# Legendre function Calculator

## Calculates the Legendre functions Pν(z) and Qν(z).

 $\normal P_\nu(z)={}_{\small 2}F_{\small 1} (-\nu,\nu+1;\ 1;\frac{1-z}{2})\\Q_\nu(z)={\large\frac{\sqrt{\pi}\Gamma(\nu+1)} {(2z)^{\nu+1}} \frac{{}_{\ \small 2}F_{\small 1} (\frac{\nu+1}{2},\frac{\nu}{2}+1;\nu+\frac{3}{2};\frac{1}{z^2})}{\Gamma(\nu+\frac{3}{2}) }}\\$
degree ν
 real number

Pν(z)
Qν(z)
 $\normal Legendre\ Polinomial\ P_\nu(z),\ Q_\nu(z)\\[10](1)\ (1-z^2)y''-2zy'+\nu(\nu+1)y=0\\\hspace{25}y=P_\nu(z),\ y=Q_\nu(z)\\[10](2)\ P_\nu(z)={}_{\small 2}F_{\small 1} (-\nu,\nu+1;\ 1;\frac{1-z}{2})\\\hspace{15}Q_\nu(z)={\large\frac{\sqrt{\pi}\Gamma(\nu+1)} {(2z)^{\nu+1}} \frac{{}_{\ \small 2}F_{\small 1} (\frac{\nu+1}{2},\frac{\nu}{2}+1;\nu+\frac{3}{2};\frac{1}{z^2})}{\Gamma(\nu+\frac{3}{2}) }}\\$

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