# Associated Legendre function Calculator

## Calculates the associated Legendre functions Pνμ(z) and Qνμ(z).

 $\normal\\\ P_\nu^\mu(z)= {\large \frac{(z+1)^{\frac{\mu}{2}}}{(z-1)^{\frac{\mu}{2}}} \frac{\ {}_{\small 2}F_{\small 1} (-\nu,\nu+1;1-\mu;\frac{1-z}{2})}{\Gamma(1-\mu)} } \\\ Q_\nu^\mu(z)= {\large \frac{e^{i\mu\pi}\sqrt{\pi}\Gamma(\nu+\mu+1)(z+1)^{\frac{\mu}{2}}(z-1)^{\frac{\mu}{2}} }{ 2^{\nu+\mu+1}z^{\nu+\mu+1}}}\\\hspace{60}\times {\large\frac{\ {}_{\small 2}F_{\small 1} (\frac{\nu+\mu}{2}+1,\frac{\nu+\mu+1}{2};\nu+\frac{3}{2};\frac{1}{z^2})}{\Gamma(\nu+\frac{3}{2})} } \\$
 degree ν real number order μ real number z 6dgt10dgt14dgt18dgt22dgt26dgt30dgt34dgt38dgt42dgt46dgt50dgt Pνμ(z) Qνμ(z)
 $\normal Associated\ Legendre\ Polinomial\\\hspace{200} P_\nu^\mu(z),\ Q_\nu^\mu(z)\\[10](1)\ (1-z^2)y''-2zy'+(\nu(\nu+1)-\frac{\mu^2}{1-z^2})y=0\\\hspace{25}y=P_\nu^\mu(z),\ y=Q_\nu^\mu(z)\\[10](2)\ P_\nu^\mu(z)= {\large \frac{(z+1)^{\frac{\mu}{2}}}{(z-1)^{\frac{\mu}{2}}} \frac{\ {}_{\small 2}F_{\small 1} (-\nu,\nu+1;1-\mu;\frac{1-z}{2})}{\Gamma(1-\mu)} } \\\hspace{20} Q_\nu^\mu(z)= {\large \frac{e^{i\mu\pi}\sqrt{\pi}\Gamma(\nu+\mu+1)(z+1)^{\frac{\mu}{2}}(z-1)^{\frac{\mu}{2}} }{ 2^{\nu+\mu+1}z^{\nu+\mu+1}}}\\\hspace{80}\times {\large\frac{\ {}_{\small 2}F_{\small 1} (\frac{\nu+\mu}{2}+1,\frac{\nu+\mu+1}{2};\nu+\frac{3}{2};\frac{1}{z^2})}{\Gamma(\nu+\frac{3}{2})} } \\$

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