Table of spherical harmonics

This is a table of orthonormalized spherical harmonics that employ the Condon-Shortley phase up to degree ${\displaystyle \ell =10}$. Some of these formulas are expressed in terms of the Cartesian expansion of the spherical harmonics into polynomials in x, y, z, and r. For purposes of this table, it is useful to express the usual spherical to Cartesian transformations that relate these Cartesian components to ${\displaystyle \theta }$ and ${\displaystyle \varphi }$ as

$${\displaystyle {\begin{cases}\cos(\theta )&=z/r\\e^{\pm i\varphi }\cdot \sin(\theta )&=(x\pm iy)/r\end{cases}}}$$