§14.23 Values on the Cut

When ,

14.23.1		$$P_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)={\mathrm{e}}^{\mp \mu \pi \mathrm{i}/2}{\mathsf{P}}_{\nu }^{\mu }\left(x\right),$$
ⓘ Symbols: : Ferrers function of the first kind, $P_{\nu }^{\mu }\left(z\right)$: 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, $x$: 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		$${\mathit{Q}}_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)=\frac{{\mathrm{e}}^{\pm \mu \pi \mathrm{i}/2}}{\mathrm{\Gamma }\left(\nu +\mu +1\right)}\left({\mathsf{Q}}_{\nu }^{\mu }\left(x\right)\mp \frac{1}{2}\pi \mathrm{i}{\mathsf{P}}_{\nu }^{\mu }\left(x\right)\right).$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, : Ferrers function of the second 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, $x$: 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		$${\mathit{Q}}_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)=\begin{array}{l}\frac{{\mathrm{e}}^{\mp \nu \pi \mathrm{i}/2}{\pi }^{3/2}{\left(1-x^2\right)}^{\mu /2}}{2^{\nu +1}}\\ \times (\begin{array}{l}\frac{x\mathbf{F}(\frac{1}{2}\mu -\frac{1}{2}\nu +\frac{1}{2},\frac{1}{2}\nu +\frac{1}{2}\mu +1;\frac{3}{2};x^2)}{\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +\frac{1}{2}\right)}\\ \mp \mathrm{i}\frac{\mathbf{F}(\frac{1}{2}\mu -\frac{1}{2}\nu ,\frac{1}{2}\nu +\frac{1}{2}\mu +\frac{1}{2};\frac{1}{2};x^2)}{\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +1\right)\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +1\right)}).\end{array}\end{array}$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathit{Q}}_{\nu }^{\mu }\left(z\right)$: 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, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ 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		${\mathsf{P}}_{\nu }^{\mu }\left(x\right)$	$={\mathrm{e}}^{\pm \mu \pi \mathrm{i}/2}{P}_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right),$
ⓘ Symbols: : Ferrers function of the first kind, $P_{\nu }^{\mu }\left(z\right)$: 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, $x$: 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		${\mathsf{Q}}_{\nu }^{\mu }\left(x\right)$	$=\frac{1}{2}\mathrm{\Gamma }\left(\nu +\mu +1\right)\left({\mathrm{e}}^{-\mu \pi \mathrm{i}/2}{\mathit{Q}}_{\nu }^{\mu }\left(x+\mathrm{i}0\right)+{\mathrm{e}}^{\mu \pi \mathrm{i}/2}{\mathit{Q}}_{\nu }^{\mu }\left(x-\mathrm{i}0\right)\right),$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second 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, $x$: 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		$${\mathsf{Q}}_{\nu }^{\mu }\left(x\right)={\mathrm{e}}^{\mp \mu \pi \mathrm{i}/2}\mathrm{\Gamma }\left(\nu +\mu +1\right){\mathit{Q}}_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)\pm \frac{1}{2}\pi \mathrm{i}{\mathrm{e}}^{\pm \mu \pi \mathrm{i}/2}{P}_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right).$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, $P_{\nu }^{\mu }\left(z\right)$: 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, $x$: 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 $(-\mathrm{\infty },-1]$ and $[1,\mathrm{\infty })$, then (14.23.4) and (14.23.6) could be used to extend the definitions of ${\mathsf{P}}_{\nu }^{\mu }\left(x\right)$ and ${\mathsf{Q}}_{\nu }^{\mu }\left(x\right)$ to complex $x$.

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

14.23.7		$${\widehat{\mathsf{Q}}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)=\frac{1}{2}{\mathrm{e}}^{3\mu \pi \mathrm{i}/2}{Q}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x-\mathrm{i}0\right)+\frac{1}{2}{\mathrm{e}}^{-3\mu \pi \mathrm{i}/2}{Q}_{-\frac{1}{2}-\mathrm{i}\tau }^{-\mu }\left(x+\mathrm{i}0\right).$$
ⓘ Symbols: ${\widehat{\mathsf{Q}}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$: conical function, $Q_{\nu }^{\mu }\left(z\right)$: 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