§14.25 Integral Representations

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

14.25.1		$$P_{\nu }^{-\mu }\left(z\right)=\frac{{\left(z^2-1\right)}^{\mu /2}}{2^{\nu }\mathrm{\Gamma }\left(\mu -\nu \right)\mathrm{\Gamma }\left(\nu +1\right)}{\int }_0^{\mathrm{\infty }}\frac{{(\sinh t)}^{2\nu +1}}{{(z+\cosh t)}^{\nu +\mu +1}}dt,$$
$\mathrm{\Re }\mu >\mathrm{\Re }\nu >-1$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $dx$: differential of $x$, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{\Re }$: 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		$${\mathit{Q}}_{\nu }^{\mu }\left(z\right)=\frac{{\pi }^{1/2}{\left(z^2-1\right)}^{\mu /2}}{2^{\mu }\mathrm{\Gamma }\left(\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\nu -\mu +1\right)}{\int }_0^{\mathrm{\infty }}\frac{{(\sinh t)}^{2\mu }}{{\left(z+{(z^2-1)}^{1/2}\cosh t\right)}^{\nu +\mu +1}}dt,$$
$\mathrm{\Re }\left(\nu +1\right)>\mathrm{\Re }\mu >-\frac{1}{2}$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathit{Q}}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{\Re }$: 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 and are continuous in $ℂ\setminus (-\mathrm{\infty },1]$.

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).