hankel1e;
pub use bessel::hankel2;
pub use bessel::hankel2e;
pub use bessel::i0;
pub use bessel::i0_prime;
pub use bessel::i0e;
pub use bessel::i1;
pub use bessel::i1_prime;
pub use bessel::i1e;
pub use bessel::iv;
pub use bessel::iv_prime;
pub use bessel::ive;
pub use bessel::ivp;
pub use bessel::j0;
pub use bessel::j0_prime;
pub use bessel::j0e;
pub use bessel::j1;
pub use bessel::j1_prime;
pub use bessel::j1e;
pub use bessel::jn;
pub use bessel::jn_prime;
pub use bessel::jne;
pub use bessel::jv;
pub use bessel::jv_prime;
pub use bessel::jve;
pub use bessel::jvp;
pub use bessel::k0;
pub use bessel::k0_prime;
pub use bessel::k0e;
pub use bessel::k1;
pub use bessel::k1_prime;
pub use bessel::k1e;
pub use bessel::kv;
pub use bessel::kv_prime;
pub use bessel::kve;
pub use bessel::kvp;
pub use bessel::spherical_jn;
pub use bessel::spherical_yn;
pub use bessel::y0;
pub use bessel::y0_prime;
pub use bessel::y0e;
pub use bessel::y1;
pub use bessel::y1_prime;
pub use bessel::y1e;
pub use bessel::yn;
pub use bessel::yn_prime;
pub use bessel::yne;
pub use bessel::yvp;
pub use distributions::bdtr;
pub use distributions::bdtr_array;
pub use distributions::bdtrc;
pub use distributions::bdtri;
pub use distributions::bdtrik;
pub use distributions::bdtrin;
pub use distributions::btdtria;
pub use distributions::btdtrib;
pub use distributions::chdtr;
pub use distributions::chdtrc;
pub use distributions::chdtri;
pub use distributions::fdtr;
pub use distributions::fdtrc;
pub use distributions::fdtridfd;
pub use distributions::gdtr;
pub use distributions::gdtrc;
pub use distributions::gdtria;
pub use distributions::gdtrib;
pub use distributions::gdtrix;
pub use distributions::kolmogi;
pub use distributions::kolmogorov;
pub use distributions::log_ndtr;
pub use distributions::nbdtr;
pub use distributions::nbdtrc;
pub use distributions::nbdtri;
pub use distributions::ndtr;
pub use distributions::ndtr_array;
pub use distributions::ndtri;
pub use distributions::ndtri_exp;
pub use distributions::pdtr;
pub use distributions::pdtrc;
pub use distributions::pdtri;
pub use distributions::pdtrik;
pub use distributions::stdtr;
pub use gamma::beta;
pub use gamma::beta;
pub use gamma::beta_safe;
pub use gamma::betainc;
pub use gamma::betainc_regularized;
pub use gamma::betaincinv;
pub use gamma::betaln;
pub use gamma::digamma;
pub use gamma::digamma;
pub use gamma::digamma_safe;
pub use gamma::gamma;
pub use gamma::gamma_safe;
pub use gamma::gammaln;
pub use gamma::loggamma;
pub use gamma::polygamma;
pub use incomplete_gamma::gammainc;
pub use incomplete_gamma::gammainc_lower;
pub use incomplete_gamma::gammainc_upper;
pub use incomplete_gamma::gammaincc;
pub use incomplete_gamma::gammainccinv;
pub use incomplete_gamma::gammaincinv;
pub use incomplete_gamma::gammasgn;
pub use incomplete_gamma::gammastar;
pub use gamma::complex::beta_complex;
pub use gamma::complex::digamma_complex;
pub use gamma::complex::gamma_complex;
pub use gamma::complex::loggamma_complex;
pub use bessel::complex::i0_complex;
pub use bessel::complex::j0_complex;
pub use bessel::complex::j1_complex;
pub use bessel::complex::jn_complex;
pub use bessel::complex::jv_complex;
pub use bessel::complex::k0_complex;
pub use erf::complex::erf_complex;
pub use erf::complex::erfc_complex;
pub use erf::complex::erfcx_complex;
pub use erf::complex::faddeeva_complex;
pub use information_theory::binary_entropy;
pub use information_theory::cross_entropy;
pub use information_theory::entr;
pub use information_theory::entr_array;
pub use information_theory::entropy;
pub use information_theory::huber;
pub use information_theory::huber_loss;
pub use information_theory::kl_div;
pub use information_theory::kl_divergence;
pub use information_theory::pseudo_huber;
pub use information_theory::rel_entr;
pub use hill::hill_floquet;
pub use hill::mathieu_hill;
