14 Legendre and Related Functions Real Arguments 14.4 Graphics 14.6 Integer Order

§14.5 Special Values

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Keywords:: Ferrers functions, special values
Permalink:: http://dlmf.nist.gov/14.5
See also:: Annotations for Ch.14

§14.5(i) $x=0$
§14.5(ii) $\mu=0$ , $\nu=0,1$
§14.5(iii) $\mu=\pm\frac{1}{2}$
§14.5(iv) $\mu=-\nu$
§14.5(v) $\mu=0$ , $\nu=\pm\frac{1}{2}$
§14.5(vi) Addendum to §14.5(ii): $\mu=0$ , $\nu=2$

§14.5(i) $x=0$

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Notes:: These results may be derived from (14.3.11)–(14.3.14) and also (14.10.4), with $\mathsf{P}$ replaced by $\mathsf{Q}$ .
Permalink:: http://dlmf.nist.gov/14.5.i
See also:: Annotations for §14.5 and Ch.14

14.5.1		$\mathsf{P}^{\mu}_{\nu}\left(0\right)=\frac{2^{\mu}\pi^{1/2}}{\Gamma\left(\frac% {1}{2}\nu-\frac{1}{2}\mu+1\right)\Gamma\left(\frac{1}{2}-\frac{1}{2}\nu-\frac{% 1}{2}\mu\right)},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\pi$ : 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		$\left.\frac{\mathrm{d}\mathsf{P}^{\mu}_{\nu}\left(x\right)}{\mathrm{d}x}\right% \|_{x=0}=-\frac{2^{\mu+1}\pi^{1/2}}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+% \frac{1}{2}\right)\Gamma\left(-\frac{1}{2}\nu-\frac{1}{2}\mu\right)},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : 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		$\mathsf{Q}^{\mu}_{\nu}\left(0\right)=-\frac{2^{\mu-1}\pi^{1/2}\sin\left(\frac{% 1}{2}(\nu+\mu)\pi\right)\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}% \right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+1\right)},$
$\nu+\mu\neq-1,-2,-3,\dots$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : sine function, $\mu$ : general order and $\nu$ : general degree A&S Ref: 8.6.2 Referenced by: Erratum (V1.1.3) for Equations (14.5.3), (14.5.4), Erratum (V1.1.3) for Equations (14.5.3), (14.5.4) Permalink: http://dlmf.nist.gov/14.5.E3 Encodings: TeX, pMML, png Correction (effective with 1.1.3): The constraint has been corrected to exclude all negative integers. See also: Annotations for §14.5(i), §14.5 and Ch.14

14.5.4		$\left.\frac{\mathrm{d}\mathsf{Q}^{\mu}_{\nu}\left(x\right)}{\mathrm{d}x}\right% \|_{x=0}=\frac{2^{\mu}\pi^{1/2}\cos\left(\frac{1}{2}(\nu+\mu)\pi\right)\Gamma% \left(\frac{1}{2}\nu+\frac{1}{2}\mu+1\right)}{\Gamma\left(\frac{1}{2}\nu-\frac% {1}{2}\mu+\frac{1}{2}\right)},$
$\nu+\mu\neq-1,-2,-3,\dots$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $x$ : real variable, $\mu$ : general order and $\nu$ : general degree A&S Ref: 8.6.4 Referenced by: Erratum (V1.1.3) for Equations (14.5.3), (14.5.4), Erratum (V1.1.3) for Equations (14.5.3), (14.5.4) Permalink: http://dlmf.nist.gov/14.5.E4 Encodings: TeX, pMML, png Correction (effective with 1.1.3): The constraint has been corrected to exclude all negative integers. See also: Annotations for §14.5(i), §14.5 and Ch.14

§14.5(ii) $\mu=0$ , $\nu=0,1$

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Notes:: Use §14.7(i).
Referenced by:: §14.5(vi), Erratum (V1.0.17) for Subsections 14.5(ii), 14.5(vi), Erratum (V1.0.17) for Subsections 14.5(ii), 14.5(vi), Erratum (V1.0.7) for Subsection 14.5(vi)
Permalink:: http://dlmf.nist.gov/14.5.ii
Clarification (effective with 1.0.17):: The title of this subsection has been changed back to its original name.
Addition (effective with 1.0.7):: The title of this subsection has been revised and the sentence at its end has been added.
See also:: Annotations for §14.5 and Ch.14

