14 Legendre and Related Functions Real Arguments 14.16 Zeros 14.18 Sums

§14.17 Integrals

Permalink:: http://dlmf.nist.gov/14.17

§14.17(i) Indefinite Integrals
§14.17(ii) Barnes’ Integral
§14.17(iii) Orthogonality Properties
§14.17(iv) Definite Integrals of Products
§14.17(v) Laplace Transforms
§14.17(vi) Mellin Transforms

§14.17(i) Indefinite Integrals

Notes:: These results may be verified by differentiation and use of the recurrence relations (§14.10).
Keywords:: Ferrers functions, integrals
Permalink:: http://dlmf.nist.gov/14.17.i

14.17.1 ${\int\left(1-x^{2}\right)^{{-\mu/2}}\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}% \nolimits\!\left(x\right)dx}={-\left(1-x^{2}\right)^{{-(\mu-1)/2}}\mathop{% \mathsf{P}^{{\mu-1}}_{{\nu}}\/}\nolimits\!\left(x\right)}.$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, : differential of , $\int$ : integral, : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.17(i)
Permalink:: http://dlmf.nist.gov/14.17.E1
Encodings:: TeX, pMML, png

14.17.2 $\int\left(1-x^{2}\right)^{{\mu/2}}\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}% \nolimits\!\left(x\right)dx=\frac{\left(1-x^{2}\right)^{{(\mu+1)/2}}}{(\nu-\mu% )(\nu+\mu+1)}\mathop{\mathsf{P}^{{\mu+1}}_{{\nu}}\/}\nolimits\!\left(x\right),$ $\mu\neq\nu$ or $-\nu-1$ .

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, : differential of , $\int$ : integral, : real variable, $\mu$ : general order and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.17.E2
Encodings:: TeX, pMML, png

14.17.3 $\int x\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)\mathop{% \mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)dx=\frac{1}{2\nu(\nu+1)}% \left((\mu^{2}-(\nu+1)(\nu+x^{2}))\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}% \nolimits\!\left(x\right)\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left% (x\right)+(\nu+1)(\nu-\mu+1)x(\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!% \left(x\right)\mathop{\mathsf{Q}^{{\mu}}_{{\nu+1}}\/}\nolimits\!\left(x\right)% +\mathop{\mathsf{P}^{{\mu}}_{{\nu+1}}\/}\nolimits\!\left(x\right)\mathop{% \mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right))-(\nu-\mu+1)^{2}\mathop% {\mathsf{P}^{{\mu}}_{{\nu+1}}\/}\nolimits\!\left(x\right)\mathop{\mathsf{Q}^{{% \mu}}_{{\nu+1}}\/}\nolimits\!\left(x\right)\right),$ $\nu\neq 0,-1$ .

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, : differential of , $\int$ : integral, : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.17(i)
Permalink:: http://dlmf.nist.gov/14.17.E3
Encodings:: TeX, pMML, png

14.17.4 $\int\frac{x}{\left(1-x^{2}\right)^{{3/2}}}\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/% }\nolimits\!\left(x\right)\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!% \left(x\right)dx=\frac{1}{\left(1-4\mu^{2}\right)\left(1-x^{2}\right)^{{1/2}}}% \left((1-2\mu^{2}+2\nu(\nu+1))\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!% \left(x\right)\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)+(% 2\nu+1)(\mu-\nu-1)x(\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x% \right)\mathop{\mathsf{Q}^{{\mu}}_{{\nu+1}}\/}\nolimits\!\left(x\right)+% \mathop{\mathsf{P}^{{\mu}}_{{\nu+1}}\/}\nolimits\!\left(x\right)\mathop{% \mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right))+2(\mu-\nu-1)^{2}% \mathop{\mathsf{P}^{{\mu}}_{{\nu+1}}\/}\nolimits\!\left(x\right)\mathop{% \mathsf{Q}^{{\mu}}_{{\nu+1}}\/}\nolimits\!\left(x\right)\right),$ $\mu\neq\pm\frac{1}{2}$ .

