# §14.24 Analytic Continuation

Let $s$ be an arbitrary integer, and $P^{-\mu}_{\nu}\left(ze^{s\pi i}\right)$ and $\boldsymbol{Q}^{\mu}_{\nu}\left(ze^{s\pi i}\right)$ denote the branches obtained from the principal branches by making $\frac{1}{2}s$ circuits, in the positive sense, of the ellipse having $\pm 1$ as foci and passing through $z$. Then

 14.24.1 $P^{-\mu}_{\nu}\left(ze^{s\pi i}\right)=e^{s\nu\pi i}P^{-\mu}_{\nu}\left(z% \right)+\frac{2i\sin\left(\left(\nu+\frac{1}{2}\right)s\pi\right)e^{-s\pi i/2}% }{\cos\left(\nu\pi\right)\Gamma\left(\mu-\nu\right)}\boldsymbol{Q}^{\mu}_{\nu}% \left(z\right),$
 14.24.2 $\boldsymbol{Q}^{\mu}_{\nu}\left(ze^{s\pi i}\right)=(-1)^{s}e^{-s\nu\pi i}% \boldsymbol{Q}^{\mu}_{\nu}\left(z\right),$

the limiting value being taken in (14.24.1) when $2\nu$ is an odd integer.

Next, let $P^{-\mu}_{\nu,s}\left(z\right)$ and $\boldsymbol{Q}^{\mu}_{\nu,s}\left(z\right)$ denote the branches obtained from the principal branches by encircling the branch point $1$ (but not the branch point $-1$) $s$ times in the positive sense. Then

 14.24.3 $\displaystyle P^{-\mu}_{\nu,s}\left(z\right)$ $\displaystyle=e^{s\mu\pi i}P^{-\mu}_{\nu}\left(z\right),$ 14.24.4 $\displaystyle\boldsymbol{Q}^{\mu}_{\nu,s}\left(z\right)$ $\displaystyle=e^{-s\mu\pi i}\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)-\frac{\pi i% \sin\left(s\mu\pi\right)}{\sin\left(\mu\pi\right)\Gamma\left(\nu-\mu+1\right)}% P^{-\mu}_{\nu}\left(z\right),$

the limiting value being taken in (14.24.4) when $\mu\in\mathbb{Z}$.

For fixed $z$, other than $\pm 1$ or $\infty$, each branch of $P^{-\mu}_{\nu}\left(z\right)$ and $\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)$ is an entire function of each parameter $\nu$ and $\mu$.

The behavior of $P^{-\mu}_{\nu}\left(z\right)$ and $\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)$ as $z\to-1$ from the left on the upper or lower side of the cut from $-\infty$ to $1$ can be deduced from (14.8.7)–(14.8.11), combined with (14.24.1) and (14.24.2) with $s=\pm 1$.