§14.8 Behavior at Singularities

As $x\to 1-$,

14.8.1		${\mathsf{P}}_{\nu }^{\mu }\left(x\right)$	$\sim {\displaystyle \frac{1}{\mathrm{\Gamma }\left(1-\mu \right)}}{\left({\displaystyle \frac{2}{1-x}}\right)}^{\mu /2},$
$\mu \ne 1,2,3,\mathrm{\dots }$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathsf{P}}_{\nu }^{\mu }\left(x\right)$: Ferrers function of the first kind, $\sim$: asymptotic equality, $x$: real variable, $\mu$: general order and $\nu$: general degree Referenced by: §14.30(ii), §14.8(i) Permalink: http://dlmf.nist.gov/14.8.E1 Encodings: TeX, pMML, png See also: Annotations for §14.8(i), §14.8 and Ch.14
14.8.2		${\mathsf{P}}_{\nu }^m\left(x\right)$	$\sim {(-1)}^m{\displaystyle \frac{{\left(\nu -m+1\right)}_{2m}}{m!}}{\left({\displaystyle \frac{1-x}{2}}\right)}^{m/2},$
$m=1,2,3,\mathrm{\dots }$, $\nu \ne m-1,m-2,\mathrm{\dots },-m$,
ⓘ Symbols: ${\mathsf{P}}_{\nu }^{\mu }\left(x\right)$: Ferrers function of the first kind, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: asymptotic equality, $!$: factorial (as in $n!$), $x$: real variable, $m$: nonnegative integer and $\nu$: general degree Referenced by: §14.30(ii), §14.8(i) Permalink: http://dlmf.nist.gov/14.8.E2 Encodings: TeX, pMML, png See also: Annotations for §14.8(i), §14.8 and Ch.14
14.8.3		${\mathsf{Q}}_{\nu }\left(x\right)$	$={\displaystyle \frac{1}{2}}\ln \left({\displaystyle \frac{2}{1-x}}\right)-\gamma -\psi \left(\nu +1\right)+O\left(1-x\right),$
$\nu \ne -1,-2,-3,\mathrm{\dots }$,
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\gamma$: Euler’s constant, $\psi \left(z\right)$: psi (or digamma) function, $\ln z$: principal branch of logarithm function, ${\mathsf{Q}}_{\nu }\left(x\right)={\mathsf{Q}}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $x$: real variable and $\nu$: general degree Referenced by: §14.8(i) Permalink: http://dlmf.nist.gov/14.8.E3 Encodings: TeX, pMML, png 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 $\mathrm{\Re }\mu >0$.

14.8.4		$${\mathsf{Q}}_{\nu }^{\mu }\left(x\right)\sim \frac{1}{2}\cos \left(\mu \pi \right)\mathrm{\Gamma }\left(\mu \right){\left(\frac{2}{1-x}\right)}^{\mu /2},$$
$\mu \ne \frac{1}{2},\frac{3}{2},\frac{5}{2},\mathrm{\dots }$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathsf{Q}}_{\nu }^{\mu }\left(x\right)$: Ferrers function of the second kind, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $x$: real variable, $\mu$: general order and $\nu$: general degree Referenced by: §14.8(i) Permalink: http://dlmf.nist.gov/14.8.E4 Encodings: TeX, pMML, png See also: Annotations for §14.8(i), §14.8 and Ch.14

14.8.5		$${\mathsf{Q}}_{\nu }^{\mu }\left(x\right)\sim {(-1)}^{\mu +(1/2)}\frac{\pi \mathrm{\Gamma }\left(\nu +\mu +1\right)}{2\mathrm{\Gamma }\left(\mu +1\right)\mathrm{\Gamma }\left(\nu -\mu +1\right)}{\left(\frac{1-x}{2}\right)}^{\mu /2},$$
$\mu =\frac{1}{2},\frac{3}{2},\frac{5}{2},\mathrm{\dots }$, $\nu \pm \mu \ne -1,-2,-3,\mathrm{\dots }$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathsf{Q}}_{\nu }^{\mu }\left(x\right)$: Ferrers function of the second kind, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $x$: real variable, $\mu$: general order and $\nu$: general degree Referenced by: §14.8(i) Permalink: http://dlmf.nist.gov/14.8.E5 Encodings: TeX, pMML, png See also: Annotations for §14.8(i), §14.8 and Ch.14