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, : differential of , $\int$ : integral, : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.17(i)
Permalink:: http://dlmf.nist.gov/14.17.E4
Encodings:: TeX, pMML, png

In (14.17.1)–(14.17.4), $\mathop{\mathsf{P}\/}\nolimits$ may be replaced by $\mathop{\mathsf{Q}\/}\nolimits$ , and in (14.17.3) and (14.17.4), $\mathop{\mathsf{Q}\/}\nolimits$ may be replaced by $\mathop{\mathsf{P}\/}\nolimits$ .

For further results, see Maximon (1955) and Prudnikov et al. (1990, pp. 37–39). See also (14.12.2), (14.12.5), and (14.12.12).

§14.17(ii) Barnes’ Integral

Notes:: See Erdélyi et al. (1953a, p. 172).
Keywords:: Barnes’ integral, Ferrers functions, associated Legendre functions, integrals
Permalink:: http://dlmf.nist.gov/14.17.ii

14.17.5 $\int_{0}^{1}x^{{\sigma}}\left(1-x^{2}\right)^{{\mu/2}}\mathop{\mathsf{P}^{{-% \mu}}_{{\nu}}\/}\nolimits\!\left(x\right)dx=\frac{\mathop{\Gamma\/}\nolimits\!% \left(\frac{1}{2}\sigma+\frac{1}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(% \frac{1}{2}\sigma+1\right)}{2^{{\mu+1}}\mathop{\Gamma\/}\nolimits\!\left(\frac% {1}{2}\sigma-\frac{1}{2}\nu+\frac{1}{2}\mu+1\right)\mathop{\Gamma\/}\nolimits% \!\left(\frac{1}{2}\sigma+\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{2}\right)},$ $\realpart{\sigma}>-1$ , $\realpart{\mu}>-1$ .

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
Permalink:: http://dlmf.nist.gov/14.17.E5
Encodings:: TeX, pMML, png

§14.17(iii) Orthogonality Properties

Notes:: For (14.17.6) and (14.17.8) see Olver (1997b, pp. 188–189). (14.17.7) and (14.17.9) may be derived from (14.9.3), (14.17.6), and (14.17.8).
Keywords:: Ferrers functions, associated Legendre functions, integrals
Permalink:: http://dlmf.nist.gov/14.17.iii

For $l,m,n=0,1,2,\dots$ ,

14.17.6 $\int_{{-1}}^{{1}}\mathop{\mathsf{P}^{{m}}_{{l}}\/}\nolimits\!\left(x\right)% \mathop{\mathsf{P}^{{m}}_{{n}}\/}\nolimits\!\left(x\right)dx=\delta_{{l,n}}% \frac{(n+m)!}{(n-m)!\left(n+\frac{1}{2}\right)},$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\delta_{{j,k}}$ : Kronecker delta, : differential of , : : factorial, $\int$ : integral, : real variable, : nonnegative integer, : nonnegative integer and : nonnegative integer
A&S Ref:: 8.14.11 8.14.13
Referenced by:: §14.17(iii)
Permalink:: http://dlmf.nist.gov/14.17.E6
Encodings:: TeX, pMML, png

14.17.7 $\int_{{-1}}^{{1}}\mathop{\mathsf{P}^{{m}}_{{l}}\/}\nolimits\!\left(x\right)% \mathop{\mathsf{P}^{{-m}}_{{n}}\/}\nolimits\!\left(x\right)dx=(-1)^{m}\delta_{% {l,n}}\frac{1}{l+\frac{1}{2}},$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\delta_{{j,k}}$ : Kronecker delta, : differential of , $\int$ : integral, : real variable, : nonnegative integer, : nonnegative integer and : nonnegative integer
Referenced by:: §14.17(iii)
Permalink:: http://dlmf.nist.gov/14.17.E7
Encodings:: TeX, pMML, png