pub use hill::CurveType;
pub use hill::HillEquation;
pub use hill::HillResult;
pub use hill::StabilityCurve;
pub use mathieu_hill::hill_characteristic_exponent;
pub use mathieu_hill::hill_periodic_solution;
pub use mathieu_hill::hill_stability_check;
pub use mathieu_hill::hill_stability_exponent;
pub use mathieu_hill::HillCoefficients;
pub use mixed_precision::auto_dispatch_bessel_j0;
pub use mixed_precision::auto_dispatch_erf;
pub use mixed_precision::auto_dispatch_gamma;
pub use mixed_precision::batch_eval_mixed;
pub use mixed_precision::MixedPrecisionConfig;
pub use utility::agm;
pub use utility::cbrt;
pub use utility::cbrt_array;
pub use utility::cosdg;
pub use utility::cosm1;
pub use utility::cotdg;
pub use utility::diric;
pub use utility::exp10;
pub use utility::exp10_array;
pub use utility::exp2;
pub use utility::expit;
pub use utility::expit_array;
pub use utility::expm1_array_utility;
pub use utility::exprel;
pub use utility::gradient;
pub use utility::log1p_array_utility;
pub use utility::log_expit;
pub use utility::logit;
pub use utility::logit_array;
pub use utility::owens_t;
pub use utility::powm1;
pub use utility::radian;
pub use utility::round;
pub use utility::round_array;
pub use utility::sindg;
pub use utility::softplus;
pub use utility::spherical_distance;
pub use utility::tandg;
pub use utility::xlog1py;
pub use utility::xlog1py_scalar;
pub use utility::xlogy;
pub use simd_ops::benchmark_simd_performance;
pub use simd_ops::erf_f32_simd;
pub use simd_ops::exp_f32_simd;
pub use simd_ops::gamma_f32_simd;
pub use simd_ops::gamma_f64_simd;
pub use simd_ops::j0_f32_simd;
pub use simd_ops::vectorized_special_ops;
pub use simd_ops::batch_bessel_j0_f32;
pub use simd_ops::batch_bessel_j0_f64;
pub use simd_ops::batch_bessel_j1_f32;
pub use simd_ops::batch_bessel_j1_f64;
pub use simd_ops::batch_bessel_y0_f32;
pub use simd_ops::batch_bessel_y0_f64;
pub use simd_ops::batch_bessel_y1_f32;
pub use simd_ops::batch_bessel_y1_f64;
pub use simd_ops::batch_beta_f32;
pub use simd_ops::batch_beta_f64;
pub use simd_ops::batch_digamma_f32;
pub use simd_ops::batch_digamma_f64;
pub use simd_ops::batch_erf_f32;
pub use simd_ops::batch_erf_f64;
pub use simd_ops::batch_erfc_f32;
pub use simd_ops::batch_erfc_f64;
pub use simd_ops::batch_gamma_f32;
pub use simd_ops::batch_gamma_f64;
pub use simd_ops::batch_lgamma_f32;
pub use simd_ops::batch_lgamma_f64;
pub use simd_ops::adaptive_gamma_processing;
pub use simd_ops::benchmark_parallel_performance;
pub use simd_ops::gamma_f64_parallel;
pub use simd_ops::j0_f64_parallel;
pub use simd_ops::gamma_f32_simd_parallel;
pub use erf::dawsn;
pub use erf::erf;
pub use erf::erfc;
pub use erf::erfcinv;
pub use erf::erfcx;
pub use erf::erfi;
pub use erf::erfinv;
pub use erf::wofz;
pub use arbitrary_precision::bessel::bessel_j_ap;
pub use arbitrary_precision::bessel::bessel_j_mp;
pub use arbitrary_precision::bessel::bessel_y_ap;
pub use arbitrary_precision::bessel::bessel_y_mp;
pub use arbitrary_precision::cleanup_cache;
pub use arbitrary_precision::error_function::erf_ap;
pub use arbitrary_precision::error_function::erf_mp;
pub use arbitrary_precision::error_function::erfc_ap;
pub use arbitrary_precision::error_function::erfc_mp;
pub use arbitrary_precision::gamma::gamma_ap;
pub use arbitrary_precision::gamma::gamma_mp;
pub use arbitrary_precision::gamma::log_gamma_ap;
pub use arbitrary_precision::gamma::log_gamma_mp;
pub use arbitrary_precision::to_complex64;
pub use arbitrary_precision::to_f64;
pub use arbitrary_precision::PrecisionContext;
pub use arbitrary_precision::bessel_j0_mpfr;
pub use arbitrary_precision::bessel_k0_mpfr;
pub use arbitrary_precision::digamma_mpfr;
pub use arbitrary_precision::erf_mpfr;
pub use arbitrary_precision::erfc_mpfr;
pub use arbitrary_precision::gamma_mpfr;
pub use arbitrary_precision::lgamma_mpfr;

