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

§14.12 Integral Representations

Referenced by:: §14.32
Permalink:: http://dlmf.nist.gov/14.12

§14.12(i)
§14.12(ii) $1<x<\infty$

§14.12(i)

Notes:: See Erdélyi et al. (1953a, pp. 158–159).
Keywords:: Ferrers functions, Mehler–Dirichlet formula
Permalink:: http://dlmf.nist.gov/14.12.i

¶ Mehler–Dirichlet Formula

Keywords:: Ferrers functions

14.12.1 $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits% \theta\right)=\frac{2^{{1/2}}(\mathop{\sin\/}\nolimits\theta)^{{\mu}}}{\pi^{{1% /2}}\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-\mu\right)}\int_{{0}}^{{% \theta}}\frac{\mathop{\cos\/}\nolimits\!\left(\left(\nu+\frac{1}{2}\right)t% \right)}{(\mathop{\cos\/}\nolimits t-\mathop{\cos\/}\nolimits\theta)^{{\mu+(1/% 2)}}}dt,$ $0<\theta<\pi$ , $\realpart{\mu}<\tfrac{1}{2}$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : 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

14.12.2 $\mathop{\mathsf{P}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{\left(1-% x^{2}\right)^{{-\mu/2}}}{\mathop{\Gamma\/}\nolimits\!\left(\mu\right)}\int_{x}% ^{1}\mathop{\mathsf{P}_{{\nu}}\/}\nolimits\!\left(t\right)(t-x)^{{\mu-1}}dt,$ $\realpart{\mu}>0$ ;

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, : differential of , $\int$ : integral, $\realpart{}$ : real part, : 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

compare (14.6.6).

14.12.3 $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits% \theta\right)=\frac{\pi^{{1/2}}\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1% \right)(\mathop{\sin\/}\nolimits\theta)^{{\mu}}}{2^{{\mu+1}}\mathop{\Gamma\/}% \nolimits\!\left(\mu+\frac{1}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(\nu-% \mu+1\right)}\*\left(\int_{{0}}^{{\infty}}\frac{(\mathop{\sinh\/}\nolimits t)^% {{2\mu}}}{(\mathop{\cos\/}\nolimits\theta+i\mathop{\sin\/}\nolimits\theta% \mathop{\cosh\/}\nolimits t)^{{\nu+\mu+1}}}dt+\int_{{0}}^{{\infty}}\frac{(% \mathop{\sinh\/}\nolimits t)^{{2\mu}}}{(\mathop{\cos\/}\nolimits\theta-i% \mathop{\sin\/}\nolimits\theta\mathop{\cosh\/}\nolimits t)^{{\nu+\mu+1}}}dt% \right),$ $0<\theta<\pi$ , $\realpart{\mu}>-\tfrac{1}{2}$ , $\realpart{(\nu\pm\mu)}>-1$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : sine function, $\mu$ : general order and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.12.E3
Encodings:: TeX, pMML, png

§14.12(ii) $1<x<\infty$

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

14.12.4 $\mathop{P^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{2^{{1/2}}\mathop{% \Gamma\/}\nolimits\!\left(\mu+\frac{1}{2}\right)\left(x^{2}-1\right)^{{\mu/2}}% }{\pi^{{1/2}}\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)\mathop{\Gamma% \/}\nolimits\!\left(\mu-\nu\right)}\*\int_{{0}}^{{\infty}}\frac{\mathop{\cosh% \/}\nolimits\!\left(\left(\nu+\frac{1}{2}\right)t\right)}{(x+\mathop{\cosh\/}% \nolimits t)^{{\mu+(1/2)}}}dt,$ $\nu+\mu\neq-1,-2,-3,\dots$ , $\realpart{(\mu-\nu)}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, : differential of , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\int$ : integral, $\realpart{}$ : real part, : real variable, $\mu$ : general order and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.12.E4
Encodings:: TeX, pMML, png

14.12.5 $\mathop{P^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{\left(x^{2}-1% \right)^{{-\mu/2}}}{\mathop{\Gamma\/}\nolimits\!\left(\mu\right)}\int_{1}^{x}% \mathop{P_{{\nu}}\/}\nolimits\!\left(t\right)(x-t)^{{\mu-1}}dt,$ $\realpart{\mu}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, : differential of , $\int$ : integral, $\realpart{}$ : real part, : 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

14.12.6 $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{\pi^{% {1/2}}\left(x^{2}-1\right)^{{\mu/2}}}{2^{{\mu}}\mathop{\Gamma\/}\nolimits\!% \left(\mu+\frac{1}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(\nu-\mu+1\right)% }\*\int_{{0}}^{{\infty}}\frac{(\mathop{\sinh\/}\nolimits t)^{{2\mu}}}{\left(x+% (x^{2}-1)^{{1/2}}\mathop{\cosh\/}\nolimits t\right)^{{\nu+\mu+1}}}dt,$ $\realpart{(\nu+1)}>\realpart{\mu}>-\tfrac{1}{2}$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, : differential of , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\realpart{}$ : real part, : 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

