# §14.8 Behavior at Singularities

## §14.8(i) $x\to 1-$ or $x\to-1+$

As $x\to 1-$,

 14.8.1 $\displaystyle\mathsf{P}^{\mu}_{\nu}\left(x\right)$ $\displaystyle\sim\frac{1}{\Gamma\left(1-\mu\right)}\left(\frac{2}{1-x}\right)^% {\mu/2},$ $\mu\neq 1,2,3,\dots$, 14.8.2 $\displaystyle\mathsf{P}^{m}_{\nu}\left(x\right)$ $\displaystyle\sim(-1)^{m}\frac{{\left(\nu-m+1\right)_{2m}}}{m!}\left(\frac{1-x% }{2}\right)^{m/2},$ $m=1,2,3,\dots$, $\nu\neq m-1,m-2,\dots,-m$, 14.8.3 $\displaystyle\mathsf{Q}^{\nu}\left(x\right)$ $\displaystyle=\frac{1}{2}\ln\left(\frac{2}{1-x}\right)-\gamma-\psi\left(\nu+1% \right)+O\left(\left(1-x\right)\ln\left(1-x\right)\right),$ $\nu\neq-1,-2,-3,\dots$, ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $\gamma$: Euler’s constant, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$: Ferrers function of the second kind, $\psi\left(\NVar{z}\right)$: psi (or digamma) function, $\ln\NVar{z}$: principal branch of logarithm function, $x$: real variable and $\nu$: general degree Referenced by: §14.8(i), Erratum (V1.1.5) for Equation (14.8.3) Permalink: http://dlmf.nist.gov/14.8.E3 Encodings: TeX, pMML, png Correction (effective with 1.1.5): The symbol $O\left(1-x\right)$ has been corrected to be $O\left(\left(1-x\right)\ln\left(1-x\right)\right)$. Suggested 2022-02-08 by Mark Ashbaugh See also: Annotations for §14.8(i), §14.8 and Ch.14

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

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

The behavior of $\mathsf{P}^{\mu}_{\nu}\left(x\right)$ and $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ 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 $\displaystyle P^{\mu}_{\nu}\left(x\right)$ $\displaystyle\sim\frac{1}{\Gamma\left(1-\mu\right)}\left(\frac{2}{x-1}\right)^% {\mu/2},$ $\mu\neq 1,2,3,\dots$, 14.8.8 $\displaystyle P^{m}_{\nu}\left(x\right)$ $\displaystyle\sim\frac{\Gamma\left(\nu+m+1\right)}{m!\Gamma\left(\nu-m+1\right% )}\left(\frac{x-1}{2}\right)^{m/2},$ $m=1,2,3,\dots$, $\nu\pm m\neq-1,-2,-3,\dots$, 14.8.9 $\displaystyle\boldsymbol{Q}^{\nu}\left(x\right)$ $\displaystyle=-\frac{\ln\left(x-1\right)}{2\Gamma\left(\nu+1\right)}+\frac{% \frac{1}{2}\ln 2-\gamma-\psi\left(\nu+1\right)}{\Gamma\left(\nu+1\right)}+O% \left(\left(x-1\right)\ln\left(x-1\right)\right),$ $\nu\neq-1,-2,-3,\dots$, ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $\Gamma\left(\NVar{z}\right)$: gamma function, $\gamma$: Euler’s constant, $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$: Olver’s associated Legendre function, $\psi\left(\NVar{z}\right)$: psi (or digamma) function, $\ln\NVar{z}$: principal branch of logarithm function, $x$: real variable and $\nu$: general degree Referenced by: Erratum (V1.1.5) for Equation (14.8.9) Permalink: http://dlmf.nist.gov/14.8.E9 Encodings: TeX, pMML, png Correction (effective with 1.1.5): The symbol $O\left(x-1\right)$ has been corrected to be $O\left(\left(x-1\right)\ln\left(x-1\right)\right)$. Suggested 2022-02-09 by Mark Ashbaugh See also: Annotations for §14.8(ii), §14.8 and Ch.14
 14.8.10 $\boldsymbol{Q}^{-n}\left(x\right)\to(-1)^{n+1}(n-1)!,$ $n=1,2,3,\dots$,
 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(iii) $x\to\infty$

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