Modules§

advanced_performance_enhancement: Advanced-high performance enhancements for critical special functions
arbitrary_precision: Arbitrary precision computation support for special functions
array_ops: Enhanced array operations for special functions
bessel: Bessel functions with enhanced numerical stability
chromatic: Chromatic Polynomials of Simple Graphs
clebsch_gordan: Clebsch-Gordan Coefficients and Related Symbols
clebsch_gordan_lie: Clebsch-Gordan series for SU(2), SU(3), SO(5), and general semisimple Lie groups.
connection_formulas: Connection formula generator for special functions.
convenience: Domain-specific convenience functions
cross_validation: Cross-validation against reference implementations
dedekind_zeta: Dedekind Zeta Functions
differentiation: Derivative rules and numerical differentiation for special functions.
distributions: Statistical distribution functions
edge_case_tests: Comprehensive edge case testing for special functions
elliptic_ext: Extended elliptic integrals and functions
elliptic_l: Elliptic curve L-functions.
elliptic_modular: Elliptic Modular Functions
erf: Error function and related functions
error: Error types for the SciRS2 special functions module
error_context: Enhanced error handling with context tracking for special functions
error_wrappers: Error handling wrappers for special functions
extended_scipy_validation: Extended validation tests for new SciPy compatibility functions
gamma: Gamma function and related implementations
gpu_context_manager: Advanced GPU context management for special functions
gpu_dispatch: GPU auto-dispatch for batch evaluation of special functions. Falls back to CPU when GPU is unavailable or array is small.
gpu_kernels: GPU kernel implementations for batch special-function evaluation.
gpu_ops: GPU-accelerated implementations of special functions
hall_polynomials: Hall polynomials for p-group extensions.
hecke_l: Hecke L-functions and Maass forms.
hill: Hill’s Equation and Generalized Periodic Stability Analysis
incomplete_gamma: Incomplete gamma and related functions
information_theory: Information theory functions
l_functions: Dirichlet L-functions and Dirichlet Characters
lame: Lamé Functions
legendre_elliptic: Legendre elliptic integrals via Carlson symmetric forms
mathieu_hill: Generalized Hill’s equation — periodic solutions and stability analysis
memory_efficient: Memory-efficient operations for large arrays
mixed_precision: Mixed-Precision Auto-Dispatch for Special Functions
nield_kuznetsov: Nield-Kuznetsov Functions
optimizations: Performance optimizations for special functions
painleve: Painleve Transcendents
performance_benchmarks: Comprehensive performance benchmarking for special functions
physics_engineering: Specialized physics and engineering functions
polylogarithm: Polylogarithm and related transcendental functions
precision: Precision enhancement utilities
python_interop: Python interoperability module for migration assistance
schur: Schur Polynomials and Symmetric Functions
selberg_zeta: Selberg Zeta Function
simd_ops: SIMD-optimized and parallel operations for special functions
stability_analysis: Numerical stability analysis for special functions
symbolic: Symbolic representation and manipulation of special functions.
theta_functions: Jacobi Theta Functions
tutte: Tutte Polynomial of Matroids
utility: Utility and convenience functions with mathematical foundations
validated: Validated numerics: ball arithmetic for certified function evaluation.
visualization: Visualization tools for special functions

