§14.8 Behavior at Singularities

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Keywords:: Ferrers functions, behavior at singularities
Permalink:: http://dlmf.nist.gov/14.8
See also:: Annotations for Ch.14

§14.8(i) $x\to 1-$ or $x\to-1+$
§14.8(ii) $x\to 1+$
§14.8(iii) $x\to\infty$

§14.8(i) $x\to 1-$ or $x\to-1+$

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Keywords:: associated Legendre functions, behavior at singularities
Notes:: For (14.8.1), (14.8.4), and (14.8.6) see Olver (1997b, p. 186). For (14.8.2) and (14.8.3) see Erdélyi et al. (1953a, p. 163). (14.8.5) may be derived from (14.8.1) and (14.9.2).
Referenced by:: §14.2(iii), §14.20(iii), §14.20(iii)
Permalink:: http://dlmf.nist.gov/14.8.i
See also:: Annotations for §14.8 and Ch.14

As $x\to 1-$ ,

14.8.1	$\displaystyle\mathsf{P}^{\mu}_{\nu}\left(x\right)$	$\displaystyle\sim\frac{1}{\Gamma\left(1-\mu\right)}\left(\frac{2}{1-x}\right)^% {\mu/2},$
$\mu\neq 1,2,3,\dots$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : 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	$\displaystyle\mathsf{P}^{m}_{\nu}\left(x\right)$	$\displaystyle\sim(-1)^{m}\frac{{\left(\nu-m+1\right)_{2m}}}{m!}\left(\frac{1-x% }{2}\right)^{m/2},$
$m=1,2,3,\dots$ , $\nu\neq m-1,m-2,\dots,-m$ ,
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : 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	$\displaystyle\mathsf{Q}_{\nu}\left(x\right)$	$\displaystyle=\frac{1}{2}\ln\left(\frac{2}{1-x}\right)-\gamma-\psi\left(\nu+1% \right)+O\left(\left(1-x\right)\ln\left(1-x\right)\right),$
$\nu\neq-1,-2,-3,\dots$ ,
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\gamma$ : Euler’s constant, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\ln\NVar{z}$ : principal branch of logarithm function, $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$ : Ferrers function of the second kind, $x$ : 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		$\mathsf{Q}^{\mu}_{\nu}\left(x\right)\sim\frac{1}{2}\cos\left(\mu\pi\right)% \Gamma\left(\mu\right)\left(\frac{2}{1-x}\right)^{\mu/2},$
$\mu\neq\tfrac{1}{2},\tfrac{3}{2},\tfrac{5}{2},\dots$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : 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		$\mathsf{Q}^{\mu}_{\nu}\left(x\right)\sim(-1)^{\mu+(1/2)}\frac{\pi\Gamma\left(% \nu+\mu+1\right)}{2\Gamma\left(\mu+1\right)\Gamma\left(\nu-\mu+1\right)}\left(% \frac{1-x}{2}\right)^{\mu/2},$
$\mu=\tfrac{1}{2},\tfrac{3}{2},\tfrac{5}{2},\dots$ , $\nu\pm\mu\neq-1,-2,-3,\dots$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : 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		$\mathsf{Q}^{-\mu}_{\nu}\left(x\right)\sim\frac{\Gamma\left(\mu\right)\Gamma% \left(\nu-\mu+1\right)}{2\Gamma\left(\nu+\mu+1\right)}\left(\frac{2}{1-x}% \right)^{\mu/2},$
$\nu\pm\mu\neq-1,-2,-3,\dots$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : 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 $\mathsf{P}^{\mu}_{\nu}\left(x\right)$ 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) $x\to 1+$

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Notes:: See Erdélyi et al. (1953a, p. 163). (14.8.10) may be derived from (14.8.7) and (14.9.12).
Permalink:: http://dlmf.nist.gov/14.8.ii
See also:: Annotations for §14.8 and Ch.14

