Module special 

Special mathematical functions for wave equations

This module provides implementations of special functions commonly used in wave equation solutions:

• Spherical Bessel functions (jₙ, yₙ)
• Spherical Hankel functions (hₙ⁽¹⁾, hₙ⁽²⁾)
• Legendre polynomials (Pₙ, Pₙᵐ)

These functions are critical for:

• Mie theory (sphere scattering)
• Spherical harmonic expansions
• FMM translation operators

Re-exports

pub use helmholtz::*;

pub use spherical::*;

Modules

helmholtz

Helmholtz Green’s function and derivatives

spherical

Spherical Bessel and Hankel functions

Functions

associated_legendre

Compute associated Legendre functions P_n^m(x) for fixed m

associated_legendre_single

Single associated Legendre function Pₙᵐ(x)

legendre_p

Single Legendre polynomial Pₙ(x)

legendre_p_derivative

Derivative of single Legendre polynomial P’ₙ(x)

legendre_polynomials

Compute Legendre polynomials P_n(x) for n = 0, 1, …, order-1

legendre_polynomials_derivative

Compute derivative of Legendre polynomials P’_n(x) for n = 0, 1, …, order-1

normalized_associated_legendre

Normalized associated Legendre functions used in spherical harmonics