# §14.25 Integral Representations

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

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).