Tesseral Harmonics

Tesseral Harmonics as defined in terms of spherical harmonics by


$\displaystyle Z_{n0}$ $\textstyle =$ $\displaystyle Y_n^0$ (142)
$\displaystyle Z_{n\alpha}\equiv Z_{n,\alpha}^c$ $\textstyle =$ $\displaystyle \frac{1}{\sqrt{2}}[Y_n^{-\alpha}+(-1)^{\alpha}Y_n^{\alpha}] \dots \alpha>0$ (143)
$\displaystyle Z_{n\alpha}\equiv Z_{n,\vert\alpha\vert}^s$ $\textstyle =$ $\displaystyle \frac{i}{\sqrt{2}}[Y_n^{\alpha}-(-1)^{\alpha}Y_n^{-\alpha}] \dots \alpha<0$ (144)

A similar table has been given in [55] on p. 238 (mind, there are errors in $Z^c_{41}$,$Z^c_{43}$,$Z^c_{52}$ ... in this reference)

\begin{eqnarray*}
Z_{00}&=&\frac{1}{\sqrt{4\pi}}\\
\hline
Z^s_{11}&=&\sqrt{\fra...
...1}{64}\sqrt{\frac{26}{231\pi}}[(x^6-15x^4y^2+15x^2y^4-y^6)/r^6]
\end{eqnarray*}




Martin Rotter 2017-01-10