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

§14.24 Analytic Continuation

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Keywords:: analytic continuation, associated Legendre functions
Notes:: See Olver (1997b, p. 179).
Permalink:: http://dlmf.nist.gov/14.24
See also:: Annotations for Ch.14

Let 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 . 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 $\boldsymbol{Q}^{\mu}_{\nu}\left(ze^{s\pi i}\right)=(-1)^{s}e^{-s\nu\pi i}% \boldsymbol{Q}^{\mu}_{\nu}\left(z\right),$

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Symbols:: $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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, : 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
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 $P^{-\mu}_{\nu,s}\left(z\right)$ 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) times in the positive sense. Then

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

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Symbols:: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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, : complex variable, $\mu$ : general order, $\nu$ : general degree and : 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 $\boldsymbol{Q}^{\mu}_{\nu,s}\left(z\right)=e^{-s\mu\pi i}\boldsymbol{Q}^{\mu}_% {\nu}\left(z\right)-\frac{\pi i\sin\left(s\mu\pi\right)}{\sin\left(\mu\pi% \right)\Gamma\left(\nu-\mu+1\right)}P^{-\mu}_{\nu}\left(z\right),$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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, $\sin\NVar{z}$ : 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
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 , 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 $s=\pm 1$ .

14.23 Values on the Cut 14.25 Integral Representations

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