§14.5 Special Values

14.5.1		$${\mathsf{P}}_{\nu }^{\mu }\left(0\right)=\frac{2^{\mu }{\pi }^{1/2}}{\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +1\right)\mathrm{\Gamma }\left(\frac{1}{2}-\frac{1}{2}\nu -\frac{1}{2}\mu \right)},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mu$: general order and $\nu$: general degree A&S Ref: 8.6.1 (modified) Referenced by: §10.59, §14.30(ii) Permalink: http://dlmf.nist.gov/14.5.E1 Encodings: TeX, pMML, png See also: Annotations for §14.5(i), §14.5 and Ch.14

14.5.2		$${\frac{d{\mathsf{P}}_{\nu }^{\mu }\left(x\right)}{dx}|}_{x=0}=-\frac{2^{\mu +1}{\pi }^{1/2}}{\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(-\frac{1}{2}\nu -\frac{1}{2}\mu \right)},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $x$: real variable, $\mu$: general order and $\nu$: general degree A&S Ref: 8.6.3 (modified) Permalink: http://dlmf.nist.gov/14.5.E2 Encodings: TeX, pMML, png See also: Annotations for §14.5(i), §14.5 and Ch.14

14.5.3		$${\mathsf{Q}}_{\nu }^{\mu }\left(0\right)=-\frac{2^{\mu -1}{\pi }^{1/2}\sin \left(\frac{1}{2}(\nu +\mu )\pi \right)\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +\frac{1}{2}\right)}{\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +1\right)},$$
$\nu +\mu \ne -1,-3,-5,\mathrm{\dots }$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $\mu$: general order and $\nu$: general degree A&S Ref: 8.6.2 Permalink: http://dlmf.nist.gov/14.5.E3 Encodings: TeX, pMML, png See also: Annotations for §14.5(i), §14.5 and Ch.14

14.5.4		$${\frac{d{\mathsf{Q}}_{\nu }^{\mu }\left(x\right)}{dx}|}_{x=0}=\frac{2^{\mu }{\pi }^{1/2}\cos \left(\frac{1}{2}(\nu +\mu )\pi \right)\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +1\right)}{\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +\frac{1}{2}\right)},$$
$\nu +\mu \ne -2,-4,-6,\mathrm{\dots }$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $x$: real variable, $\mu$: general order and $\nu$: general degree A&S Ref: 8.6.4 Permalink: http://dlmf.nist.gov/14.5.E4 Encodings: TeX, pMML, png See also: Annotations for §14.5(i), §14.5 and Ch.14

14.5.5		$${\mathsf{P}}_0\left(x\right)=P_0\left(x\right)=1,$$
ⓘ Symbols: ${\mathsf{P}}_{\nu }\left(x\right)={\mathsf{P}}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind and $x$: real variable A&S Ref: 8.4.1 Permalink: http://dlmf.nist.gov/14.5.E5 Encodings: TeX, pMML, png See also: Annotations for §14.5(ii), §14.5 and Ch.14

14.5.6		$${\mathsf{P}}_1\left(x\right)=P_1\left(x\right)=x.$$
ⓘ Symbols: ${\mathsf{P}}_{\nu }\left(x\right)={\mathsf{P}}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind and $x$: real variable A&S Ref: 8.4.3 Permalink: http://dlmf.nist.gov/14.5.E6 Encodings: TeX, pMML, png See also: Annotations for §14.5(ii), §14.5 and Ch.14

14.5.7		${\mathsf{Q}}_0\left(x\right)$	$={\displaystyle \frac{1}{2}}\ln \left({\displaystyle \frac{1+x}{1-x}}\right),$
ⓘ Symbols: $\ln z$: principal branch of logarithm function, ${\mathsf{Q}}_{\nu }\left(x\right)={\mathsf{Q}}_{\nu }^0\left(x\right)$: Ferrers function of the second kind and $x$: real variable A&S Ref: 8.4.2 Permalink: http://dlmf.nist.gov/14.5.E7 Encodings: TeX, pMML, png See also: Annotations for §14.5(ii), §14.5 and Ch.14
14.5.8		${\mathsf{Q}}_1\left(x\right)$	$={\displaystyle \frac{x}{2}}\ln \left({\displaystyle \frac{1+x}{1-x}}\right)-1.$
ⓘ Symbols: $\ln z$: principal branch of logarithm function, ${\mathsf{Q}}_{\nu }\left(x\right)={\mathsf{Q}}_{\nu }^0\left(x\right)$: Ferrers function of the second kind and $x$: real variable A&S Ref: 8.4.4 Permalink: http://dlmf.nist.gov/14.5.E8 Encodings: TeX, pMML, png See also: Annotations for §14.5(ii), §14.5 and Ch.14