14.5.5		$\mathsf{P}_{0}\left(x\right)=P_{0}\left(x\right)=1,$
ⓘ Symbols: $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$ : 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: $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$ : 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	$\displaystyle\mathsf{Q}_{0}\left(x\right)$	$\displaystyle=\frac{1}{2}\ln\left(\frac{1+x}{1-x}\right),$
ⓘ Symbols: $\ln\NVar{z}$ : principal branch of logarithm function, $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$ : 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	$\displaystyle\mathsf{Q}_{1}\left(x\right)$	$\displaystyle=\frac{x}{2}\ln\left(\frac{1+x}{1-x}\right)-1.$
ⓘ Symbols: $\ln\NVar{z}$ : principal branch of logarithm function, $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$ : 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	$\displaystyle\boldsymbol{Q}_{0}\left(x\right)$	$\displaystyle=\frac{1}{2}\ln\left(\frac{x+1}{x-1}\right),$
ⓘ Symbols: $\ln\NVar{z}$ : principal branch of logarithm function, $\boldsymbol{Q}_{\NVar{\nu}}\left(\NVar{z}\right)=\boldsymbol{Q}^{0}_{\nu}\left% (z\right)$ : 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	$\displaystyle\boldsymbol{Q}_{1}\left(x\right)$	$\displaystyle=\frac{x}{2}\ln\left(\frac{x+1}{x-1}\right)-1.$
ⓘ Symbols: $\ln\NVar{z}$ : principal branch of logarithm function, $\boldsymbol{Q}_{\NVar{\nu}}\left(\NVar{z}\right)=\boldsymbol{Q}^{0}_{\nu}\left% (z\right)$ : 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).

§14.5(iii) $\mu=\pm\frac{1}{2}$

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Keywords:: associated Legendre functions, special values
Notes:: See Erdélyi et al. (1953a, p. 150).
Permalink:: http://dlmf.nist.gov/14.5.iii
See also:: Annotations for §14.5 and Ch.14

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

14.5.11	$\displaystyle\mathsf{P}^{1/2}_{\nu}\left(\cos\theta\right)$	$\displaystyle=\left(\frac{2}{\pi\sin\theta}\right)^{1/2}\cos\left(\left(\nu+% \tfrac{1}{2}\right)\theta\right),$
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\nu$ : general degree and $0<\theta<\pi$ : 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	$\displaystyle\mathsf{P}^{-1/2}_{\nu}\left(\cos\theta\right)$	$\displaystyle=\left(\frac{2}{\pi\sin\theta}\right)^{1/2}\frac{\sin\left(\left(% \nu+\frac{1}{2}\right)\theta\right)}{\nu+\frac{1}{2}},$
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\nu$ : general degree and $0<\theta<\pi$ : 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	$\displaystyle\mathsf{Q}^{1/2}_{\nu}\left(\cos\theta\right)$	$\displaystyle=-\left(\frac{\pi}{2\sin\theta}\right)^{1/2}\sin\left(\left(\nu+% \tfrac{1}{2}\right)\theta\right),$
ⓘ Symbols: $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\nu$ : general degree and $0<\theta<\pi$ : 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		$\mathsf{Q}^{-1/2}_{\nu}\left(\cos\theta\right)=\left(\frac{\pi}{2\sin\theta}% \right)^{1/2}\frac{\cos\left(\left(\nu+\frac{1}{2}\right)\theta\right)}{\nu+% \frac{1}{2}}.$
ⓘ Symbols: $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\nu$ : general degree and $0<\theta<\pi$ : 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	$\displaystyle P^{1/2}_{\nu}\left(\cosh\xi\right)$	$\displaystyle=\left(\frac{2}{\pi\sinh\xi}\right)^{1/2}\cosh\left(\left(\nu+% \tfrac{1}{2}\right)\xi\right),$
ⓘ 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, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : 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	$\displaystyle P^{-1/2}_{\nu}\left(\cosh\xi\right)$	$\displaystyle=\left(\frac{2}{\pi\sinh\xi}\right)^{1/2}\frac{\sinh\left(\left(% \nu+\frac{1}{2}\right)\xi\right)}{\nu+\frac{1}{2}},$
ⓘ 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, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : 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	$\displaystyle\boldsymbol{Q}^{\pm 1/2}_{\nu}\left(\cosh\xi\right)$	$\displaystyle=\left(\frac{\pi}{2\sinh\xi}\right)^{1/2}\frac{\exp\left(-\left(% \nu+\frac{1}{2}\right)\xi\right)}{\Gamma\left(\nu+\frac{3}{2}\right)}.$
ⓘ 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, $\exp\NVar{z}$ : exponential function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : 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(iv) $\mu=-\nu$

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Notes:: See Erdélyi et al. (1953a, p. 150).
Permalink:: http://dlmf.nist.gov/14.5.iv
See also:: Annotations for §14.5 and Ch.14

14.5.18	$\displaystyle\mathsf{P}^{-\nu}_{\nu}\left(\cos\theta\right)$	$\displaystyle=\frac{(\sin\theta)^{\nu}}{2^{\nu}\Gamma\left(\nu+1\right)},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\nu$ : general degree and $0<\theta<\pi$ : 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	$\displaystyle P^{-\nu}_{\nu}\left(\cosh\xi\right)$	$\displaystyle=\frac{(\sinh\xi)^{\nu}}{2^{\nu}\Gamma\left(\nu+1\right)}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : 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

