29 Numerics library [numerics]

29.7 Mathematical functions for floating-point types [c.math]

29.7.6 Mathematical special functions [sf.cmath]

29.7.6.21 Spherical associated Legendre functions [sf.cmath.sph.legendre]

floating-point-type sph_legendre(unsigned l, unsigned m, floating-point-type theta);
float        sph_legendref(unsigned l, unsigned m, float theta);
long double  sph_legendrel(unsigned l, unsigned m, long double theta);

Effects: These functions compute the spherical associated Legendre functions of their respective arguments l, m, and theta (theta measured in radians).

Returns:

Y_{ℓ}^{m} (θ, 0)

, where

Y_{ℓ}^{m}

is given by Formula 29.41, l is l, m is m, and θ is theta.

\begin{matrix} Y_{ℓ}^{m} (θ, ϕ) = (- 1)^{m} {[\frac{(2 ℓ + 1)}{4 π} \frac{(ℓ - m)!}{(ℓ + m)!}]}^{1 / 2} P_{ℓ}^{m} (cos θ) e^{i m ϕ}, for | m | \leq ℓ \\ (29.41) \end{matrix}

Remarks: The effect of calling each of these functions is implementation-defined if l >= 128.