14 Legendre and Related Functions Complex Arguments 14.22 Graphics 14.24 Analytic Continuation

§14.23 Values on the Cut

ⓘ

Keywords:: Ferrers functions, analytic continuation, associated Legendre functions, values on the cut
Notes:: For (14.23.1) and (14.23.5) see Olver (1997b, p. 185), and for (14.23.7) see Dunster (1991). (14.23.2) may be derived from (14.23.4) and (14.24.2). (14.23.3) may be derived from (14.3.11), (14.3.12), and (14.9.14). (14.23.4) may be derived from (14.23.1). (14.23.6) may be derived from (14.23.1) and (14.23.2).
Permalink:: http://dlmf.nist.gov/14.23
See also:: Annotations for Ch.14

When $-1<x<1$ ,

14.23.1		$P^{\mu}_{\nu}\left(x\pm i0\right)=e^{\mp\mu\pi i/2}\mathsf{P}^{\mu}_{\nu}\left% (x\right),$
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree Referenced by: §14.23 Permalink: http://dlmf.nist.gov/14.23.E1 Encodings: TeX, pMML, png See also: Annotations for §14.23 and Ch.14

14.23.2		$\boldsymbol{Q}^{\mu}_{\nu}\left(x\pm i0\right)=\frac{e^{\pm\mu\pi i/2}}{\Gamma% \left(\nu+\mu+1\right)}\left(\mathsf{Q}^{\mu}_{\nu}\left(x\right)\mp\tfrac{1}{% 2}\pi i\mathsf{P}^{\mu}_{\nu}\left(x\right)\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second 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, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree Referenced by: §14.23 Permalink: http://dlmf.nist.gov/14.23.E2 Encodings: TeX, pMML, png See also: Annotations for §14.23 and Ch.14

In terms of the hypergeometric function $\mathbf{F}$ (§14.3(i))

14.23.3		$\boldsymbol{Q}^{\mu}_{\nu}\left(x\pm i0\right)=\frac{e^{\mp\nu\pi i/2}\pi^{3/2% }\left(1-x^{2}\right)^{\mu/2}}{2^{\nu+1}}\left(\frac{x\mathbf{F}\left(\frac{1}% {2}\mu-\frac{1}{2}\nu+\frac{1}{2},\frac{1}{2}\nu+\frac{1}{2}\mu+1;\frac{3}{2};% x^{2}\right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\right)% \Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}\right)}\mp i\frac{% \mathbf{F}\left(\frac{1}{2}\mu-\frac{1}{2}\nu,\frac{1}{2}\nu+\frac{1}{2}\mu+% \frac{1}{2};\frac{1}{2};x^{2}\right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}% \mu+1\right)\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+1\right)}\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree Referenced by: §14.23 Permalink: http://dlmf.nist.gov/14.23.E3 Encodings: TeX, pMML, png See also: Annotations for §14.23 and Ch.14

Conversely,

14.23.4	$\displaystyle\mathsf{P}^{\mu}_{\nu}\left(x\right)$	$\displaystyle=e^{\pm\mu\pi i/2}P^{\mu}_{\nu}\left(x\pm i0\right),$
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree A&S Ref: 8.3.2 (corrected) Referenced by: §14.23, §14.23, §14.3(iv) Permalink: http://dlmf.nist.gov/14.23.E4 Encodings: TeX, pMML, png See also: Annotations for §14.23 and Ch.14
14.23.5	$\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(x\right)$	$\displaystyle=\tfrac{1}{2}\Gamma\left(\nu+\mu+1\right)\left(e^{-\mu\pi i/2}% \boldsymbol{Q}^{\mu}_{\nu}\left(x+i0\right)+e^{\mu\pi i/2}\boldsymbol{Q}^{\mu}% _{\nu}\left(x-i0\right)\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second 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, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree A&S Ref: 8.3.4 (modified) Referenced by: §14.23 Permalink: http://dlmf.nist.gov/14.23.E5 Encodings: TeX, pMML, png See also: Annotations for §14.23 and Ch.14

or equivalently,

14.23.6		$\mathsf{Q}^{\mu}_{\nu}\left(x\right)=e^{\mp\mu\pi i/2}\Gamma\left(\nu+\mu+1% \right)\boldsymbol{Q}^{\mu}_{\nu}\left(x\pm i0\right)\pm\tfrac{1}{2}\pi ie^{% \pm\mu\pi i/2}P^{\mu}_{\nu}\left(x\pm i0\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree Referenced by: §14.23, §14.23 Permalink: http://dlmf.nist.gov/14.23.E6 Encodings: TeX, pMML, png See also: Annotations for §14.23 and Ch.14

If cuts are introduced along the intervals $(-\infty,-1]$ and $[1,\infty)$ , then (14.23.4) and (14.23.6) could be used to extend the definitions of $\mathsf{P}^{\mu}_{\nu}\left(x\right)$ and $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ to complex $x$ .

The conical function defined by (14.20.2) can be represented similarly by

14.23.7		$\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)=\tfrac{1}{2}e^{% 3\mu\pi i/2}Q^{-\mu}_{-\frac{1}{2}+i\tau}\left(x-i0\right)+\tfrac{1}{2}e^{-3% \mu\pi i/2}Q^{-\mu}_{-\frac{1}{2}-i\tau}\left(x+i0\right).$
ⓘ Symbols: $\widehat{\mathsf{Q}}^{-\NVar{\mu}}_{\NVar{-\frac{1}{2}+i\tau}}\left(\NVar{x}\right)$ : conical function, $Q^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $x$ : real variable, $\tau$ : real variable and $\mu$ : general order Referenced by: §14.23 Permalink: http://dlmf.nist.gov/14.23.E7 Encodings: TeX, pMML, png See also: Annotations for §14.23 and Ch.14

14.22 Graphics 14.24 Analytic Continuation

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§14.23 Values on the Cut

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