Legendre associated functions
Compute the Legendre associated polynomials Pνμ(z), Qνμ(z). The variable z also supports complex numbers.
The Legendre associated polynomials use the following definitions.
| \(\normalsize\\ \ P_\nu^\mu(z)= {\large \frac{(z+1)^{\frac{\mu}{2}}}{(z-1)^{\frac{\mu}{2}}} \frac{\ {}_{\small 2}F_{\small 1} (-\nu,\nu+1;1-\mu;\frac{1-z}{2})}{\Gamma(1-\mu)} } \\ \ Q_\nu^\mu(z)= {\large \frac{e^{i\mu\pi}\sqrt{\pi}\Gamma(\nu+\mu+1)(z+1)^{\frac{\mu}{2}}(z-1)^{\frac{\mu}{2}} }{ 2^{\nu+1}z^{\nu+\mu+1}}}\\ \hspace{60px}\times {\large\frac{\ {}_{\small 2}F_{\small 1} (\frac{\nu+\mu}{2}+1,\frac{\nu+\mu+1}{2};\nu+\frac{3}{2};\frac{1}{z^2})}{\Gamma(\nu+\frac{3}{2})} } \\ \) |
| Pνμ(z) | |
| Qνμ(z) |
| \(\normalsize Associated\ Legendre\ Polinomial\\ \hspace{200px} P_\nu^\mu(z),\ Q_\nu^\mu(z)\\ (1)\ (1-z^2)y''-2zy'+(\nu(\nu+1)-\frac{\mu^2}{1-z^2})y=0\\ \hspace{25px}y=P_\nu^\mu(z),\ y=Q_\nu^\mu(z)\\ (2)\ P_\nu^\mu(z)= {\large \frac{(z+1)^{\frac{\mu}{2}}}{(z-1)^{\frac{\mu}{2}}} \frac{\ {}_{\small 2}F_{\small 1} (-\nu,\nu+1;1-\mu;\frac{1-z}{2})}{\Gamma(1-\mu)} } \\ \hspace{20px} Q_\nu^\mu(z)= {\large \frac{e^{i\mu\pi}\sqrt{\pi}\Gamma(\nu+\mu+1)(z+1)^{\frac{\mu}{2}}(z-1)^{\frac{\mu}{2}} }{ 2^{\nu+1}z^{\nu+\mu+1}}}\\ \hspace{80px}\times {\large\frac{\ {}_{\small 2}F_{\small 1} (\frac{\nu+\mu}{2}+1,\frac{\nu+\mu+1}{2};\nu+\frac{3}{2};\frac{1}{z^2})}{\Gamma(\nu+\frac{3}{2})} } \\\) |
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