14.8.6		$${\mathsf{Q}}_{\nu }^{-\mu }\left(x\right)\sim \frac{\mathrm{\Gamma }\left(\mu \right)\mathrm{\Gamma }\left(\nu -\mu +1\right)}{2\mathrm{\Gamma }\left(\nu +\mu +1\right)}{\left(\frac{2}{1-x}\right)}^{\mu /2},$$
$\nu \pm \mu \ne -1,-2,-3,\mathrm{\dots }$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathsf{Q}}_{\nu }^{\mu }\left(x\right)$: Ferrers function of the second kind, $\sim$: asymptotic equality, $x$: real variable, $\mu$: general order and $\nu$: general degree Referenced by: §14.8(i) Permalink: http://dlmf.nist.gov/14.8.E6 Encodings: TeX, pMML, png See also: Annotations for §14.8(i), §14.8 and Ch.14

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

14.8.7		$P_{\nu }^{\mu }\left(x\right)$	$\sim {\displaystyle \frac{1}{\mathrm{\Gamma }\left(1-\mu \right)}}{\left({\displaystyle \frac{2}{x-1}}\right)}^{\mu /2},$
$\mu \ne 1,2,3,\mathrm{\dots }$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\sim$: asymptotic equality, $x$: real variable, $\mu$: general order and $\nu$: general degree Referenced by: §14.21(iii), §14.24, §14.8(ii) Permalink: http://dlmf.nist.gov/14.8.E7 Encodings: TeX, pMML, png See also: Annotations for §14.8(ii), §14.8 and Ch.14
14.8.8		$P_{\nu }^m\left(x\right)$	$\sim {\displaystyle \frac{\mathrm{\Gamma }\left(\nu +m+1\right)}{m!\mathrm{\Gamma }\left(\nu -m+1\right)}}{\left({\displaystyle \frac{x-1}{2}}\right)}^{m/2},$
$m=1,2,3,\mathrm{\dots }$, $\nu \pm m\ne -1,-2,-3,\mathrm{\dots }$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\sim$: asymptotic equality, $!$: factorial (as in $n!$), $x$: real variable, $m$: nonnegative integer and $\nu$: general degree Permalink: http://dlmf.nist.gov/14.8.E8 Encodings: TeX, pMML, png See also: Annotations for §14.8(ii), §14.8 and Ch.14
14.8.9		${\mathit{Q}}_{\nu }\left(x\right)$	$=-{\displaystyle \frac{\ln \left(x-1\right)}{2\mathrm{\Gamma }\left(\nu +1\right)}}+{\displaystyle \frac{\frac{1}{2}\ln 2-\gamma -\psi \left(\nu +1\right)}{\mathrm{\Gamma }\left(\nu +1\right)}}+O\left(x-1\right),$
$\nu \ne -1,-2,-3,\mathrm{\dots }$,
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\psi \left(z\right)$: psi (or digamma) function, $\ln z$: principal branch of logarithm function, ${\mathit{Q}}_{\nu }\left(z\right)={\mathit{Q}}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function, $x$: real variable and $\nu$: general degree Permalink: http://dlmf.nist.gov/14.8.E9 Encodings: TeX, pMML, png See also: Annotations for §14.8(ii), §14.8 and Ch.14

14.8.10		$${\mathit{Q}}_{-n}\left(x\right)\to {(-1)}^{n+1}(n-1)!,$$
$n=1,2,3,\mathrm{\dots }$,
ⓘ Symbols: $!$: factorial (as in $n!$), ${\mathit{Q}}_{\nu }\left(z\right)={\mathit{Q}}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function, $x$: real variable and $n$: nonnegative integer Referenced by: §14.8(ii) Permalink: http://dlmf.nist.gov/14.8.E10 Encodings: TeX, pMML, png See also: Annotations for §14.8(ii), §14.8 and Ch.14