Macros§

special_error: Macro for consistent error creation with context
validate_with_context: Macro for consistent validation with error context

Structs§

LentzResult: Result of a modified-Lentz continued-fraction evaluation.
SpheroidalConfig: Configuration for spheroidal wave function computations.
SpheroidalEigenvalue: Result of a spheroidal eigenvalue computation.

Enums§

SphericalBesselKind: Spherical Bessel function selector for the radial function expansion.
SpheroidType: Prolate vs. oblate spheroidal geometry.
SpheroidalKind: Spheroidal geometry type.
SpheroidalParity: Sign tag distinguishing prolate (+c² in coupling matrix) from oblate (-c² coupling, via the standard c → ic substitution).

Constants§

DEFAULT_D_LEN: Default truncation length for the d-coefficient series. Empirically large enough for |c| ≤ 50 and n ≤ 30.

Functions§

acosdg: Inverse cosine in degrees.
ai: Airy function of the first kind, Ai(x).
ai_complex: Complex Airy function of the first kind, Ai(z)
ai_zeros: Compute zeros of Airy function Ai(x)
aie: Exponentially scaled Airy function Ai(x) * exp(2/3 * x^(3/2)) for x >= 0
aip: Derivative of the Airy function of the first kind, Ai’(x).
aip_complex: Complex derivative of Airy function of the first kind, Ai’(z)
airye: Exponentially scaled Airy functions and their derivatives
anger_j: Anger function J_v(x).
anger_weber: Relation between Anger, Weber, and Bessel functions.
angular_function: Evaluate the spheroidal angular function S_{m,n}(c, η) of the first kind using the Flammer / Bouwkamp d-coefficient expansion, normalised to match SciPy’s scipy.special.pro_ang1 / obl_ang1 convention.
angular_spheroidal: Compute the angular spheroidal wave function S_{mn}(c, x) for |x| ≤ 1.
asindg: Inverse sine in degrees.
associated_legendre: Compute the associated Legendre polynomial P_l^m(x) via forward recurrence.
atandg: Inverse tangent in degrees.
bei: Compute the bei function, the imaginary part of the Kelvin function
beip: Compute the derivative of the bei function
bell_number: Computes the Bell number B(n).
ber: Compute the ber function, the real part of the Kelvin function
bernoulli_number: Computes the Bernoulli number B(n).
bernoulli_poly: Bernoulli polynomial B_n(x).
berp: Compute the derivative of the ber function
besselpoly: Compute Bessel polynomial
bi: Airy function of the second kind, Bi(x).
bi_complex: Complex Airy function of the second kind, Bi(z)
bi_zeros: Compute zeros of Airy function Bi(x)
bie: Exponentially scaled Airy function Bi(x) * exp(-2/3 * |x|^(3/2)) for x >= 0
binomial: Computes the binomial coefficient “n choose k”.
bip: Derivative of the Airy function of the second kind, Bi’(x).
bip_complex: Complex derivative of Airy function of the second kind, Bi’(z)
boxcox: Box-Cox transformation
boxcox1p: Box-Cox transformation of 1 + x
boxcox1p_array: Box-Cox1p transformation for arrays
boxcox_array: Box-Cox transformation for arrays
cf_modified_lentz: Evaluate a generic continued fraction of the form
chebyshev: Computes the value of the Chebyshev polynomial of the first kind T_n(x) of degree n.
chi: Hyperbolic cosine integral, Chi(x) = γ + ln(x) + ∫₀ˣ (cosh(t)-1)/t dt
ci: Cosine integral, Ci(x) = -∫ₓ^∞ cos(t)/t dt = γ + ln(x) + ∫₀ˣ (cos(t)-1)/t dt
clausen: Clausen function Cl_2(x).
comb: Combination function - alias for binomial coefficient
complete_elliptic_pi: Complete elliptic integral of the third kind Pi(n, m).
coulomb_f: Computes the regular Coulomb wave function F_L(η,ρ)
coulomb_g: Computes the irregular Coulomb wave function G_L(η,ρ)
