§14.12 Integral Representations

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

§14.12(i) $-1<x<1$
§14.12(ii) $1<x<\infty$

§14.12(i) $-1<x<1$

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, $dx$ : differential of $x$ , $\int$ : 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, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $x$ : 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

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, $dx$ : differential of $x$ , $\mathop{\cosh\/}\nolimits z$ : 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
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§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, $dx$ : differential of $x$ , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\int$ : integral, $\realpart{}$ : real part, $x$ : 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, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : 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

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, $dx$ : differential of $x$ , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\realpart{}$ : 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

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, $dx$ : 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

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, $dx$ : differential of $x$ , $!$ : $n!$ : factorial, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable, $m$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/14.12.E8
Encodings:: TeX, pMML, png

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, $dx$ : differential of $x$ , $!$ : $n!$ : factorial, $\mathop{\cosh\/}\nolimits z$ : 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

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, $x$ : real variable and $u$
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, $dx$ : differential of $x$ , $!$ : $n!$ : factorial, $\int$ : integral, $x$ : real variable, $m$ : nonnegative integer and $n$ : 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, $dx$ : differential of $x$ , $!$ : $n!$ : factorial, $\int$ : 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

¶ 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, $dx$ : differential of $x$ , $!$ : $n!$ : factorial, $\int$ : integral, $x$ : 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

¶ 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, $dx$ : differential of $x$ , $!$ : $n!$ : factorial, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\int$ : integral, $x$ : real variable and $n$ : 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.

§14.12 Integral Representations

Contents

§14.12(i)

¶ Mehler–Dirichlet Formula

§14.12(ii)

¶ Neumann’s Integral

¶ Heine’s Integral

§14.12(i) $-1<x<1$

§14.12(ii) $1<x<\infty$