14.17.8 $\int_{{-1}}^{{1}}\frac{\mathop{\mathsf{P}^{{l}}_{{n}}\/}\nolimits\!\left(x% \right)\mathop{\mathsf{P}^{{m}}_{{n}}\/}\nolimits\!\left(x\right)}{1-x^{2}}dx=% \delta_{{l,m}}\frac{(n+m)!}{(n-m)!m},$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\delta_{{j,k}}$ : Kronecker delta, : differential of , : : factorial, $\int$ : integral, : real variable, : nonnegative integer, : nonnegative integer and : nonnegative integer
A&S Ref:: 8.14.12 8.14.14
Referenced by:: §14.17(iii)
Permalink:: http://dlmf.nist.gov/14.17.E8
Encodings:: TeX, pMML, png

14.17.9 $\int_{{-1}}^{{1}}\frac{\mathop{\mathsf{P}^{{l}}_{{n}}\/}\nolimits\!\left(x% \right)\mathop{\mathsf{P}^{{-m}}_{{n}}\/}\nolimits\!\left(x\right)}{1-x^{2}}dx% =(-1)^{l}\delta_{{l,m}}\frac{1}{l},$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\delta_{{j,k}}$ : Kronecker delta, : differential of , $\int$ : integral, : real variable, : nonnegative integer, : nonnegative integer and : nonnegative integer
Referenced by:: §14.17(iii)
Permalink:: http://dlmf.nist.gov/14.17.E9
Encodings:: TeX, pMML, png

§14.17(iv) Definite Integrals of Products

Notes:: See Erdélyi et al. (1953a, pp. 170–171). The version of (14.17.16) given in Erdélyi et al. (1953a, p. 171(18)) is incorrect. For (14.17.17) see Din (1981).
Keywords:: Ferrers functions, associated Legendre functions, integrals
Permalink:: http://dlmf.nist.gov/14.17.iv

With $\mathop{\psi\/}\nolimits\!\left(x\right)={\mathop{\Gamma\/}\nolimits^{{\prime}% }}\!\left(x\right)/\mathop{\Gamma\/}\nolimits\!\left(x\right)$ (§5.2(i)),

14.17.10 $\int_{{-1}}^{{1}}\mathop{\mathsf{P}_{{\nu}}\/}\nolimits\!\left(x\right)\mathop% {\mathsf{P}_{{\lambda}}\/}\nolimits\!\left(x\right)dx=\frac{2\left(2\mathop{% \sin\/}\nolimits\!\left(\nu\pi\right)\mathop{\sin\/}\nolimits\!\left(\lambda% \pi\right)\left(\mathop{\psi\/}\nolimits\!\left(\nu+1\right)-\mathop{\psi\/}% \nolimits\!\left(\lambda+1\right)\right)+\pi\mathop{\sin\/}\nolimits\!\left((% \lambda-\nu)\pi\right)\right)}{\pi^{2}(\lambda-\nu)(\lambda+\nu+1)},$ $\lambda\neq\nu$ or $-\nu-1$ .

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, : differential of , $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : real variable and $\nu$ : general degree
A&S Ref:: 8.14.4
Permalink:: http://dlmf.nist.gov/14.17.E10
Encodings:: TeX, pMML, png

14.17.11 $\int_{{-1}}^{{1}}\left(\mathop{\mathsf{P}_{{\nu}}\/}\nolimits\!\left(x\right)% \right)^{2}dx=\frac{\pi^{2}-2{\mathop{\sin\/}\nolimits^{{2}}}\!\left(\nu\pi% \right){\mathop{\psi\/}\nolimits^{{\prime}}}\!\left(\nu+1\right)}{\pi^{2}\left% (\nu+\frac{1}{2}\right)},$ $\nu\neq-\frac{1}{2}$ .

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, : differential of , $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : real variable and $\nu$ : general degree
A&S Ref:: 8.14.5 (corrected)
Permalink:: http://dlmf.nist.gov/14.17.E11
Encodings:: TeX, pMML, png

14.17.12 $\int_{{-1}}^{{1}}\mathop{\mathsf{Q}_{{\nu}}\/}\nolimits\!\left(x\right)\mathop% {\mathsf{Q}_{{\lambda}}\/}\nolimits\!\left(x\right)dx=\frac{\left((\mathop{% \psi\/}\nolimits\!\left(\nu+1\right)-\mathop{\psi\/}\nolimits\!\left(\lambda+1% \right))(1+\mathop{\cos\/}\nolimits\!\left(\nu\pi\right)\mathop{\cos\/}% \nolimits\!\left(\lambda\pi\right))+\frac{1}{2}\pi\mathop{\sin\/}\nolimits\!% \left((\lambda-\nu)\pi\right)\right)}{(\lambda-\nu)(\lambda+\nu+1)},$ $\lambda\neq\nu$ or $-\nu-1$ , $\lambda\text{ and }\nu\neq-1,-2,-3,\dots$ .

