A Julia implementation for computing angular and radial spheroidal wave functions ($S_{mn}$, $R_{mn}^{(i)}$) and their associated eigenvalues ($\lambda_{mn}$).

This package was created to support the development of AnalyticEMModes.jl, where spheroidal wave functions are needed for electromagnetic mode calculations. Since no native Julia implementation was available at the time (or at least none could be found after searching through various channels), this package provides the necessary functionality based on standard references and established numerical methods.

The goal is not to provide cutting-edge or highly optimized methods, but simply to offer the minimum set of functionalities needed for practical use in Julia.

• Angular spheroidal wave functions: Both Legendre expansion and power series methods
• Radial spheroidal wave functions: First and second kind
• Characteristic values (eigenvalues): Individual and sequence computation
• Prolate and oblate cases: Support for both coordinate systems
• Pure Julia implementation: No external dependencies for core computations

using Pkg
Pkg.add("SpheroidalWaveFunctions.jl")

using SpheroidalWaveFunctions

# Compute angular function (prolate case)
m, n, c = 1, 2, 0.5
x = 0.3
S, dS = prolate_angular_leg(m, n, c, x)

# Compute characteristic value
λ = prolate_cv(m, n, c)

# Compute radial function (first kind)
ξ = 1.5
R, dR = prolate_radial1(m, n, c, ξ)

# Oblate case
S_obl, dS_obl = oblate_angular_ps(m, n, c, x)
λ_obl = oblate_cv(m, n, c)

Some implementation details (such as matrix sizes and initial values) were informed by examining SciPy's routines and Zhang & Jin's Fortran code.