Finding large order spherical harmonics

[alemsalem]

is there an approximation for spherical harmonics for very large l and m in closed form?

 

[lurflurf]

sure see The Theory of Spherical and Ellipsoidal Harmonics by E. W. Hobson
and learn such things as
$$l^{-m}\mathrm{P}_l^m(\cos(\theta)=\sqrt{\frac{2}{l \pi \sin(\theta)}}\cos \left( \left( l+\frac{1}{2} \right)\theta-\frac{\pi}{4}+m\frac{\pi}{2} \right)+{O}(l^{-3/2}) \\
l^{-m}\mathrm{Q}_l^m(\cos(\theta)=\sqrt{\frac{2}{l \pi \sin(\theta)}}\cos \left( \left( l+\frac{1}{2} \right)\theta+\frac{\pi}{4}+m\frac{\pi}{2} \right)+{O}(l^{-3/2}) \\
\theta \in (\epsilon,\pi-\epsilon) \\
m<<l$$
of course there are endless variations if you need more accuracy or l or theta complex and so on.

 

[alemsalem]

Thanks!