14.5.9		${\mathit{Q}}_0\left(x\right)$	$={\displaystyle \frac{1}{2}}\ln \left({\displaystyle \frac{x+1}{x-1}}\right),$
ⓘ Symbols: $\ln z$: principal branch of logarithm function, ${\mathit{Q}}_{\nu }\left(z\right)={\mathit{Q}}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function and $x$: real variable A&S Ref: 8.4.2 (modified) Permalink: http://dlmf.nist.gov/14.5.E9 Encodings: TeX, pMML, png See also: Annotations for §14.5(ii), §14.5 and Ch.14
14.5.10		${\mathit{Q}}_1\left(x\right)$	$={\displaystyle \frac{x}{2}}\ln \left({\displaystyle \frac{x+1}{x-1}}\right)-1.$
ⓘ Symbols: $\ln z$: principal branch of logarithm function, ${\mathit{Q}}_{\nu }\left(z\right)={\mathit{Q}}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function and $x$: real variable A&S Ref: 8.4.4 (modified) Permalink: http://dlmf.nist.gov/14.5.E10 Encodings: TeX, pMML, png See also: Annotations for §14.5(ii), §14.5 and Ch.14

For the corresponding formulas when $\nu =2$ see §14.5(vi).

In this subsection and the next two, and $\xi >0$.

14.5.11		${\mathsf{P}}_{\nu }^{1/2}\left(\cos \theta \right)$	$={\left({\displaystyle \frac{2}{\pi \sin \theta }}\right)}^{1/2}\cos \left(\left(\nu +\frac{1}{2}\right)\theta \right),$
ⓘ Symbols: : Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\nu$: general degree and : variable A&S Ref: 8.6.12 Permalink: http://dlmf.nist.gov/14.5.E11 Encodings: TeX, pMML, png See also: Annotations for §14.5(iii), §14.5 and Ch.14
14.5.12		${\mathsf{P}}_{\nu }^{-1/2}\left(\cos \theta \right)$	$={\left({\displaystyle \frac{2}{\pi \sin \theta }}\right)}^{1/2}{\displaystyle \frac{\sin \left(\left(\nu +\frac{1}{2}\right)\theta \right)}{\nu +\frac{1}{2}}},$
ⓘ Symbols: : Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\nu$: general degree and : variable A&S Ref: 8.6.14 Permalink: http://dlmf.nist.gov/14.5.E12 Encodings: TeX, pMML, png See also: Annotations for §14.5(iii), §14.5 and Ch.14
14.5.13		${\mathsf{Q}}_{\nu }^{1/2}\left(\cos \theta \right)$	$=-{\left({\displaystyle \frac{\pi }{2\sin \theta }}\right)}^{1/2}\sin \left(\left(\nu +\frac{1}{2}\right)\theta \right),$
ⓘ Symbols: : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\nu$: general degree and : variable A&S Ref: 8.6.13 Permalink: http://dlmf.nist.gov/14.5.E13 Encodings: TeX, pMML, png See also: Annotations for §14.5(iii), §14.5 and Ch.14

14.5.14		$${\mathsf{Q}}_{\nu }^{-1/2}\left(\cos \theta \right)={\left(\frac{\pi }{2\sin \theta }\right)}^{1/2}\frac{\cos \left(\left(\nu +\frac{1}{2}\right)\theta \right)}{\nu +\frac{1}{2}}.$$
ⓘ Symbols: : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\nu$: general degree and : variable A&S Ref: 8.6.15 (corrected) Referenced by: Erratum (V1.0.16) for Equation (14.5.14) Permalink: http://dlmf.nist.gov/14.5.E14 Encodings: TeX, pMML, png Errata (effective with 1.0.16): Originally this equation was incorrect because of a minus sign in front of the right-hand side. Reported 2017-04-10 by André Greiner-Petter See also: Annotations for §14.5(iii), §14.5 and Ch.14