coulomb_h_plus: Computes the outgoing Coulomb wave function H⁺_L(η,ρ)
coulomb_hminus: Computes the incoming Coulomb wave function H⁻_L(η,ρ)
coulomb_phase_shift: Computes the Coulomb phase shift σ_L(η)
d_coefficients: Compute the Flammer / Bouwkamp expansion coefficients d_r^{m,n}(c) for the spheroidal angular function S_{m,n}(c, η) = Σ_r d_r · P_{m+r}^m(η) where r = 2p + parity, parity = (n - m) mod 2.
d_coefficients_with_len: Variant of d_coefficients with an explicit truncation length.
debye1: First Debye function D_1(x) = (1/x) * integral_0^x t/(e^t - 1) dt
debye2: Second Debye function D_2(x) = (2/x^2) * integral_0^x t^2/(e^t - 1) dt
debye3: Third Debye function D_3(x) = (3/x^3) * integral_0^x t^3/(e^t - 1) dt
debye4: Fourth Debye function D_4(x) = (4/x^4) * integral_0^x t^4/(e^t - 1) dt
debye5: Fifth Debye function D_5(x) = (5/x^5) * integral_0^x t^5/(e^t - 1) dt
dirichlet_eta: Dirichlet eta function η(s) = (1 - 2^{1-s}) ζ(s).
double_factorial: Computes the double factorial of a non-negative integer.
e1: Exponential integral E₁(x) = ∫ₓ^∞ e⁻ᵗ/t dt
ellip_harm: Ellipsoidal harmonic functions E_n^m(h²)
ellip_harm_2: Second-order ellipsoidal harmful functions F_n^m(h²)
ellip_harm_array: Array version of ellipsoidal harmonics
ellip_harm_coefficients: Ellipsoidal harmonic expansion coefficients
ellip_harm_complex: Complex ellipsoidal harmonics
ellip_normal: Ellipsoidal harmonic normalization constants
ellipe: Complete elliptic integral of the second kind (alternative interface)
ellipeinc: Incomplete elliptic integral of the second kind (alternative interface)
ellipj: Jacobian elliptic functions with all three functions returned at once
ellipk: Complete elliptic integral of the first kind (alternative interface)
ellipkinc: Incomplete elliptic integral of the first kind (alternative interface)
ellipkm1: Complete elliptic integral of the first kind K(1-m)
elliprc: Carlson elliptic integral RC(x, y)
elliprd: Carlson elliptic integral RD(x, y, z)
elliprf: Carlson elliptic integral RF(x, y, z)
elliprf_array: Array versions of Carlson elliptic integrals
elliprg: Carlson elliptic integral RG(x, y, z)
elliprj: Carlson elliptic integral RJ(x, y, z, p)
elliptic_e: Complete elliptic integral of the second kind
elliptic_e_inc: Incomplete elliptic integral of the second kind
elliptic_f: Incomplete elliptic integral of the first kind
elliptic_k: Complete elliptic integral of the first kind
elliptic_nome: Inverse elliptic nome q(m).
elliptic_pi: Incomplete elliptic integral of the third kind
euler_number: Computes the Euler number E(n).
euler_poly: Euler polynomial E_n(x).
exp1: Exponential integral E_1(x).
expint: Exponential integral of order n, E₍ₙ₎(x) = ∫ₓ^∞ e⁻ᵗ/t^n dt
expm1_array: Computes exp(x) - 1 with improved numerical stability for small x.
expn: Generalized exponential integral E_n(x).
factorial: Computes the factorial of a non-negative integer.
factorial2: Computes the double factorial n!! (alias for SciPy compatibility).
factorialk: Computes the k-factorial (multi-factorial) of n.
flammer_eigenvalue: Compute the spheroidal characteristic value λ_{mn}(c) directly from the Flammer recurrence tridiagonal eigenproblem.
fresnel: Compute the Fresnel sine and cosine integrals.
fresnel_complex: Compute the Fresnel sine and cosine integrals for complex argument.
fresnelc: Compute the Fresnel cosine integral.
fresnels: Compute the Fresnel sine integral.
gegenbauer: Computes the value of the Gegenbauer (advancedspherical) polynomial C_n^(λ)(x).
