# Spherical harmonics Calculator

## Calculates the spherical harmonics Ynm(θ,φ).

 [ difinition type Atype B refer to lower ] n n=0,1,2,... m m= -n ~ n zenith θ degreeradian azimuth φ same unit as θ 6digit10digit14digit18digit22digit26digit30digit34digit38digit42digit46digit50digit Ynm(θ,φ)
 $\normal Spherical\ harmonic\ function\ Y_n^m(\theta,\phi)\\[10](1)\ {\large\frac{sin\theta}{\Theta}\frac{d}{d\theta}}(sin\theta{\large\frac{d\Theta}{d\theta}})+n(n+1)sin^2\theta+{\large\frac{1}{\Phi}\frac{d^2\Phi}{d\phi^2}}=0\\\hspace{25} Y_n^m(\theta,\phi)=\Theta(\theta)\Phi(\phi)\\[10](2)\hspace{0}{\large\int_{\small 0}^{\hspace{25}\small{\pi}}\int_{\small 0}^{\hspace{25}\small{2\pi}}}Y_n^m(\theta,\phi)Y_{n'}^{m'*}(\theta,\phi)sin\theta d\theta d\phi\\\hspace{200}=\delta_{nn'}\delta_{mm'}\\[20](3)\ Y_n^m(\theta,\phi)\ has\ several\ definitions.\\[10]type\ A:\ used\ by\ Wolfram,\ etc\\\hspace{5} Y_n^m(\theta,\phi)=\sqrt{\large\frac{2n+1}{4\pi}\frac{(n-m)!}{(n+m)!}}P_n^m(cos\theta)e^{im\phi}\\[5]\hspace{5} P_n^m(x)= {\large \frac{(1+x)^{\frac{m}{2}}}{(1-x)^{\frac{m}{2}}} \frac{\ {}_{\small 2}F_{\small 1} (-n,n+1;1-m;\frac{1-x}{2})}{\Gamma(1-m)} } \\[20]type\ B:\ by\ used\ Maple,\ etc\\ \hspace{5} Y_n^m(\theta,\phi)=\sqrt{\large\frac{2n+1}{4\pi}\frac{(n-m)!}{(n+m)!}}P_n^m(cos\theta)e^{im(\phi+\pi)}\\[5]\hspace{5} P_n^m(x)= {\large \frac{(x+1)^{\frac{m}{2}}}{(x-1)^{\frac{m}{2}}} \frac{\ {}_{\small 2}F_{\small 1} (-n,n+1;1-m;\frac{1-x}{2})}{\Gamma(1-m)} } \\$

Spherical harmonics
 [1-1] /1 Disp-Num5103050100200
[1]  2010/06/22 20:10   Male / 20 level / A specialized student / Very /
Purpose of use
TO STUDY THE BEAVIOUR COMLICATED MATHEMATICL FUNCTION AND APPLY IT TO PHYSICAL PROBLEMS
Comment/Request
I THINK THAT IS A USEFULL WORK FOR BEGNERS AND SPACLISTS

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