14.5.15		$P_{\nu }^{1/2}\left(\cosh \xi \right)$	$={\left({\displaystyle \frac{2}{\pi \sinh \xi }}\right)}^{1/2}\cosh \left(\left(\nu +\frac{1}{2}\right)\xi \right),$
ⓘ 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, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\nu$: general degree and $\xi >0$: variable A&S Ref: 8.6.8 (modified) Permalink: http://dlmf.nist.gov/14.5.E15 Encodings: TeX, pMML, png See also: Annotations for §14.5(iii), §14.5 and Ch.14
14.5.16		$P_{\nu }^{-1/2}\left(\cosh \xi \right)$	$={\left({\displaystyle \frac{2}{\pi \sinh \xi }}\right)}^{1/2}{\displaystyle \frac{\sinh \left(\left(\nu +\frac{1}{2}\right)\xi \right)}{\nu +\frac{1}{2}}},$
ⓘ 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, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\nu$: general degree and $\xi >0$: variable A&S Ref: 8.6.9 (modified) Permalink: http://dlmf.nist.gov/14.5.E16 Encodings: TeX, pMML, png See also: Annotations for §14.5(iii), §14.5 and Ch.14
14.5.17		${\mathit{Q}}_{\nu }^{\pm 1/2}\left(\cosh \xi \right)$	$={\left({\displaystyle \frac{\pi }{2\sinh \xi }}\right)}^{1/2}{\displaystyle \frac{\exp \left(-\left(\nu +\frac{1}{2}\right)\xi \right)}{\mathrm{\Gamma }\left(\nu +\frac{3}{2}\right)}}.$
ⓘ 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, $\exp z$: exponential function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\nu$: general degree and $\xi >0$: variable A&S Ref: 8.6.10 (corrected) 8.6.11 (corrected) Permalink: http://dlmf.nist.gov/14.5.E17 Encodings: TeX, pMML, png See also: Annotations for §14.5(iii), §14.5 and Ch.14

14.5.18		${\mathsf{P}}_{\nu }^{-\nu }\left(\cos \theta \right)$	$={\displaystyle \frac{{(\sin \theta )}^{\nu }}{2^{\nu }\mathrm{\Gamma }\left(\nu +1\right)}},$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $\cos z$: cosine function, $\sin z$: sine function, $\nu$: general degree and : variable A&S Ref: 8.6.17 Permalink: http://dlmf.nist.gov/14.5.E18 Encodings: TeX, pMML, png See also: Annotations for §14.5(iv), §14.5 and Ch.14
14.5.19		$P_{\nu }^{-\nu }\left(\cosh \xi \right)$	$={\displaystyle \frac{{(\sinh \xi )}^{\nu }}{2^{\nu }\mathrm{\Gamma }\left(\nu +1\right)}}.$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\nu$: general degree and $\xi >0$: variable A&S Ref: 8.6.16 (modified) Permalink: http://dlmf.nist.gov/14.5.E19 Encodings: TeX, pMML, png See also: Annotations for §14.5(iv), §14.5 and Ch.14

In this subsection $K\left(k\right)$ and $E\left(k\right)$ denote the complete elliptic integrals of the first and second kinds; see §19.2(ii).

14.5.20		$${\mathsf{P}}_{\frac{1}{2}}\left(\cos \theta \right)=\frac{2}{\pi }\left(2E\left(\sin \left(\frac{1}{2}\theta \right)\right)-K\left(\sin \left(\frac{1}{2}\theta \right)\right)\right),$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\cos z$: cosine function, ${\mathsf{P}}_{\nu }\left(x\right)={\mathsf{P}}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $\sin z$: sine function and : variable A&S Ref: 8.13.11 (in slightly different form) Permalink: http://dlmf.nist.gov/14.5.E20 Encodings: TeX, pMML, png See also: Annotations for §14.5(v), §14.5 and Ch.14

14.5.21		${\mathsf{P}}_{-\frac{1}{2}}\left(\cos \theta \right)$	$={\displaystyle \frac{2}{\pi }}K\left(\sin \left(\frac{1}{2}\theta \right)\right),$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\cos z$: cosine function, ${\mathsf{P}}_{\nu }\left(x\right)={\mathsf{P}}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $\sin z$: sine function and : variable A&S Ref: 8.13.9 Permalink: http://dlmf.nist.gov/14.5.E21 Encodings: TeX, pMML, png See also: Annotations for §14.5(v), §14.5 and Ch.14
14.5.22		${\mathsf{Q}}_{\frac{1}{2}}\left(\cos \theta \right)$	$=K\left(\cos \left(\frac{1}{2}\theta \right)\right)-2E\left(\cos \left(\frac{1}{2}\theta \right)\right),$
ⓘ Symbols: $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\cos z$: cosine function, ${\mathsf{Q}}_{\nu }\left(x\right)={\mathsf{Q}}_{\nu }^0\left(x\right)$: Ferrers function of the second kind and : variable A&S Ref: 8.13.12 (in slightly different form) Permalink: http://dlmf.nist.gov/14.5.E22 Encodings: TeX, pMML, png See also: Annotations for §14.5(v), §14.5 and Ch.14
14.5.23		${\mathsf{Q}}_{-\frac{1}{2}}\left(\cos \theta \right)$	$=K\left(\cos \left(\frac{1}{2}\theta \right)\right).$
ⓘ Symbols: $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\cos z$: cosine function, ${\mathsf{Q}}_{\nu }\left(x\right)={\mathsf{Q}}_{\nu }^0\left(x\right)$: Ferrers function of the second kind and : variable A&S Ref: 8.13.10 (in slightly different form) Permalink: http://dlmf.nist.gov/14.5.E23 Encodings: TeX, pMML, png See also: Annotations for §14.5(v), §14.5 and Ch.14

