# Legendre functions

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## 11—20 of 151 matching pages

##### 14: 14.5 Special Values
###### §14.5(v) $\mu=0$, $\nu=\pm\frac{1}{2}$
14.5.30 $\boldsymbol{Q}_{2}\left(x\right)=\frac{3x^{2}-1}{8}\ln\left(\frac{x+1}{x-1}% \right)-\frac{3}{4}x.$
##### 16: 14.25 Integral Representations
###### §14.25 Integral Representations
14.25.1 $P^{-\mu}_{\nu}\left(z\right)=\frac{\left(z^{2}-1\right)^{\mu/2}}{2^{\nu}\Gamma% \left(\mu-\nu\right)\Gamma\left(\nu+1\right)}\int_{0}^{\infty}\frac{(\sinh t)^% {2\nu+1}}{(z+\cosh t)^{\nu+\mu+1}}\mathrm{d}t,$ $\Re\mu>\Re\nu>-1$,
14.25.2 $\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)=\frac{\pi^{1/2}\left(z^{2}-1\right)^{% \mu/2}}{2^{\mu}\Gamma\left(\mu+\frac{1}{2}\right)\Gamma\left(\nu-\mu+1\right)}% \*\int_{0}^{\infty}\frac{(\sinh t)^{2\mu}}{\left(z+(z^{2}-1)^{1/2}\cosh t% \right)^{\nu+\mu+1}}\mathrm{d}t,$ $\Re\left(\nu+1\right)>\Re\mu>-\tfrac{1}{2}$,
##### 17: 14.12 Integral Representations
###### §14.12(ii) $1
14.12.5 $P^{-\mu}_{\nu}\left(x\right)=\frac{\left(x^{2}-1\right)^{-\mu/2}}{\Gamma\left(% \mu\right)}\int_{1}^{x}P_{\nu}\left(t\right)(x-t)^{\mu-1}\mathrm{d}t,$ $\Re\mu>0$.
14.12.12 $\boldsymbol{Q}^{m}_{n}\left(x\right)=\frac{1}{(n-m)!}P^{m}_{n}\left(x\right)% \int_{x}^{\infty}\frac{\mathrm{d}t}{\left(t^{2}-1\right)\left(\displaystyle P^% {m}_{n}\left(t\right)\right)^{2}},$ $n\geq m$.
##### 18: 14.24 Analytic Continuation
###### §14.24 Analytic Continuation
14.24.1 $P^{-\mu}_{\nu}\left(ze^{s\pi i}\right)=e^{s\nu\pi i}P^{-\mu}_{\nu}\left(z% \right)+\frac{2i\sin\left(\left(\nu+\frac{1}{2}\right)s\pi\right)e^{-s\pi i/2}% }{\cos\left(\nu\pi\right)\Gamma\left(\mu-\nu\right)}\boldsymbol{Q}^{\mu}_{\nu}% \left(z\right),$
14.24.3 $P^{-\mu}_{\nu,s}\left(z\right)=e^{s\mu\pi i}P^{-\mu}_{\nu}\left(z\right),$
14.24.4 $\boldsymbol{Q}^{\mu}_{\nu,s}\left(z\right)=e^{-s\mu\pi i}\boldsymbol{Q}^{\mu}_% {\nu}\left(z\right)-\frac{\pi i\sin\left(s\mu\pi\right)}{\sin\left(\mu\pi% \right)\Gamma\left(\nu-\mu+1\right)}P^{-\mu}_{\nu}\left(z\right),$
For fixed $z$, other than $\pm 1$ or $\infty$, each branch of $P^{-\mu}_{\nu}\left(z\right)$ and $\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)$ is an entire function of each parameter $\nu$ and $\mu$. …
##### 19: 14.2 Differential Equations
###### §14.2(ii) Associated Legendre Equation
Ferrers functions and the associated Legendre functions are related to the Legendre functions by the equations $\mathsf{P}^{0}_{\nu}\left(x\right)=\mathsf{P}_{\nu}\left(x\right)$, $\mathsf{Q}^{0}_{\nu}\left(x\right)=\mathsf{Q}_{\nu}\left(x\right)$, $P^{0}_{\nu}\left(x\right)=P_{\nu}\left(x\right)$, $Q^{0}_{\nu}\left(x\right)=Q_{\nu}\left(x\right)$, $\boldsymbol{Q}^{0}_{\nu}\left(x\right)=\boldsymbol{Q}_{\nu}\left(x\right)=Q_{% \nu}\left(x\right)/\Gamma\left(\nu+1\right)$. … Unless stated otherwise in §§14.214.20 it is assumed that the arguments of the functions $\mathsf{P}^{\mu}_{\nu}\left(x\right)$ and $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ lie in the interval $(-1,1)$, and the arguments of the functions $P^{\mu}_{\nu}\left(x\right)$, $Q^{\mu}_{\nu}\left(x\right)$, and $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)$ lie in the interval $(1,\infty)$. …
14.2.10 $\mathscr{W}\left\{P^{\mu}_{\nu}\left(x\right),Q^{\mu}_{\nu}\left(x\right)% \right\}=-e^{\mu\pi i}\frac{\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+% 1\right)\left(x^{2}-1\right)},$
14.2.11 $P^{\mu}_{\nu+1}\left(x\right)Q^{\mu}_{\nu}\left(x\right)-P^{\mu}_{\nu}\left(x% \right)Q^{\mu}_{\nu+1}\left(x\right)=e^{\mu\pi i}\frac{\Gamma\left(\nu+\mu+1% \right)}{\Gamma\left(\nu-\mu+2\right)}.$
##### 20: 14.8 Behavior at Singularities
14.8.7 $P^{\mu}_{\nu}\left(x\right)\sim\frac{1}{\Gamma\left(1-\mu\right)}\left(\frac{2% }{x-1}\right)^{\mu/2},$ $\mu\neq 1,2,3,\dots$,
14.8.8 $P^{m}_{\nu}\left(x\right)\sim\frac{\Gamma\left(\nu+m+1\right)}{m!\Gamma\left(% \nu-m+1\right)}\left(\frac{x-1}{2}\right)^{m/2},$ $m=1,2,3,\dots$, $\nu\pm m\neq-1,-2,-3,\dots$,
14.8.11 $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)\sim\frac{\Gamma\left(\mu\right)}{2% \Gamma\left(\nu+\mu+1\right)}\left(\frac{2}{x-1}\right)^{\mu/2},$ $\Re\mu>0$, $\nu+\mu\neq-1,-2,-3,\dots$.
14.8.14 $P^{\mu}_{-1/2}\left(x\right)\sim\frac{1}{\Gamma\left(\frac{1}{2}-\mu\right)}% \left(\frac{2}{\pi x}\right)^{1/2}\ln x,$ $\mu\neq\tfrac{1}{2},\tfrac{3}{2},\tfrac{5}{2},\dots$,