# Associated Legendre polynomial (chart) Calculator

## Calculates a table of the associated Legendre polynomial Pnm(x) and draws the chart.

 The associated Legendre function Pnm(x) has several definitions.$\normal\\type\ A:\ used\ by\ Wolfram(type2),\ etc\\\ P_n^m(x)= {\large \frac{(1+x)^{\frac{m}{2}}}{(1-x)^{\frac{m}{2}}} \frac{\ {}_{\small 2}F_{\small 1} (-n,n+1;1-m;\frac{1-x}{2})}{\Gamma(1-m)} } \\[20]type\ B:\ used\ by\ Maple,\ Wolfram(type3),\ etc\\\ P_n^m(x)= {\large \frac{(x+1)^{\frac{m}{2}}}{(x-1)^{\frac{m}{2}}} \frac{\ {}_{\small 2}F_{\small 1} (-n,n+1;1-m;\frac{1-x}{2})}{\Gamma(1-m)} } \\\vspace{10}$
difinition
 type A type B
 n=0,1,2,...
 -n≦m≦n
[ initial value x
 -1≦x≦1
increment
 repetition ]
 $\normal Associated\ Legendre\ polinomial\ P_n^m(x)\\[10](1)\ (1-x^2)y''-2xy'+(n(n+1)-\frac{m^2}{1-x^2})y=0\\\hspace{25}y=P_n^m(x)\\[10](2)\ {\large\int_{\tiny -1}^{\hspace{25}\tiny 1}}P_n^m(x)P_{n'}^m(x)dx={\large\frac{2}{2n+1}\frac{(n+m)!}{(n-m)!}}\delta_{nn'}\\(3)\ {\large\int_{\tiny -1}^{\hspace{25}\tiny 1}}P_n^m(x)P_n^{m'}(x){\large\frac{dx}{1-x^2}}={\large\frac {(n+m)!}{m(n-m)!}}\delta_{mm'}\\(4)\ P_n^{-m}(x)=(-1)^m{\large\frac{\Gamma(n-m+1)}{\Gamma(n+m+1)}}P_n^m(x)\\[10](5)\ P_n^m(x)\ has\ several\ definitions.\\[10]\hspace{10} type\ A:\ used\ by\ Wolfram(type2),\ etc\\\hspace{15} P_n^m(x)= {\large \frac{(1+x)^{\frac{m}{2}}}{(1-x)^{\frac{m}{2}}} \frac{\ {}_{\small 2}F_{\small 1} (-n,n+1;1-m;\frac{1-x}{2})}{\Gamma(1-m)} } \\[20]\hspace{10} type\ B:\ Maple,\ Wolfram(type3),\ etc\\ \hspace{15} P_n^m(x)= {\large \frac{(x+1)^{\frac{m}{2}}}{(x-1)^{\frac{m}{2}}} \frac{\ {}_{\small 2}F_{\small 1} (-n,n+1;1-m;\frac{1-x}{2})}{\Gamma(1-m)} } \\$

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