Function angular_function 

pub fn angular_function(
    parity_kind: SpheroidalParity,
    m: i32,
    n: i32,
    c: f64,
    eta: f64,
) -> SpecialResult<(f64, f64)>

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.

The convention is non-Condon–Shortley: at c = 0, S_{m,n}(0, η) = (-1)^m · [CS-Legendre P_n^m(η)] = [un-CS Legendre P_n^m(η)], which matches the Meixner–Schäfke normalisation of Flammer (1957) §3.1.

Algorithm

1. Compute the Flammer eigenvalue λ and d-coefficients (with d[k_target] = 1).
2. Form the partial sums S_raw(η) = Σ_p d_p · P_{m+r_p}^m(η), S_raw'(η) = Σ_p d_p · (P_{m+r_p}^m)'(η) in the Condon–Shortley basis.
3. Determine the Meixner–Schäfke normalisation factor K so that
   • even parity: K · S_raw(0) = P_n^m(0) (Condon–Shortley);
   • odd parity: K · S_raw'(0) = (P_n^m)'(0) (Condon–Shortley).
4. Apply (-1)^m for the SciPy non-CS convention.

Returns (S, S').

Errors

• SpecialError::DomainError if (m, n) invalid or |η| > 1.
• SpecialError::ComputationError if d-coefficients fail to converge.