# §14.24 Analytic Continuation

Let $s$ be an arbitrary integer, and $\mathop{P^{-\mu}_{\nu}\/}\nolimits\!\left(ze^{s\pi i}\right)$ and $\mathop{\boldsymbol{Q}^{\mu}_{\nu}\/}\nolimits\!\left(ze^{s\pi i}\right)$ 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 $\mathop{P^{-\mu}_{\nu}\/}\nolimits\!\left(ze^{s\pi i}\right)=e^{s\nu\pi i}% \mathop{P^{-\mu}_{\nu}\/}\nolimits\!\left(z\right)+\frac{2i\mathop{\sin\/}% \nolimits\!\left(\left(\nu+\frac{1}{2}\right)s\pi\right)e^{-s\pi i/2}}{\mathop% {\cos\/}\nolimits\!\left(\nu\pi\right)\mathop{\Gamma\/}\nolimits\!\left(\mu-% \nu\right)}\mathop{\boldsymbol{Q}^{\mu}_{\nu}\/}\nolimits\!\left(z\right),$
 14.24.2 $\mathop{\boldsymbol{Q}^{\mu}_{\nu}\/}\nolimits\!\left(ze^{s\pi i}\right)=(-1)^% {s}e^{-s\nu\pi i}\mathop{\boldsymbol{Q}^{\mu}_{\nu}\/}\nolimits\!\left(z\right),$

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

Next, let $\mathop{P^{-\mu}_{\nu,s}\/}\nolimits\!\left(z\right)$ and $\mathop{\boldsymbol{Q}^{\mu}_{\nu,s}\/}\nolimits\!\left(z\right)$ 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 $\displaystyle\mathop{P^{-\mu}_{\nu,s}\/}\nolimits\!\left(z\right)$ $\displaystyle=e^{s\mu\pi i}\mathop{P^{-\mu}_{\nu}\/}\nolimits\!\left(z\right),$ 14.24.4 $\displaystyle\mathop{\boldsymbol{Q}^{\mu}_{\nu,s}\/}\nolimits\!\left(z\right)$ $\displaystyle=e^{-s\mu\pi i}\mathop{\boldsymbol{Q}^{\mu}_{\nu}\/}\nolimits\!% \left(z\right)-\frac{\pi i\mathop{\sin\/}\nolimits\!\left(s\mu\pi\right)}{% \mathop{\sin\/}\nolimits\!\left(\mu\pi\right)\mathop{\Gamma\/}\nolimits\!\left% (\nu-\mu+1\right)}\mathop{P^{-\mu}_{\nu}\/}\nolimits\!\left(z\right),$

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 $\mathop{P^{-\mu}_{\nu}\/}\nolimits\!\left(z\right)$ and $\mathop{\boldsymbol{Q}^{\mu}_{\nu}\/}\nolimits\!\left(z\right)$ is an entire function of each parameter $\nu$ and $\mu$.

The behavior of $\mathop{P^{-\mu}_{\nu}\/}\nolimits\!\left(z\right)$ and $\mathop{\boldsymbol{Q}^{\mu}_{\nu}\/}\nolimits\!\left(z\right)$ 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 $s=\pm 1$.