\(\normalsize Spherical\ harmonic\ function\ Y_n^m(\theta,\phi)\\
(1)\ {\large\frac{\sin\theta}{\Theta}\frac{d}{d\theta}}( \sin\theta{\large\frac{d\Theta}{d\theta}})+n(n+1) \sin^2\theta+{\large\frac{1}{\Phi}\frac{d^2\Phi}{d\phi^2}}=0\\
\hspace{25px} Y_n^m(\theta,\phi)=\Theta(\theta)\Phi(\phi)\\
(2){\large\int_{\small 0}^{\small{\pi}}\int_{\small 0}^{\small{2\pi}}}Y_n^m(\theta,\phi)Y_{n'}^{m'*}(\theta,\phi) \sin\theta d\theta d\phi\\
\hspace{200px}=\delta_{nn'}\delta_{mm'}\\
(3)\ Y_n^m(\theta,\phi)\ has\ several\ definitions.\\
type\ A:\ used\ by\ Wolfram,\ etc\\
\hspace{5px} Y_n^m(\theta,\phi)=\sqrt{\large\frac{2n+1}{4\pi}\frac{(n-m)!}{(n+m)!}}P_n^m( \cos\theta)e^{im\phi}\\
\hspace{5px} P_n^m(x)= {\large \frac{(1+x)^{\frac{m}{2}}}{(1-x)^{\frac{m}{2}}} \frac{\ {}_{\small 2}F_{\small 1} (-n,n+1;1-m;\frac{1-x}{2})}{\Gamma(1-m)} } \\
type\ B:\ by\ used\ Maple,\ etc\\
\hspace{5px} Y_n^m(\theta,\phi)=\sqrt{\large\frac{2n+1}{4\pi}\frac{(n-m)!}{(n+m)!}}P_n^m( \cos\theta)e^{im(\phi+\pi)}\\
\hspace{5px} P_n^m(x)= {\large \frac{(x+1)^{\frac{m}{2}}}{(x-1)^{\frac{m}{2}}} \frac{\ {}_{\small 2}F_{\small 1} (-n,n+1;1-m;\frac{1-x}{2})}{\Gamma(1-m)} } \\
\) |
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