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DLMF: §14.12 Integral Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions

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14 Legendre and Related Functions Real Arguments 14.11 Derivatives with Respect to Degree or Order 14.13 Trigonometric Expansions

§14.12 Integral Representations

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Referenced by:: 3rd item
Permalink:: http://dlmf.nist.gov/14.12
See also:: Annotations for Ch.14

Contents

§14.12(i) $-1<x<1$

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Keywords:: Ferrers functions, Mehler–Dirichlet formula, integral representations
Notes:: See Erdélyi et al. (1953a, pp. 158–159).
Permalink:: http://dlmf.nist.gov/14.12.i
See also:: Annotations for §14.12 and Ch.14

Mehler–Dirichlet Formula

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Keywords:: Ferrers functions, integral representations
See also:: Annotations for §14.12(i), §14.12 and Ch.14

14.12.1		$\displaystyle\mathsf{P}^{\mu}_{\nu}\left(\cos\theta\right)$	$\displaystyle=\frac{2^{1/2}(\sin\theta)^{\mu}}{\pi^{1/2}\Gamma\left(\frac{1}{2% }-\mu\right)}\int_{0}^{\theta}\frac{\cos\left(\left(\nu+\frac{1}{2}\right)t% \right)}{(\cos t-\cos\theta)^{\mu+(1/2)}}\,\mathrm{d}t,$
$0<\theta<\pi$ , $\Re\mu<\tfrac{1}{2}$ .
ⓘ 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, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $\mu$ : general order and $\nu$ : general degree Referenced by: §14.20(iv), §18.10(i) Permalink: http://dlmf.nist.gov/14.12.E1 Encodings: TeX, pMML, png See also: Annotations for §14.12(i), §14.12(i), §14.12 and Ch.14
14.12.2		$\displaystyle\mathsf{P}^{-\mu}_{\nu}\left(x\right)$	$\displaystyle=\frac{\left(1-x^{2}\right)^{-\mu/2}}{\Gamma\left(\mu\right)}\int% _{x}^{1}\mathsf{P}_{\nu}\left(t\right)(t-x)^{\mu-1}\,\mathrm{d}t,$
$\Re\mu>0$ ;
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree Referenced by: §14.17(i) Permalink: http://dlmf.nist.gov/14.12.E2 Encodings: TeX, pMML, png See also: Annotations for §14.12(i), §14.12(i), §14.12 and Ch.14

compare (14.6.6).

14.12.3		$\mathsf{Q}^{\mu}_{\nu}\left(\cos\theta\right)=\frac{\pi^{1/2}\Gamma\left(\nu+% \mu+1\right)(\sin\theta)^{\mu}}{2^{\mu+1}\Gamma\left(\mu+\frac{1}{2}\right)% \Gamma\left(\nu-\mu+1\right)}\*\left(\int_{0}^{\infty}\frac{(\sinh t)^{2\mu}}{% (\cos\theta+i\sin\theta\cosh t)^{\nu+\mu+1}}\,\mathrm{d}t+\int_{0}^{\infty}% \frac{(\sinh t)^{2\mu}}{(\cos\theta-i\sin\theta\cosh t)^{\nu+\mu+1}}\,\mathrm{% d}t\right),$
$0<\theta<\pi$ , $\Re\mu>-\tfrac{1}{2}$ , $\Re\nu\pm\mu>-1$ .
ⓘ 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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $\mu$ : general order and $\nu$ : general degree Permalink: http://dlmf.nist.gov/14.12.E3 Encodings: TeX, pMML, png See also: Annotations for §14.12(i), §14.12(i), §14.12 and Ch.14

§14.12(ii) $1<x<\infty$

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Keywords:: associated Legendre functions, integral representations
Notes:: See Erdélyi et al. (1953a, pp. 155–156 and 159) and Olver (1997b, pp. 181–183 and 185).
Permalink:: http://dlmf.nist.gov/14.12.ii
See also:: Annotations for §14.12 and Ch.14

