14 Legendre and Related Functions Complex Arguments 14.23 Values on the Cut 14.25 Integral Representations

§14.24 Analytic Continuation

Notes:: See Olver (1997b, p. 179).
Keywords:: analytic continuation, associated Legendre functions
Permalink:: http://dlmf.nist.gov/14.24

Let 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 . Then

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, $\mathop{\cos\/}\nolimits z$ : cosine function, : base of exponential function, $\mathop{\sin\/}\nolimits z$ : sine function, : complex variable, $\mu$ : general order, $\nu$ : general degree and : arbitrary integer
Referenced by:: §14.24, §14.24
Permalink:: http://dlmf.nist.gov/14.24.E1
Encodings:: TeX, pMML, png

14.24.2 $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(ze^{{s\pi i}}\right)% =(-1)^{s}e^{{-s\nu\pi i}}\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!% \left(z\right),$

Symbols:: $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, : base of exponential function, : complex variable, $\mu$ : general order, $\nu$ : general degree and : arbitrary integer
Referenced by:: §14.23, §14.24
Permalink:: http://dlmf.nist.gov/14.24.E2
Encodings:: TeX, pMML, png

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) times in the positive sense. Then

14.24.3 $\mathop{P^{{-\mu}}_{{\nu,s}}\/}\nolimits\!\left(z\right)=e^{{s\mu\pi i}}% \mathop{P^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, : base of exponential function, : complex variable, $\mu$ : general order, $\nu$ : general degree and : arbitrary integer
Permalink:: http://dlmf.nist.gov/14.24.E3
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, : base of exponential function, $\mathop{\sin\/}\nolimits z$ : sine function, : complex variable, $\mu$ : general order, $\nu$ : general degree and : arbitrary integer
Referenced by:: §14.24
Permalink:: http://dlmf.nist.gov/14.24.E4
Encodings:: TeX, pMML, png

the limiting value being taken in (14.24.4) when $\mu\in\Integer$ .

For fixed , 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 $s=\pm 1$ .

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