# §14.25 Integral Representations

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

where the multivalued functions have their principal values when $1 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).