14.12.7 $\mathop{P^{{m}}_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{\left(\nu+1\right)_{% {m}}}{\pi}\*\int_{{0}}^{{\pi}}\left(x+\left(x^{2}-1\right)^{{1/2}}\mathop{\cos% \/}\nolimits\phi\right)^{\nu}\mathop{\cos\/}\nolimits\!\left(m\phi\right)d\phi,$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\int$ : integral, : real variable, : nonnegative integer and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.12.E7
Encodings:: TeX, pMML, png

14.12.8 $\mathop{P^{{m}}_{{n}}\/}\nolimits\!\left(x\right)=\frac{2^{m}m!(n+m)!\left(x^{% 2}-1\right)^{{m/2}}}{(2m)!(n-m)!\pi}\int_{{0}}^{{\pi}}\left(x+\left(x^{2}-1% \right)^{{1/2}}\mathop{\cos\/}\nolimits\phi\right)^{{n-m}}(\mathop{\sin\/}% \nolimits\phi)^{{2m}}d\phi,$ $n\geq m$ .

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , : : factorial, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : real variable, : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/14.12.E8
Encodings:: TeX, pMML, png

14.12.9 $\mathop{\boldsymbol{Q}^{{m}}_{{n}}\/}\nolimits\!\left(x\right)=\frac{1}{n!}% \int_{0}^{u}\left(x-\left(x^{2}-1\right)^{{1/2}}\mathop{\cosh\/}\nolimits t% \right)^{n}\mathop{\cosh\/}\nolimits\!\left(mt\right)dt,$

Symbols:: $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, : differential of , : : factorial, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\int$ : integral, : real variable, : nonnegative integer, : nonnegative integer and
Permalink:: http://dlmf.nist.gov/14.12.E9
Encodings:: TeX, pMML, png

where

14.12.10 $u=\frac{1}{2}\mathop{\ln\/}\nolimits\!\left(\frac{x+1}{x-1}\right).$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : real variable and
Referenced by:: §14.15(i)
Permalink:: http://dlmf.nist.gov/14.12.E10
Encodings:: TeX, pMML, png

14.12.11 $\mathop{\boldsymbol{Q}^{{m}}_{{n}}\/}\nolimits\!\left(x\right)=\frac{\left(x^{% 2}-1\right)^{{m/2}}}{2^{{n+1}}n!}\int_{{-1}}^{{1}}\frac{\left(1-t^{2}\right)^{% n}}{(x-t)^{{n+m+1}}}dt,$

Symbols:: $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, : differential of , : : factorial, $\int$ : integral, : real variable, : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/14.12.E11
Encodings:: TeX, pMML, png

14.12.12 $\mathop{\boldsymbol{Q}^{{m}}_{{n}}\/}\nolimits\!\left(x\right)=\frac{1}{(n-m)!% }\mathop{P^{{m}}_{{n}}\/}\nolimits\!\left(x\right)\int_{{x}}^{{\infty}}\frac{% dt}{\left(t^{2}-1\right)\left(\displaystyle\mathop{P^{{m}}_{{n}}\/}\nolimits\!% \left(t\right)\right)^{2}},$ $n\geq m$ .

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, : differential of , : : factorial, $\int$ : integral, : real variable, : nonnegative integer and : nonnegative integer
Referenced by:: §14.17(i)
Permalink:: http://dlmf.nist.gov/14.12.E12
Encodings:: TeX, pMML, png

¶ Neumann’s Integral

Keywords:: Legendre functions, Neumann’s integral

14.12.13 $\mathop{\boldsymbol{Q}_{{n}}\/}\nolimits\!\left(x\right)=\frac{1}{2(n!)}\int_{% {-1}}^{{1}}\frac{\mathop{P_{{n}}\/}\nolimits\!\left(t\right)}{x-t}dt.$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, : differential of , : : factorial, $\int$ : integral, : real variable and : nonnegative integer
A&S Ref:: 8.8.3 (modified)
Permalink:: http://dlmf.nist.gov/14.12.E13
Encodings:: TeX, pMML, png

¶ Heine’s Integral

Keywords:: Heine’s integral, Legendre functions

14.12.14 $\mathop{\boldsymbol{Q}_{{n}}\/}\nolimits\!\left(x\right)=\frac{1}{n!}\int_{{0}% }^{{\infty}}\frac{dt}{\left(x+(x^{2}-1)^{{1/2}}\mathop{\cosh\/}\nolimits t% \right)^{{n+1}}}.$

Symbols:: $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, : differential of , : : factorial, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\int$ : integral, : real variable and : nonnegative integer
Permalink:: http://dlmf.nist.gov/14.12.E14
Encodings:: TeX, pMML, png

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.

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 14.11 Derivatives with Respect to Degree or Order 14.13 Trigonometric Expansions