§14.24 Analytic Continuation

Let $s$ be an arbitrary integer, and $P_{\nu }^{-\mu }\left(z{\mathrm{e}}^{s\pi \mathrm{i}}\right)$ and ${\mathit{Q}}_{\nu }^{\mu }\left(z{\mathrm{e}}^{s\pi \mathrm{i}}\right)$ 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		$$P_{\nu }^{-\mu }\left(z{\mathrm{e}}^{s\pi \mathrm{i}}\right)={\mathrm{e}}^{s\nu \pi \mathrm{i}}{P}_{\nu }^{-\mu }\left(z\right)+\frac{2\mathrm{i}\sin \left(\left(\nu +\frac{1}{2}\right)s\pi \right){\mathrm{e}}^{-s\pi \mathrm{i}/2}}{\cos \left(\nu \pi \right)\mathrm{\Gamma }\left(\mu -\nu \right)}{\mathit{Q}}_{\nu }^{\mu }\left(z\right),$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${\mathit{Q}}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $z$: complex variable, $\mu$: general order, $\nu$: general degree and $s$: arbitrary integer Referenced by: §14.24, §14.24 Permalink: http://dlmf.nist.gov/14.24.E1 Encodings: TeX, pMML, png See also: Annotations for §14.24 and Ch.14

14.24.2		$${\mathit{Q}}_{\nu }^{\mu }\left(z{\mathrm{e}}^{s\pi \mathrm{i}}\right)={(-1)}^s{\mathrm{e}}^{-s\nu \pi \mathrm{i}}{\mathit{Q}}_{\nu }^{\mu }\left(z\right),$$
ⓘ Symbols: ${\mathit{Q}}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $\mu$: general order, $\nu$: general degree and $s$: arbitrary integer Referenced by: §14.23, §14.24 Permalink: http://dlmf.nist.gov/14.24.E2 Encodings: TeX, pMML, png See also: Annotations for §14.24 and Ch.14

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

Next, let $P_{\nu ,s}^{-\mu }\left(z\right)$ and ${\mathit{Q}}_{\nu ,s}^{\mu }\left(z\right)$ 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		$P_{\nu ,s}^{-\mu }\left(z\right)$	$={\mathrm{e}}^{s\mu \pi \mathrm{i}}{P}_{\nu }^{-\mu }\left(z\right),$
ⓘ Symbols: $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $\mu$: general order, $\nu$: general degree and $s$: arbitrary integer Permalink: http://dlmf.nist.gov/14.24.E3 Encodings: TeX, pMML, png See also: Annotations for §14.24 and Ch.14
14.24.4		${\mathit{Q}}_{\nu ,s}^{\mu }\left(z\right)$	$={\mathrm{e}}^{-s\mu \pi \mathrm{i}}{\mathit{Q}}_{\nu }^{\mu }\left(z\right)-{\displaystyle \frac{\pi \mathrm{i}\sin \left(s\mu \pi \right)}{\sin \left(\mu \pi \right)\mathrm{\Gamma }\left(\nu -\mu +1\right)}}{P}_{\nu }^{-\mu }\left(z\right),$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${\mathit{Q}}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $z$: complex variable, $\mu$: general order, $\nu$: general degree and $s$: arbitrary integer Referenced by: §14.24 Permalink: http://dlmf.nist.gov/14.24.E4 Encodings: TeX, pMML, png See also: Annotations for §14.24 and Ch.14

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

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

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