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

§14.25 Integral Representations

Notes:: For (14.25.1) see Olver (1997b, p. 179); for (14.25.2) see Erdélyi et al. (1953a, p. 155).
Keywords:: associated Legendre functions
Referenced by:: ¶ ‣ §14.12(ii), §14.32
Permalink:: http://dlmf.nist.gov/14.25

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

14.25.1 $\mathop{P^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{\left(z^{2}-1% \right)^{{\mu/2}}}{2^{{\nu}}\mathop{\Gamma\/}\nolimits\!\left(\mu-\nu\right)% \mathop{\Gamma\/}\nolimits\!\left(\nu+1\right)}\int_{{0}}^{{\infty}}\frac{(% \mathop{\sinh\/}\nolimits t)^{{2\nu+1}}}{(z+\mathop{\cosh\/}\nolimits t)^{{\nu% +\mu+1}}}dt,$ $\realpart{\mu}>\realpart{\nu}>-1$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, : differential of , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\realpart{}$ : real part, : 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

14.25.2 $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{\pi^{% {1/2}}\left(z^{2}-1\right)^{{\mu/2}}}{2^{{\mu}}\mathop{\Gamma\/}\nolimits\!% \left(\mu+\frac{1}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(\nu-\mu+1\right)% }\*\int_{{0}}^{{\infty}}\frac{(\mathop{\sinh\/}\nolimits t)^{{2\mu}}}{\left(z+% (z^{2}-1)^{{1/2}}\mathop{\cosh\/}\nolimits t\right)^{{\nu+\mu+1}}}dt,$ $\realpart{(\nu+1)}>\realpart{\mu}>-\tfrac{1}{2},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, : differential of , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\realpart{}$ : real part, : 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

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

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).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 14.24 Analytic Continuation 14.26 Uniform Asymptotic Expansions