hermite: Computes the value of the Hermite polynomial H_n(x) of degree n.
hermite_prob: Computes the value of the Hermite function (probabilists’ Hermite polynomial) He_n(x).
hurwitz_zeta: Hurwitz zeta function.
hyp0f1: Confluent hypergeometric function 1F1(a;b;z)
hyp0f1_enhanced: Enhanced confluent hypergeometric limit function ₀F₁(;a;z)
hyp1f1
hyp1f1_enhanced: Enhanced confluent hypergeometric function ₁F₁(a;b;z)
hyp1f1_regularized: Regularized confluent hypergeometric function ₁F₁(a;b;z) / Γ(b)
hyp2f1: Gaussian (ordinary) hypergeometric function 2F1(a,b;c;z)
hyp2f1_enhanced: Enhanced hypergeometric function ₂F₁(a,b;c;z) with convergence acceleration
hyp2f1_regularized: Regularized hypergeometric function ₂F₁(a,b;c;z) / Γ(c)
hyperu: Confluent hypergeometric function U(a,b,x) of the second kind
inv_boxcox: Inverse Box-Cox transformation
inv_boxcox1p: Inverse Box-Cox transformation of 1 + x form
inv_boxcox1p_array: Inverse Box-Cox1p transformation for arrays
inv_boxcox_array: Inverse Box-Cox transformation for arrays
it2_struve0: Compute the second integration of the Struve function of order 0
it_mod_struve0: Compute the integrated modified Struve function of order 0
it_struve0: Compute the integrated Struve function of order 0
itairy: Integral of Airy functions: ∫₀^x Ai(t) dt and ∫₀^x Bi(t) dt
itj0y0: Compute integrals ∫₀^∞ tⁿ J₀(t) Y₀(xt) dt and ∫₀^∞ tⁿ J₀(t) Y₀(xt) J₀(t) dt
j0_zeros: Compute the k-th zero of J₀(x)
j1_zeros: Compute the k-th zero of J₁(x)
jacobi: Computes the value of the Jacobi polynomial P_n^(α,β)(x).
jacobi_cn: Jacobi elliptic function cn(u, m)
jacobi_dn: Jacobi elliptic function dn(u, m)
jacobi_sn: Jacobi elliptic function sn(u, m)
jn_zeros: Compute the k-th zero of Jₙ(x)
jnjnp_zeros: Compute zeros where Jₙ(x) and J’ₙ(x) cross zero simultaneously
jnp_zeros: Compute the k-th zero of the derivative J’ₙ(x)
jnyn_zeros: Compute zeros of Jₙ(x) and Yₙ(x) simultaneously
kei: Compute the kei function, the imaginary part of the Kelvin K function
keip: Compute the derivative of the kei function
kelvin: Compute all Kelvin functions and their derivatives at once.
ker: Compute the ker function, the real part of the Kelvin K function
kerp: Compute the derivative of the ker function
laguerre: Computes the value of the Laguerre polynomial L_n(x) of degree n.
laguerre_generalized: Computes the value of the generalized Laguerre polynomial L_n^(α)(x).
lambert_w: Lambert W function for real and complex arguments.
lambert_w_real: Lambert W function for real arguments on the principal branch (k=0).
legendre: Computes the value of the Legendre polynomial P_n(x) of degree n.
legendre_assoc: Computes the associated Legendre function P_n^m(x).
lerch_phi: Lerch transcendent Φ(z, s, a).
li: Calculates the logarithmic integral Li(x) = ∫₂ˣ dt/ln(t)
li_complex: Calculates the logarithmic integral for complex values, Li(z) = ∫₀ᶻ dt/ln(t)
ln_pochhammer: Compute the logarithm of the Pochhammer symbol (rising factorial)
log1p_array: Computes log(1 + x) with improved numerical stability for small x.
log_abs_gamma: Computes the log of the absolute value of the gamma function.
log_softmax: Computes the log-softmax function for a vector of inputs.
log_wright_bessel: Logarithm of Wright’s generalized Bessel function
loggamma_sign: Logarithm of the absolute value of the gamma function with sign.
logistic: Computes the logistic (sigmoid) function.
logistic_derivative: Computes the derivative of the logistic function.
logsumexp: Computes the LogSumExp function for numerical stability.
