# Spherical harmonics Calculator

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

 [ difinition type A type B refer to lower ] n n=0,1,2,... m m= -n ~ n zenith θ degree radian 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

Sending completion

To improve this 'Spherical harmonics Calculator', please fill in questionnaire.
Male or Female ?
Male Female
Age
Under 20 years old 20 years old level 30 years old level
40 years old level 50 years old level 60 years old level or over
Occupation
Elementary school/ Junior high-school student
High-school/ University/ Grad student A homemaker An office worker / A public employee
Self-employed people An engineer A teacher / A researcher
A retired person Others
Useful?
Very Useful A little Not at All
Purpose of use?