§14.5(v) $\mu=0$ , $\nu=\pm\frac{1}{2}$

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Keywords:: Ferrers functions, associated Legendre functions, elliptic integrals, relations to other functions, special values
Notes:: See Magnus et al. (1966, p. 173). These results may be verified by comparing the hypergeometric representations of the Legendre functions and elliptic integrals (§§14.3 and 19.5).
Referenced by:: §14.19(i)
Permalink:: http://dlmf.nist.gov/14.5.v
See also:: Annotations for §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		$\mathsf{P}_{\frac{1}{2}}\left(\cos\theta\right)=\frac{2}{\pi}\left(2E\left(% \sin\left(\tfrac{1}{2}\theta\right)\right)-K\left(\sin\left(\tfrac{1}{2}\theta% \right)\right)\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $\cos\NVar{z}$ : cosine function, $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $\sin\NVar{z}$ : sine function and $0<\theta<\pi$ : 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	$\displaystyle\mathsf{P}_{-\frac{1}{2}}\left(\cos\theta\right)$	$\displaystyle=\frac{2}{\pi}K\left(\sin\left(\tfrac{1}{2}\theta\right)\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\cos\NVar{z}$ : cosine function, $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $\sin\NVar{z}$ : sine function and $0<\theta<\pi$ : 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	$\displaystyle\mathsf{Q}_{\frac{1}{2}}\left(\cos\theta\right)$	$\displaystyle=K\left(\cos\left(\tfrac{1}{2}\theta\right)\right)-2E\left(\cos% \left(\tfrac{1}{2}\theta\right)\right),$
ⓘ Symbols: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $\cos\NVar{z}$ : cosine function, $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$ : Ferrers function of the second kind and $0<\theta<\pi$ : 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	$\displaystyle\mathsf{Q}_{-\frac{1}{2}}\left(\cos\theta\right)$	$\displaystyle=K\left(\cos\left(\tfrac{1}{2}\theta\right)\right).$
ⓘ Symbols: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\cos\NVar{z}$ : cosine function, $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$ : Ferrers function of the second kind and $0<\theta<\pi$ : 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	$\displaystyle P_{\frac{1}{2}}\left(\cosh\xi\right)$	$\displaystyle=\frac{2}{\pi}e^{\xi/2}E\left(\left(1-e^{-2\xi}\right)^{1/2}% \right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$ : 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	$\displaystyle P_{-\frac{1}{2}}\left(\cosh\xi\right)$	$\displaystyle=\frac{2}{\pi\cosh\left(\frac{1}{2}\xi\right)}K\left(\tanh\left(% \tfrac{1}{2}\xi\right)\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\cosh\NVar{z}$ : hyperbolic cosine function, $\tanh\NVar{z}$ : hyperbolic tangent function, $P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$ : 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		$\boldsymbol{Q}_{\frac{1}{2}}\left(\cosh\xi\right)=2\pi^{-1/2}\cosh\xi% \operatorname{sech}\left(\tfrac{1}{2}\xi\right)K\left(\operatorname{sech}\left% (\tfrac{1}{2}\xi\right)\right)-4\pi^{-1/2}\cosh\left(\tfrac{1}{2}\xi\right)E% \left(\operatorname{sech}\left(\tfrac{1}{2}\xi\right)\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $\cosh\NVar{z}$ : hyperbolic cosine function, $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\boldsymbol{Q}_{\NVar{\nu}}\left(\NVar{z}\right)=\boldsymbol{Q}^{0}_{\nu}\left% (z\right)$ : 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		$\boldsymbol{Q}_{-\frac{1}{2}}\left(\cosh\xi\right)=2\pi^{-1/2}e^{-\xi/2}K\left% (e^{-\xi}\right).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\boldsymbol{Q}_{\NVar{\nu}}\left(\NVar{z}\right)=\boldsymbol{Q}^{0}_{\nu}\left% (z\right)$ : 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(vi) Addendum to §14.5(ii): $\mu=0$ , $\nu=2$

ⓘ

Referenced by:: §14.5(ii), Erratum (V1.0.17) for Subsections 14.5(ii), 14.5(vi), Erratum (V1.0.7) for Subsection 14.5(vi)
Permalink:: http://dlmf.nist.gov/14.5.vi
Clarification (effective with 1.0.17):: The title of this subsection has been restricted to be more descriptive.
Addition (effective with 1.0.7):: This subsection has been added.
See also:: Annotations for §14.5 and Ch.14

14.5.28	$\displaystyle\mathsf{P}_{2}\left(x\right)$	$\displaystyle=P_{2}\left(x\right)=\frac{3x^{2}-1}{2},$
ⓘ Symbols: $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$ : 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	$\displaystyle\mathsf{Q}_{2}\left(x\right)$	$\displaystyle=\frac{3x^{2}-1}{4}\ln\left(\frac{1+x}{1-x}\right)-\frac{3}{2}x,$
ⓘ Symbols: $\ln\NVar{z}$ : principal branch of logarithm function, $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$ : 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	$\displaystyle\boldsymbol{Q}_{2}\left(x\right)$	$\displaystyle=\frac{3x^{2}-1}{8}\ln\left(\frac{x+1}{x-1}\right)-\frac{3}{4}x.$
ⓘ Symbols: $\ln\NVar{z}$ : principal branch of logarithm function, $\boldsymbol{Q}_{\NVar{\nu}}\left(\NVar{z}\right)=\boldsymbol{Q}^{0}_{\nu}\left% (z\right)$ : 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