lommel_s1: Lommel function of the first kind s_{mu,nu}(z).
lommel_s2: Lommel function of the second kind S_{mu,nu}(z).
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 functions
mathieu_odd_coef: Fourier coefficients for odd Mathieu functions
mathieu_sem: Odd Mathieu function and its derivative
mod_fresnel_plus: Compute the modified Fresnel plus integrals.
mod_fresnelminus: Compute the modified Fresnel minus integrals.
mod_struve: Compute the Modified Struve function L_v(x)
multinomial: Multinomial coefficient.
obl_ang1: Computes the oblate spheroidal angular function of the first kind, S_{m,n}(c, η).
obl_cv: Computes the characteristic value for oblate spheroidal wave functions.
obl_cv_seq: Computes a sequence of characteristic values for oblate spheroidal wave functions.
obl_rad1: Computes the oblate spheroidal radial function of the first kind, R_{m,n}^{(1)}(c, ξ).
obl_rad2: Computes the oblate spheroidal radial function of the second kind, R_{m,n}^{(2)}(c, ξ).
pbdv: Parabolic cylinder function D_v(x) and its derivative.
pbdv_seq: Compute a sequence of parabolic cylinder functions D_v(x) for v = 0, 1, …, vmax.
pbvv: Parabolic cylinder function V_v(x) and its derivative.
pbvv_seq: Compute a sequence of parabolic cylinder functions V_v(x) for v = 0, 1, …, vmax.
pbwa: Compute parabolic cylinder function W(a,x) and its derivative.
perm: Computes the number of permutations (alias for SciPy compatibility).
permutations: Computes the number of permutations of n items taken k at a time.
pochhammer: Compute the Pochhammer symbol (rising factorial)
polylog: Polylogarithm function Li_s(z) = Σ_{k=1}^∞ z^k / k^s
pro_ang1: Computes the prolate spheroidal angular function of the first kind, S_{m,n}(c, η).
pro_cv: Computes the characteristic value λ_{m,n}(c) for prolate spheroidal wave functions.
pro_cv_seq: Computes a sequence of characteristic values for prolate spheroidal wave functions.
pro_rad1: Computes the prolate spheroidal radial function of the first kind, R_{m,n}^{(1)}(c, ξ).
pro_rad2: Computes the prolate spheroidal radial function of the second kind, R_{m,n}^{(2)}(c, ξ).
pseudo_voigt: Pseudo-Voigt approximation
radial_function: Evaluate the spheroidal radial function R_{mn}^{(j)}(c, ξ) of the first or second kind via the Flammer §4.4 / A&S 21.9 spherical Bessel expansion:
radial_spheroidal_1: Compute the radial spheroidal wave function of the first kind R_{mn}^{(1)}(c, ξ) for ξ ≥ 1.
riccati_jn: Riccati-Bessel function S_n(x) = x * j_n(x).
riccati_yn: Riccati-Bessel function C_n(x) = -x * y_n(x).
riemann_zeta: Riemann zeta function for real argument (f64 specialisation).
riemann_zeta_complex: Riemann zeta function for complex argument using Euler-Maclaurin.
scaled_recurrence_step: One-step rescale helper: given a “register” of running values that may all drift toward overflow or underflow together, apply a multiplicative rescale with log_scale bookkeeping so the magnitudes return to a safe range.
shi: Hyperbolic sine integral, Shi(x) = ∫₀ˣ sinh(t)/t dt
shichi: Computes both hyperbolic sine and cosine integrals simultaneously: (Shi(x), Chi(x))
si: Sine integral, Si(x) = ∫₀ˣ sin(t)/t dt
sici: Computes both sine and cosine integrals simultaneously: (Si(x), Ci(x))
sinc: Computes the normalized sinc function.
sinc_array: Computes the normalized sinc function for an array of inputs.
softmax: Computes the softmax function for a vector of inputs.
solid_harmonic_irregular: Irregular solid harmonic I_l^m(r, theta, phi).