14.12.4		$\displaystyle P^{-\mu}_{\nu}\left(x\right)$	$\displaystyle=\frac{2^{1/2}\Gamma\left(\mu+\frac{1}{2}\right)\left(x^{2}-1% \right)^{\mu/2}}{\pi^{1/2}\Gamma\left(\nu+\mu+1\right)\Gamma\left(\mu-\nu% \right)}\*\int_{0}^{\infty}\frac{\cosh\left(\left(\nu+\frac{1}{2}\right)t% \right)}{(x+\cosh t)^{\mu+(1/2)}}\,\mathrm{d}t,$
$\nu+\mu\neq-1,-2,-3,\dots$ , $\Re\left(\mu-\nu\right)>0$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $\Re$ : real part, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree Permalink: http://dlmf.nist.gov/14.12.E4 Encodings: TeX, pMML, png See also: Annotations for §14.12(ii), §14.12 and Ch.14
14.12.5		$\displaystyle P^{-\mu}_{\nu}\left(x\right)$	$\displaystyle=\frac{\left(x^{2}-1\right)^{-\mu/2}}{\Gamma\left(\mu\right)}\int% _{1}^{x}P_{\nu}\left(t\right)(x-t)^{\mu-1}\,\mathrm{d}t,$
$\Re\mu>0$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree Referenced by: §14.17(i) Permalink: http://dlmf.nist.gov/14.12.E5 Encodings: TeX, pMML, png See also: Annotations for §14.12(ii), §14.12 and Ch.14
14.12.6		$\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)$	$\displaystyle=\frac{\pi^{1/2}\left(x^{2}-1\right)^{\mu/2}}{2^{\mu}\Gamma\left(% \mu+\frac{1}{2}\right)\Gamma\left(\nu-\mu+1\right)}\*\int_{0}^{\infty}\frac{(% \sinh t)^{2\mu}}{\left(x+(x^{2}-1)^{1/2}\cosh t\right)^{\nu+\mu+1}}\,\mathrm{d% }t,$
$\Re\left(\nu+1\right)>\Re\mu>-\tfrac{1}{2}$ .
ⓘ 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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $\Re$ : real part, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree A&S Ref: 8.8.2 (modified) Permalink: http://dlmf.nist.gov/14.12.E6 Encodings: TeX, pMML, png See also: Annotations for §14.12(ii), §14.12 and Ch.14
14.12.7		$\displaystyle P^{m}_{\nu}\left(x\right)$	$\displaystyle=\frac{{\left(\nu+1\right)_{m}}}{\pi}\*\int_{0}^{\pi}\left(x+% \left(x^{2}-1\right)^{1/2}\cos\phi\right)^{\nu}\cos\left(m\phi\right)\,\mathrm% {d}\phi,$
ⓘ Symbols: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real variable, $m$ : nonnegative integer and $\nu$ : general degree Permalink: http://dlmf.nist.gov/14.12.E7 Encodings: TeX, pMML, png See also: Annotations for §14.12(ii), §14.12 and Ch.14
14.12.8		$\displaystyle P^{m}_{n}\left(x\right)$	$\displaystyle=\frac{2^{m}m!(n+m)!\left(x^{2}-1\right)^{m/2}}{(2m)!(n-m)!\pi}% \int_{0}^{\pi}\left(x+\left(x^{2}-1\right)^{1/2}\cos\phi\right)^{n-m}(\sin\phi% )^{2m}\,\mathrm{d}\phi,$
$n\geq m$ .
ⓘ 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, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $\sin\NVar{z}$ : sine function, $x$ : real variable, $m$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/14.12.E8 Encodings: TeX, pMML, png See also: Annotations for §14.12(ii), §14.12 and Ch.14

