Function axial_cylindrical_reciprocal 

fn axial_cylindrical_reciprocal(
    order: i32,
    beta: Complex,
    z: Complex,
    eta: Complex,
    sheet: Option<Sheet>,
) -> Complex

The reciprocal term of a value-only 1D cylindrical sum on its axis, which needs half of the polynomial terms: adjacent half-integer gamma functions share one evaluation, recurring downward near zero and upward at large arguments, where a downward step would subtract the leading asymptotic term. On a sheet (reduced_gamma) the endpoint term takes the phase e^(-i pi n) of the lower side.