14 Legendre and Related Functions Complex Arguments 14.24 Analytic Continuation 14.26 Uniform Asymptotic Expansions

§14.25 Integral Representations

ⓘ

Keywords:: associated Legendre functions, integral representations
Notes:: For (14.25.1) see Olver (1997b, p. 179); for (14.25.2) see Erdélyi et al. (1953a, p. 155).
Referenced by:: §14.12(ii), 3rd item
Permalink:: http://dlmf.nist.gov/14.25
See also:: Annotations for Ch.14

The principal values of $P^{-\mu}_{\nu}\left(z\right)$ and $\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)$ (§14.21(i)) are given by

14.25.1		$P^{-\mu}_{\nu}\left(z\right)=\frac{\left(z^{2}-1\right)^{\mu/2}}{2^{\nu}\Gamma% \left(\mu-\nu\right)\Gamma\left(\nu+1\right)}\int_{0}^{\infty}\frac{(\sinh t)^% {2\nu+1}}{(z+\cosh t)^{\nu+\mu+1}}\,\mathrm{d}t,$
$\Re\mu>\Re\nu>-1$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $\Re$ : real part, $z$ : complex variable, $\mu$ : general order and $\nu$ : general degree Referenced by: §14.25 Permalink: http://dlmf.nist.gov/14.25.E1 Encodings: TeX, pMML, png See also: Annotations for §14.25 and Ch.14

14.25.2		$\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)=\frac{\pi^{1/2}\left(z^{2}-1\right)^{% \mu/2}}{2^{\mu}\Gamma\left(\mu+\frac{1}{2}\right)\Gamma\left(\nu-\mu+1\right)}% \*\int_{0}^{\infty}\frac{(\sinh t)^{2\mu}}{\left(z+(z^{2}-1)^{1/2}\cosh t% \right)^{\nu+\mu+1}}\,\mathrm{d}t,$
$\Re\left(\nu+1\right)>\Re\mu>-\tfrac{1}{2}$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $\Re$ : real part, $z$ : complex variable, $\mu$ : general order and $\nu$ : general degree Referenced by: §14.25 Permalink: http://dlmf.nist.gov/14.25.E2 Encodings: TeX, pMML, png See also: Annotations for §14.25 and Ch.14

where the multivalued functions have their principal values when $1<z<\infty$ and are continuous in $\mathbb{C}\setminus(-\infty,1]$ .

For corresponding contour integrals, with less restrictions on $\mu$ and $\nu$ , see Olver (1997b, pp. 174–179), and for further integral representations see Magnus et al. (1966, §4.6.1).

14.24 Analytic Continuation 14.26 Uniform Asymptotic Expansions

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