14.12.9		$\boldsymbol{Q}^{m}_{n}\left(x\right)=\frac{1}{n!}\int_{0}^{u}\left(x-\left(x^{% 2}-1\right)^{1/2}\cosh t\right)^{n}\cosh\left(mt\right)\,\mathrm{d}t,$
ⓘ Symbols: $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $x$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $u$ Permalink: http://dlmf.nist.gov/14.12.E9 Encodings: TeX, pMML, png See also: Annotations for §14.12(ii), §14.12 and Ch.14

where

14.12.10		$u=\frac{1}{2}\ln\left(\frac{x+1}{x-1}\right).$
ⓘ Symbols: $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real variable and $u$ Referenced by: §14.15(i) Permalink: http://dlmf.nist.gov/14.12.E10 Encodings: TeX, pMML, png See also: Annotations for §14.12(ii), §14.12 and Ch.14

14.12.11		$\boldsymbol{Q}^{m}_{n}\left(x\right)=\frac{\left(x^{2}-1\right)^{m/2}}{2^{n+1}% n!}\int_{-1}^{1}\frac{\left(1-t^{2}\right)^{n}}{(x-t)^{n+m+1}}\,\mathrm{d}t,$
ⓘ Symbols: $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $x$ : real variable, $m$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/14.12.E11 Encodings: TeX, pMML, png See also: Annotations for §14.12(ii), §14.12 and Ch.14

14.12.12		$\boldsymbol{Q}^{m}_{n}\left(x\right)=\frac{1}{(n-m)!}P^{m}_{n}\left(x\right)% \int_{x}^{\infty}\frac{\,\mathrm{d}t}{\left(t^{2}-1\right)\left(\displaystyle P% ^{m}_{n}\left(t\right)\right)^{2}},$
$n\geq m$ .
ⓘ Symbols: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $x$ : real variable, $m$ : nonnegative integer and $n$ : nonnegative integer Referenced by: §14.17(i) Permalink: http://dlmf.nist.gov/14.12.E12 Encodings: TeX, pMML, png See also: Annotations for §14.12(ii), §14.12 and Ch.14

Neumann’s Integral

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Keywords:: Legendre functions, Neumann’s integral
See also:: Annotations for §14.12(ii), §14.12 and Ch.14

14.12.13		$\boldsymbol{Q}_{n}\left(x\right)=\frac{1}{2(n!)}\int_{-1}^{1}\frac{P_{n}\left(% t\right)}{x-t}\,\mathrm{d}t.$
ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $\boldsymbol{Q}_{\NVar{\nu}}\left(\NVar{z}\right)=\boldsymbol{Q}^{0}_{\nu}\left% (z\right)$ : Olver’s associated Legendre function, $x$ : real variable and $n$ : nonnegative integer A&S Ref: 8.8.3 (modified) Permalink: http://dlmf.nist.gov/14.12.E13 Encodings: TeX, pMML, png See also: Annotations for §14.12(ii), §14.12(ii), §14.12 and Ch.14

Heine’s Integral

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Keywords:: Heine’s integral, Legendre functions
See also:: Annotations for §14.12(ii), §14.12 and Ch.14

14.12.14		$\boldsymbol{Q}_{n}\left(x\right)=\frac{1}{n!}\int_{0}^{\infty}\frac{\,\mathrm{% d}t}{\left(x+(x^{2}-1)^{1/2}\cosh t\right)^{n+1}}.$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $\boldsymbol{Q}_{\NVar{\nu}}\left(\NVar{z}\right)=\boldsymbol{Q}^{0}_{\nu}\left% (z\right)$ : Olver’s associated Legendre function, $x$ : real variable and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/14.12.E14 Encodings: TeX, pMML, png See also: Annotations for §14.12(ii), §14.12(ii), §14.12 and Ch.14

For further integral representations see Erdélyi et al. (1953a, pp. 158–159) and Magnus et al. (1966, pp. 184–190), and for contour integrals and other representations see §14.25.

14.11 Derivatives with Respect to Degree or Order 14.13 Trigonometric Expansions

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