14.5.24		$P_{\frac{1}{2}}\left(\cosh \xi \right)$	$={\displaystyle \frac{2}{\pi }}{\mathrm{e}}^{\xi /2}E\left({\left(1-{\mathrm{e}}^{-2\xi }\right)}^{1/2}\right),$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind and $\xi >0$: variable A&S Ref: 8.13.6 Permalink: http://dlmf.nist.gov/14.5.E24 Encodings: TeX, pMML, png See also: Annotations for §14.5(v), §14.5 and Ch.14
14.5.25		$P_{-\frac{1}{2}}\left(\cosh \xi \right)$	$={\displaystyle \frac{2}{\pi \cosh \left(\frac{1}{2}\xi \right)}}K\left(\tanh \left(\frac{1}{2}\xi \right)\right),$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind and $\xi >0$: variable A&S Ref: 8.13.2 Permalink: http://dlmf.nist.gov/14.5.E25 Encodings: TeX, pMML, png See also: Annotations for §14.5(v), §14.5 and Ch.14

14.5.26		$${\mathit{Q}}_{\frac{1}{2}}\left(\cosh \xi \right)=\begin{array}{l}2{\pi }^{-1/2}\cosh \xi \mathrm{sech}\left(\frac{1}{2}\xi \right)K\left(\mathrm{sech}\left(\frac{1}{2}\xi \right)\right)\\ -4{\pi }^{-1/2}\cosh \left(\frac{1}{2}\xi \right)E\left(\mathrm{sech}\left(\frac{1}{2}\xi \right)\right),\end{array}$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, ${\mathit{Q}}_{\nu }\left(z\right)={\mathit{Q}}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function and $\xi >0$: variable A&S Ref: 8.13.7 (in slightly different form) Permalink: http://dlmf.nist.gov/14.5.E26 Encodings: TeX, pMML, png See also: Annotations for §14.5(v), §14.5 and Ch.14

14.5.27		$${\mathit{Q}}_{-\frac{1}{2}}\left(\cosh \xi \right)=2{\pi }^{-1/2}{\mathrm{e}}^{-\xi /2}K\left({\mathrm{e}}^{-\xi }\right).$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, ${\mathit{Q}}_{\nu }\left(z\right)={\mathit{Q}}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function and $\xi >0$: variable A&S Ref: 8.13.4 (modified) Permalink: http://dlmf.nist.gov/14.5.E27 Encodings: TeX, pMML, png See also: Annotations for §14.5(v), §14.5 and Ch.14

14.5.28		${\mathsf{P}}_2\left(x\right)$	$=P_2\left(x\right)={\displaystyle \frac{3x^2-1}{2}},$
ⓘ Symbols: ${\mathsf{P}}_{\nu }\left(x\right)={\mathsf{P}}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind and $x$: real variable Referenced by: (14.5.28) Permalink: http://dlmf.nist.gov/14.5.E28 Encodings: TeX, pMML, png Addition (effective with 1.0.7): (14.5.28) has been added to this section. See also: Annotations for §14.5(vi), §14.5 and Ch.14
14.5.29		${\mathsf{Q}}_2\left(x\right)$	$={\displaystyle \frac{3x^2-1}{4}}\ln \left({\displaystyle \frac{1+x}{1-x}}\right)-{\displaystyle \frac{3}{2}}x,$
ⓘ Symbols: $\ln z$: principal branch of logarithm function, ${\mathsf{Q}}_{\nu }\left(x\right)={\mathsf{Q}}_{\nu }^0\left(x\right)$: Ferrers function of the second kind and $x$: real variable Referenced by: (14.5.29) Permalink: http://dlmf.nist.gov/14.5.E29 Encodings: TeX, pMML, png Addition (effective with 1.0.7): (14.5.29) has been added to this section. See also: Annotations for §14.5(vi), §14.5 and Ch.14
14.5.30		${\mathit{Q}}_2\left(x\right)$	$={\displaystyle \frac{3x^2-1}{8}}\ln \left({\displaystyle \frac{x+1}{x-1}}\right)-{\displaystyle \frac{3}{4}}x.$
ⓘ Symbols: $\ln z$: principal branch of logarithm function, ${\mathit{Q}}_{\nu }\left(z\right)={\mathit{Q}}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function and $x$: real variable Referenced by: (14.5.30) Permalink: http://dlmf.nist.gov/14.5.E30 Encodings: TeX, pMML, png Addition (effective with 1.0.7): (14.5.30) has been added to this section. See also: Annotations for §14.5(vi), §14.5 and Ch.14