14.8.7	$\displaystyle P^{\mu}_{\nu}\left(x\right)$	$\displaystyle\sim\frac{1}{\Gamma\left(1-\mu\right)}\left(\frac{2}{x-1}\right)^% {\mu/2},$
$\mu\neq 1,2,3,\dots$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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	$\displaystyle P^{m}_{\nu}\left(x\right)$	$\displaystyle\sim\frac{\Gamma\left(\nu+m+1\right)}{m!\Gamma\left(\nu-m+1\right% )}\left(\frac{x-1}{2}\right)^{m/2},$
$m=1,2,3,\dots$ , $\nu\pm m\neq-1,-2,-3,\dots$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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	$\displaystyle\boldsymbol{Q}_{\nu}\left(x\right)$	$\displaystyle=-\frac{\ln\left(x-1\right)}{2\Gamma\left(\nu+1\right)}+\frac{% \frac{1}{2}\ln 2-\gamma-\psi\left(\nu+1\right)}{\Gamma\left(\nu+1\right)}+O% \left(\left(x-1\right)\ln\left(x-1\right)\right),$
$\nu\neq-1,-2,-3,\dots$ ,
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\gamma$ : Euler’s constant, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\ln\NVar{z}$ : principal branch of logarithm function, $\boldsymbol{Q}_{\NVar{\nu}}\left(\NVar{z}\right)=\boldsymbol{Q}^{0}_{\nu}\left% (z\right)$ : Olver’s associated Legendre 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		$\boldsymbol{Q}_{-n}\left(x\right)\to(-1)^{n+1}(n-1)!,$
$n=1,2,3,\dots$ ,
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $\boldsymbol{Q}_{\NVar{\nu}}\left(\NVar{z}\right)=\boldsymbol{Q}^{0}_{\nu}\left% (z\right)$ : 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		$\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)\sim\frac{\Gamma\left(\mu\right)}{2% \Gamma\left(\nu+\mu+1\right)}\left(\frac{2}{x-1}\right)^{\mu/2},$
$\Re\mu>0$ , $\nu+\mu\neq-1,-2,-3,\dots$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, $\sim$ : asymptotic equality, $\Re$ : 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(iii) $x\to\infty$

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Keywords:: Ferrers functions, behavior at singularities
Notes:: See Olver (1997b, pp. 171 and 173). (14.8.16) may be derived from (14.8.15) and (14.9.12).
Permalink:: http://dlmf.nist.gov/14.8.iii
See also:: Annotations for §14.8 and Ch.14

14.8.12	$\displaystyle P^{\mu}_{\nu}\left(x\right)$	$\displaystyle\sim\frac{\Gamma\left(\nu+\frac{1}{2}\right)}{\pi^{1/2}\Gamma% \left(\nu-\mu+1\right)}(2x)^{\nu},$
$\Re\nu>-\tfrac{1}{2}$ , $\mu-\nu\neq 1,2,3,\dots$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\Re$ : 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	$\displaystyle P^{\mu}_{\nu}\left(x\right)$	$\displaystyle\sim\frac{\Gamma\left(-\nu-\frac{1}{2}\right)}{\pi^{1/2}\Gamma% \left(-\mu-\nu\right)(2x)^{\nu+1}},$
$\Re\nu<-\tfrac{1}{2}$ , $\nu+\mu\neq 0,1,2,\dots$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\Re$ : 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	$\displaystyle P^{\mu}_{-1/2}\left(x\right)$	$\displaystyle\sim\frac{1}{\Gamma\left(\frac{1}{2}-\mu\right)}\left(\frac{2}{% \pi x}\right)^{1/2}\ln x,$
$\mu\neq\tfrac{1}{2},\tfrac{3}{2},\tfrac{5}{2},\dots$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$ : 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		$\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)\sim\frac{\pi^{1/2}}{\Gamma\left(\nu+% \frac{3}{2}\right)(2x)^{\nu+1}},$
$\nu\neq-\tfrac{3}{2},-\tfrac{5}{2},-\tfrac{7}{2},\dots$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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		${\boldsymbol{Q}^{\mu}_{-n-(1/2)}\left(x\right)\sim\frac{\pi^{1/2}\Gamma\left(% \mu+n+\frac{1}{2}\right)}{n!\Gamma\left(\mu-n+\frac{1}{2}\right)(2x)^{n+(1/2)}% }},$
$n=1,2,3,\dots$ , $\mu-n+\frac{1}{2}\neq 0,-1,-2,\dots$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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

§14.8 Behavior at Singularities

Contents

§14.8(i) x→1− or x→−1+

§14.8(ii) x→1+

§14.8(iii) x→∞

§14.8(i) $x\to 1-$ or $x\to-1+$

§14.8(ii) $x\to 1+$

§14.8(iii) $x\to\infty$