14.8.11		$${\mathit{Q}}_{\nu }^{\mu }\left(x\right)\sim \frac{\mathrm{\Gamma }\left(\mu \right)}{2\mathrm{\Gamma }\left(\nu +\mu +1\right)}{\left(\frac{2}{x-1}\right)}^{\mu /2},$$
$\mathrm{\Re }\mu >0$, $\nu +\mu \ne -1,-2,-3,\mathrm{\dots }$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathit{Q}}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\sim$: asymptotic equality, $\mathrm{\Re }$: real part, $x$: real variable, $\mu$: general order and $\nu$: general degree Referenced by: §14.24 Permalink: http://dlmf.nist.gov/14.8.E11 Encodings: TeX, pMML, png See also: Annotations for §14.8(ii), §14.8 and Ch.14

14.8.12		$P_{\nu }^{\mu }\left(x\right)$	$\sim {\displaystyle \frac{\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}{{\pi }^{1/2}\mathrm{\Gamma }\left(\nu -\mu +1\right)}}{(2x)}^{\nu },$
$\mathrm{\Re }\nu >-\frac{1}{2}$, $\mu -\nu \ne 1,2,3,\mathrm{\dots }$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Re }$: real part, $x$: real variable, $\mu$: general order and $\nu$: general degree Permalink: http://dlmf.nist.gov/14.8.E12 Encodings: TeX, pMML, png See also: Annotations for §14.8(iii), §14.8 and Ch.14
14.8.13		$P_{\nu }^{\mu }\left(x\right)$	$\sim {\displaystyle \frac{\mathrm{\Gamma }\left(-\nu -\frac{1}{2}\right)}{{\pi }^{1/2}\mathrm{\Gamma }\left(-\mu -\nu \right){(2x)}^{\nu +1}}},$
, $\nu +\mu \ne 0,1,2,\mathrm{\dots }$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Re }$: real part, $x$: real variable, $\mu$: general order and $\nu$: general degree Permalink: http://dlmf.nist.gov/14.8.E13 Encodings: TeX, pMML, png See also: Annotations for §14.8(iii), §14.8 and Ch.14
14.8.14		$P_{-1/2}^{\mu }\left(x\right)$	$\sim {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\frac{1}{2}-\mu \right)}}{\left({\displaystyle \frac{2}{\pi x}}\right)}^{1/2}\ln x,$
$\mu \ne \frac{1}{2},\frac{3}{2},\frac{5}{2},\mathrm{\dots }$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $x$: real variable and $\mu$: general order Permalink: http://dlmf.nist.gov/14.8.E14 Encodings: TeX, pMML, png See also: Annotations for §14.8(iii), §14.8 and Ch.14

14.8.15		$${\mathit{Q}}_{\nu }^{\mu }\left(x\right)\sim \frac{{\pi }^{1/2}}{\mathrm{\Gamma }\left(\nu +\frac{3}{2}\right){(2x)}^{\nu +1}},$$
$\nu \ne -\frac{3}{2},-\frac{5}{2},-\frac{7}{2},\mathrm{\dots }$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathit{Q}}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $x$: real variable, $\mu$: general order and $\nu$: general degree Referenced by: §14.8(iii) Permalink: http://dlmf.nist.gov/14.8.E15 Encodings: TeX, pMML, png See also: Annotations for §14.8(iii), §14.8 and Ch.14

14.8.16		$${\mathit{Q}}_{-n-(1/2)}^{\mu }\left(x\right)\sim \frac{{\pi }^{1/2}\mathrm{\Gamma }\left(\mu +n+\frac{1}{2}\right)}{n!\mathrm{\Gamma }\left(\mu -n+\frac{1}{2}\right){(2x)}^{n+(1/2)}},$$
$n=1,2,3,\mathrm{\dots }$, $\mu -n+\frac{1}{2}\ne 0,-1,-2,\mathrm{\dots }$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathit{Q}}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $x$: real variable, $n$: nonnegative integer and $\mu$: general order Referenced by: §14.21(iii), §14.8(iii) Permalink: http://dlmf.nist.gov/14.8.E16 Encodings: TeX, pMML, png See also: Annotations for §14.8(iii), §14.8 and Ch.14