FLINT3.7.0-dev
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acb_hypgeom_spherical_y

Synopsis#include <flint/acb_hypgeom.h>
void acb_hypgeom_spherical_y(acb_t res, slong n, slong m, const acb_t theta, const acb_t phi, slong prec)source

Sets res to the spherical harmonic of degree nn and order mm at latitude angle θ\theta and longitude angle ϕ\phi, normalized so that

Ynm(θ,ϕ)=2n+14π(n−m)!(n+m)! eimϕPnm(cos⁡(θ)).Y_n^m(\theta, \phi) = \sqrt{\frac{2n+1}{4\pi} \frac{(n-m)!}{(n+m)!}}\, e^{im\phi} P_n^m(\cos(\theta)).

The definition is extended to negative mm and nn by symmetry. The function is a polynomial in cos⁡θ\cos\theta and sin⁡θ\sin\theta and is evaluated with acb_hypgeom_legendre_p_uiui_rec.

Cite this page

The FLINT team, “acb_hypgeom_spherical_y (acb_hypgeom.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/acb_hypgeom/acb_hypgeom_spherical_y.html (generated 2026-11-02).