Symbols:: $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : real variable and $\nu$ : general degree
A&S Ref:: 8.14.6
Permalink:: http://dlmf.nist.gov/14.17.E12
Encodings:: TeX, pMML, png

14.17.13 $\int_{{-1}}^{{1}}\left(\mathop{\mathsf{Q}_{{\nu}}\/}\nolimits\!\left(x\right)% \right)^{2}dx=\frac{\pi^{2}-2\left(1+{\mathop{\cos\/}\nolimits^{{2}}}\!\left(% \nu\pi\right)\right){\mathop{\psi\/}\nolimits^{{\prime}}}\!\left(\nu+1\right)}% {2(2\nu+1)},$ $\nu\neq-\frac{1}{2}$ or $-1,-2,-3,\dots$ .

Symbols:: $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\int$ : integral, : real variable and $\nu$ : general degree
A&S Ref:: 8.14.7
Permalink:: http://dlmf.nist.gov/14.17.E13
Encodings:: TeX, pMML, png

14.17.14 $\int_{{-1}}^{{1}}\mathop{\mathsf{P}_{{\nu}}\/}\nolimits\!\left(x\right)\mathop% {\mathsf{Q}_{{\lambda}}\/}\nolimits\!\left(x\right)dx=\frac{2\mathop{\sin\/}% \nolimits\!\left(\nu\pi\right)\mathop{\cos\/}\nolimits\!\left(\lambda\pi\right% )\left(\mathop{\psi\/}\nolimits\!\left(\nu+1\right)-\mathop{\psi\/}\nolimits\!% \left(\lambda+1\right)\right)+\pi\mathop{\cos\/}\nolimits\!\left((\lambda-\nu)% \pi\right)-\pi}{\pi(\lambda-\nu)(\lambda+\nu+1)},$ $\realpart{\lambda}>0$ , $\realpart{\nu}>0$ , $\lambda\neq\nu$ .

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : sine function, : real variable and $\nu$ : general degree
A&S Ref:: 8.14.8
Permalink:: http://dlmf.nist.gov/14.17.E14
Encodings:: TeX, pMML, png

14.17.15 $\int_{{-1}}^{{1}}\mathop{\mathsf{P}_{{\nu}}\/}\nolimits\!\left(x\right)\mathop% {\mathsf{Q}_{{\nu}}\/}\nolimits\!\left(x\right)dx=-\frac{\mathop{\sin\/}% \nolimits\!\left(2\nu\pi\right){\mathop{\psi\/}\nolimits^{{\prime}}}\!\left(% \nu+1\right)}{\pi(2\nu+1)},$ $\realpart{\nu}>0$ .

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, : differential of , $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : sine function, : real variable and $\nu$ : general degree
A&S Ref:: 8.14.9
Permalink:: http://dlmf.nist.gov/14.17.E15
Encodings:: TeX, pMML, png

14.17.16 $\int_{{-1}}^{{1}}\mathop{\mathsf{P}^{{m}}_{{l}}\/}\nolimits\!\left(x\right)% \mathop{\mathsf{Q}^{{m}}_{{n}}\/}\nolimits\!\left(x\right)dx=\frac{\left(1-(-1% )^{{l+n}}\right)(l+m)!}{(l-n)(l+n+1)(l-m)!},$ $l,m,n=0,1,2,\dots$ , $l\neq n$ .

