ルジャンドル陪多項式（グラフ）

\(\normalsize Associated\ Legendre\ polynomial\ P_n^m(x)\\ (1)\ (1-x^2)y''-2xy'+(n(n+1)-\frac{m^2}{1-x^2})y=0\\ \hspace{25px}y=P_n^m(x)\\ (2)\ {\large\int_{\tiny -1}^{\tiny 1}}P_n^m(x)P_{n'}^m(x)dx={\large\frac{2}{2n+1}\frac{(n+m)!}{(n-m)!}}\delta_{nn'}\\ (3)\ {\large\int_{\tiny -1}^{\tiny 1}}P_n^m(x)P_n^{m'}(x){\large\frac{dx}{1-x^2}}={\large\frac {(n+m)!}{m(n-m)!}}\delta_{mm'}\\ (4)\ P_n^{-m}(x)=(-1)^m{\large\frac{\Gamma(n-m+1)}{\Gamma(n+m+1)}}P_n^m(x)\\ (5)\ P_n^m(x)\ has\ several\ definitions.\\ \hspace{10px} type\ A:\ used\ by\ Wolfram(type2),\ etc\\ \hspace{15px} P_n^m(x)= {\large \frac{(1+x)^{\frac{m}{2}}}{(1-x)^{\frac{m}{2}}} \frac{\ {}_{\small 2}F_{\small 1} (-n,n+1;1-m;\frac{1-x}{2})}{\Gamma(1-m)} } \\ \hspace{10px} type\ B:\ Maple,\ Wolfram(type3),\ etc\\ \hspace{15px} P_n^m(x)= {\large \frac{(x+1)^{\frac{m}{2}}}{(x-1)^{\frac{m}{2}}} \frac{\ {}_{\small 2}F_{\small 1} (-n,n+1;1-m;\frac{1-x}{2})}{\Gamma(1-m)} } \\ \)