solid_harmonic_regular: Regular solid harmonic R_l^m(r, theta, phi).
spence: Spence’s function (dilogarithm): Li₂(x) = -∫₀ˣ ln(1-t)/t dt
sph_harm: Computes the value of the real spherical harmonic Y_l^m(θ, φ) function.
sph_harm_complex: Computes the value of the complex spherical harmonic Y_l^m(θ, φ) function.
sph_harm_cos_angle: Spherical harmonic addition theorem coefficient.
sph_harm_normalization: Compute the normalization factor for spherical harmonics.
spherical_bessel_j: Compute the spherical Bessel function of the first kind j_l(x).
spherical_in: Modified spherical Bessel function of the first kind i_n(x).
spherical_in_derivative: Derivative of the modified spherical Bessel function of the first kind.
spherical_kn: Modified spherical Bessel function of the second kind k_n(x).
spherical_kn_derivative: Derivative of the modified spherical Bessel function of the second kind.
spheroidal_eigenvalue_mn: Compute the spheroidal characteristic value λ_{mn}(c).
spheroidal_eigenvalue_wf: Compute the spheroidal eigenvalue λ_{mn}(c) for the angular spheroidal wave function.
spheroidal_ps: Compute the angular spheroidal wave function of the first kind S_{mn}^{(1)}(c, x).
spheroidal_wronskian: Evaluate the (approximate) Wronskian of the spheroidal wave functions.
stirling2: Computes the Stirling number of the second kind (alias for SciPy compatibility).
stirling_first: Computes the Stirling number of the first kind.
stirling_second: Computes the Stirling number of the second kind.
struve: Compute the Struve function H_v(x)
tail_ratio_lentz: Estimate a pessimistic upper bound on the next-coefficient ratio |d_{r+2} / d_r| for very large r via Lentz-CF. Used by callers that want a fast convergence check before committing to a full backward sweep.
voigt_profile: Voigt profile function
voigt_profile_array: Voigt profile for arrays
voigt_profile_fwhm: Voigt profile with different parameterization
voigt_profile_fwhm_array: Voigt profile FWHM for arrays
voigt_profile_normalized: Normalized Voigt profile
weber_e: Weber function E_v(x).
weierstrass_p: Weierstrass elliptic function wp(z; g2, g3).
weierstrass_p_prime: Derivative of the Weierstrass elliptic function wp’(z; g2, g3).
weierstrass_sigma: Weierstrass sigma function sigma_W(z; g2, g3).
weierstrass_zeta: Weierstrass zeta function zeta_W(z; g2, g3).
whittaker_m: Whittaker function M_{kappa,mu}(z)
whittaker_w: Whittaker function W_{kappa,mu}(z)
wright_bessel: Computes the Wright Bessel function J_{rho, beta}(z)
wright_bessel_complex: Computes the Wright Bessel function for complex arguments J_{rho, beta}(z)
wright_bessel_zeros: Computes the first n zeros of the Wright Bessel function J_{rho, beta}(z)
wright_omega: Computes the Wright Omega function for a complex argument.
wright_omega_optimized: Optimized Wright Omega function calculation for complex arguments.
wright_omega_real: Computes the Wright Omega function for a real argument.
wright_omega_real_optimized: Optimized Wright Omega function calculation for real arguments.
y0_zeros: Compute the k-th zero of Y₀(x)
y1_zeros: Compute the k-th zero of Y₁(x)
yn_zeros: Compute the k-th zero of Yₙ(x)
zeta: Riemann zeta function.
zetac: Riemann zeta function minus 1.

Crate scirs2_special

Crate scirs2_special Copy item path

§SciRS2 Special - Special Mathematical Functions

§🎯 Key Features

§📦 Module Overview

§🚀 Quick Start

§🏗️ Architecture

§📊 Performance

§🔒 Version Information

Re-exports§

Modules§

Macros§

Structs§

Enums§

Constants§

Functions§

Crate scirs2_special