Spherical harmonics

A function $Y$ defined on a sphere is said to be a spherical harmonic if it is an eigenfunction of the Laplace-Beltrami operator on the spherical surface $r=$ const:

In a spherical coordinate system $(\vartheta, \phi)$ where $\vartheta$ is the polar angle (or colatitude in radians) and $\phi$ is the azimuthal angle, the complete set of spherical harmonics is given by ¹

where $P_{n}^{m},$ the Legendre functions of the first kind, solve the differential equation $\frac{1}{\sin \theta} \frac{d}{d \theta}\left(\sin \theta \frac{d P_{n}^{m}}{d \theta}\right)+\left[n(n+1)-\frac{m^{2}}{\sin ^{2} \theta}\right] P_{n}^{m}=0 .$ The integers $m$ and $n$ satisfy $0 \leq m \leq n,$ and the superscript $z=\mathrm{e}, \mathrm{o},$ which stands for "even" or "odd" refers to the parity with respect to azimuthal angle $\phi .$ The eigenvalue of $Y_{m n}$ is

irrespective of the index $m$ and the the parity $z$.