# §14.8 Behavior at Singularities

## §14.8(i) $x\to 1-$ or $x\to-1+$

As $x\to 1-$,

 14.8.1 $\displaystyle\mathop{\mathsf{P}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle\sim\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(1-\mu\right)}\left% (\frac{2}{1-x}\right)^{\mu/2},$ $\mu\neq 1,2,3,\dots$, 14.8.2 $\displaystyle\mathop{\mathsf{P}^{m}_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle\sim(-1)^{m}\frac{{\left(\nu-m+1\right)_{2m}}}{m!}\left(\frac{1-x% }{2}\right)^{m/2},$ $m=1,2,3,\dots$, $\nu\neq m-1,m-2,\dots,-m$, 14.8.3 $\displaystyle\mathop{\mathsf{Q}_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle=\frac{1}{2}\mathop{\ln\/}\nolimits\!\left(\frac{2}{1-x}\right)-% \gamma-\mathop{\psi\/}\nolimits\!\left(\nu+1\right)+\mathop{O\/}\nolimits\!% \left(1-x\right),$ $\nu\neq-1,-2,-3,\dots$,

where $\gamma$ is Euler’s constant (§5.2(ii)). In the next three relations $\Re{\mu}>0$.

 14.8.4 $\mathop{\mathsf{Q}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)\sim\frac{1}{2}% \mathop{\cos\/}\nolimits\!\left(\mu\pi\right)\mathop{\Gamma\/}\nolimits\!\left% (\mu\right)\left(\frac{2}{1-x}\right)^{\mu/2},$ $\mu\neq\tfrac{1}{2},\tfrac{3}{2},\tfrac{5}{2},\dots$,
 14.8.5 $\mathop{\mathsf{Q}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)\sim(-1)^{\mu+(1/2)}% \frac{\pi\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}{2\mathop{\Gamma\/% }\nolimits\!\left(\mu+1\right)\mathop{\Gamma\/}\nolimits\!\left(\nu-\mu+1% \right)}\left(\frac{1-x}{2}\right)^{\mu/2},$ $\mu=\tfrac{1}{2},\tfrac{3}{2},\tfrac{5}{2},\dots$, $\nu\pm\mu\neq-1,-2,-3,\dots$,
 14.8.6 $\mathop{\mathsf{Q}^{-\mu}_{\nu}\/}\nolimits\!\left(x\right)\sim\frac{\mathop{% \Gamma\/}\nolimits\!\left(\mu\right)\mathop{\Gamma\/}\nolimits\!\left(\nu-\mu+% 1\right)}{2\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}\left(\frac{2}{1% -x}\right)^{\mu/2},$ $\nu\pm\mu\neq-1,-2,-3,\dots$.

The behavior of $\mathop{\mathsf{P}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$ and $\mathop{\mathsf{Q}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$ as $x\to-1+$ follows from the above results and the connection formulas (14.9.8) and (14.9.10).

## §14.8(ii) $x\to 1+$

 14.8.7 $\displaystyle\mathop{P^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle\sim\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(1-\mu\right)}\left% (\frac{2}{x-1}\right)^{\mu/2},$ $\mu\neq 1,2,3,\dots$, 14.8.8 $\displaystyle\mathop{P^{m}_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle\sim\frac{\mathop{\Gamma\/}\nolimits\!\left(\nu+m+1\right)}{m!% \mathop{\Gamma\/}\nolimits\!\left(\nu-m+1\right)}\left(\frac{x-1}{2}\right)^{m% /2},$ $m=1,2,3,\dots$, $\nu\pm m\neq-1,-2,-3,\dots$, 14.8.9 $\displaystyle\mathop{\boldsymbol{Q}_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle=-\frac{\mathop{\ln\/}\nolimits\!\left(x-1\right)}{2\mathop{% \Gamma\/}\nolimits\!\left(\nu+1\right)}+\frac{\frac{1}{2}\mathop{\ln\/}% \nolimits 2-\gamma-\mathop{\psi\/}\nolimits\!\left(\nu+1\right)}{\mathop{% \Gamma\/}\nolimits\!\left(\nu+1\right)}+\mathop{O\/}\nolimits\!\left(x-1\right),$ $\nu\neq-1,-2,-3,\dots$,
 14.8.10 $\mathop{\boldsymbol{Q}_{-n}\/}\nolimits\!\left(x\right)\to(-1)^{n+1}(n-1)!,$ $n=1,2,3,\dots$,
 14.8.11 $\mathop{\boldsymbol{Q}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)\sim\frac{% \mathop{\Gamma\/}\nolimits\!\left(\mu\right)}{2\mathop{\Gamma\/}\nolimits\!% \left(\nu+\mu+1\right)}\left(\frac{2}{x-1}\right)^{\mu/2},$ $\Re{\mu}>0$, $\nu+\mu\neq-1,-2,-3,\dots$.

## §14.8(iii) $x\to\infty$

 14.8.12 $\displaystyle\mathop{P^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle\sim\frac{\mathop{\Gamma\/}\nolimits\!\left(\nu+\frac{1}{2}\right% )}{\pi^{1/2}\mathop{\Gamma\/}\nolimits\!\left(\nu-\mu+1\right)}(2x)^{\nu},$ $\Re{\nu}>-\tfrac{1}{2}$, $\mu-\nu\neq 1,2,3,\dots$, 14.8.13 $\displaystyle\mathop{P^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle\sim\frac{\mathop{\Gamma\/}\nolimits\!\left(-\nu-\frac{1}{2}% \right)}{\pi^{1/2}\mathop{\Gamma\/}\nolimits\!\left(-\mu-\nu\right)(2x)^{\nu+1% }},$ $\Re{\nu}<-\tfrac{1}{2}$, $\nu+\mu\neq 0,1,2,\dots$, 14.8.14 $\displaystyle\mathop{P^{\mu}_{-1/2}\/}\nolimits\!\left(x\right)$ $\displaystyle\sim\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-\mu% \right)}\left(\frac{2}{\pi x}\right)^{1/2}\mathop{\ln\/}\nolimits x,$ $\mu\neq\tfrac{1}{2},\tfrac{3}{2},\tfrac{5}{2},\dots$,
 14.8.15 $\mathop{\boldsymbol{Q}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)\sim\frac{\pi^{1% /2}}{\mathop{\Gamma\/}\nolimits\!\left(\nu+\frac{3}{2}\right)(2x)^{\nu+1}},$ $\nu\neq-\tfrac{3}{2},-\tfrac{5}{2},-\tfrac{7}{2},\dots$,
 14.8.16 ${\mathop{\boldsymbol{Q}^{\mu}_{-n-(1/2)}\/}\nolimits\!\left(x\right)\sim\frac{% \pi^{1/2}\mathop{\Gamma\/}\nolimits\!\left(\mu+n+\frac{1}{2}\right)}{n!\mathop% {\Gamma\/}\nolimits\!\left(\mu-n+\frac{1}{2}\right)(2x)^{n+(1/2)}}},$ $n=1,2,3,\dots$, $\mu-n+\frac{1}{2}\neq 0,-1,-2,\dots$.