Spherical harmonics Calculator

Calculates the spherical harmonics Ynm(θ,φ).

[ difinition
 type A type B refer to lower ]
 n=0,1,2,...
 m= -n ~ n
zenith θ
azimuth φ
 same unit as θ

Ynm(θ,φ)
 $\normal Spherical\ harmonic\ function\ Y_n^m(\theta,\phi)\\[10](1)\ {\large\frac{sin\theta}{\Theta}\frac{d}{d\theta}}(sin\theta{\large\frac{d\Theta}{d\theta}})+n(n+1)sin^2\theta+{\large\frac{1}{\Phi}\frac{d^2\Phi}{d\phi^2}}=0\\\hspace{25} Y_n^m(\theta,\phi)=\Theta(\theta)\Phi(\phi)\\[10](2)\hspace{0}{\large\int_{\small 0}^{\hspace{25}\small{\pi}}\int_{\small 0}^{\hspace{25}\small{2\pi}}}Y_n^m(\theta,\phi)Y_{n'}^{m'*}(\theta,\phi)sin\theta d\theta d\phi\\\hspace{200}=\delta_{nn'}\delta_{mm'}\\[20](3)\ Y_n^m(\theta,\phi)\ has\ several\ definitions.\\[10]type\ A:\ used\ by\ Wolfram,\ etc\\\hspace{5} Y_n^m(\theta,\phi)=\sqrt{\large\frac{2n+1}{4\pi}\frac{(n-m)!}{(n+m)!}}P_n^m(cos\theta)e^{im\phi}\\[5]\hspace{5} P_n^m(x)= {\large \frac{(1+x)^{\frac{m}{2}}}{(1-x)^{\frac{m}{2}}} \frac{\ {}_{\small 2}F_{\small 1} (-n,n+1;1-m;\frac{1-x}{2})}{\Gamma(1-m)} } \\[20]type\ B:\ by\ used\ Maple,\ etc\\ \hspace{5} Y_n^m(\theta,\phi)=\sqrt{\large\frac{2n+1}{4\pi}\frac{(n-m)!}{(n+m)!}}P_n^m(cos\theta)e^{im(\phi+\pi)}\\[5]\hspace{5} P_n^m(x)= {\large \frac{(x+1)^{\frac{m}{2}}}{(x-1)^{\frac{m}{2}}} \frac{\ {}_{\small 2}F_{\small 1} (-n,n+1;1-m;\frac{1-x}{2})}{\Gamma(1-m)} } \\$

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