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, : differential of , : : factorial, $\int$ : integral, : real variable, : nonnegative integer and : nonnegative integer
A&S Ref:: 8.14.10 (corrected)
Referenced by:: §14.17(iv)
Permalink:: http://dlmf.nist.gov/14.17.E16
Encodings:: TeX, pMML, png

14.17.17 $\int_{{0}}^{{\pi}}\mathop{\mathsf{Q}_{{l}}\/}\nolimits\!\left(\mathop{\cos\/}% \nolimits\theta\right)\mathop{\mathsf{P}_{{m}}\/}\nolimits\!\left(\mathop{\cos% \/}\nolimits\theta\right)\mathop{\mathsf{P}_{{n}}\/}\nolimits\!\left(\mathop{% \cos\/}\nolimits\theta\right)\mathop{\sin\/}\nolimits\theta d\theta=0,$ $l,m,n=1,2,3,\dots$ ,

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : nonnegative integer and : nonnegative integer
Referenced by:: §14.17(iv)
Permalink:: http://dlmf.nist.gov/14.17.E17
Encodings:: TeX, pMML, png

(When is even the condition $\left|m-n\right|<l<m+n$ is not needed.) Next,

14.17.18 $\int_{{1}}^{{\infty}}\mathop{P_{{\nu}}\/}\nolimits\!\left(x\right)\mathop{Q_{{% \lambda}}\/}\nolimits\!\left(x\right)dx=\frac{1}{(\lambda-\nu)(\nu+\lambda+1)},$ $\realpart{\lambda}>\realpart{\nu}>0$ .

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the second kind, : differential of , $\int$ : integral, $\realpart{}$ : real part, : real variable and $\nu$ : general degree
A&S Ref:: 8.14.1
Permalink:: http://dlmf.nist.gov/14.17.E18
Encodings:: TeX, pMML, png

14.17.19 $\int_{{1}}^{{\infty}}\mathop{Q_{{\nu}}\/}\nolimits\!\left(x\right)\mathop{Q_{{% \lambda}}\/}\nolimits\!\left(x\right)dx=\frac{\mathop{\psi\/}\nolimits\!\left(% \lambda+1\right)-\mathop{\psi\/}\nolimits\!\left(\nu+1\right)}{(\lambda-\nu)(% \lambda+\nu+1)},$ $\realpart{(\lambda+\nu)}>-1$ , $\lambda\neq\nu$ , $\lambda$ and $\nu\neq-1,-2,-3,\dots$ .

Symbols:: $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the second kind, : differential of , $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\int$ : integral, $\realpart{}$ : real part, : real variable and $\nu$ : general degree
A&S Ref:: 8.14.2
Permalink:: http://dlmf.nist.gov/14.17.E19
Encodings:: TeX, pMML, png

14.17.20 $\int_{{1}}^{{\infty}}(\mathop{Q_{{\nu}}\/}\nolimits\!\left(x\right))^{2}dx=% \frac{{\mathop{\psi\/}\nolimits^{{\prime}}}\!\left(\nu+1\right)}{2\nu+1},$ $\realpart{\nu}>-\tfrac{1}{2}$ .

Symbols:: $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the second kind, : differential of , $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\int$ : integral, $\realpart{}$ : real part, : real variable and $\nu$ : general degree
A&S Ref:: 8.14.3
Permalink:: http://dlmf.nist.gov/14.17.E20
Encodings:: TeX, pMML, png

For further results, see Prudnikov et al. (1990, pp. 194–240); also (34.3.21).

§14.17(v) Laplace Transforms

Keywords:: Ferrers functions, associated Legendre functions, integrals
Permalink:: http://dlmf.nist.gov/14.17.v

For Laplace transforms and inverse Laplace transforms involving associated Legendre functions, see Erdélyi et al. (1954a, pp. 179–181, 270–272), Oberhettinger and Badii (1973, pp. 113–118, 317–324), Prudnikov et al. (1992a, §§3.22, 3.32, and 3.33), and Prudnikov et al. (1992b, §§3.20, 3.30, and 3.31).

§14.17(vi) Mellin Transforms

Keywords:: Ferrers functions, associated Legendre functions, integrals
Permalink:: http://dlmf.nist.gov/14.17.vi

For Mellin transforms involving associated Legendre functions see Oberhettinger (1974, pp. 69–82) and Marichev (1983, pp. 247–283), and for inverse transforms see Oberhettinger (1974, pp. 205–215).