§14.17 Integrals

14.17.1		$$\int {\left(1-x^2\right)}^{-\mu /2}{\mathsf{P}}_{\nu }^{\mu }\left(x\right)dx=-{\left(1-x^2\right)}^{-(\mu -1)/2}{\mathsf{P}}_{\nu }^{\mu -1}\left(x\right).$$
ⓘ Symbols: : Ferrers function of the first kind, $dx$: differential of $x$, $\int$: integral, $x$: 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 See also: Annotations for §14.17(i), §14.17 and Ch.14

14.17.2		$$\int {\left(1-x^2\right)}^{\mu /2}{\mathsf{P}}_{\nu }^{\mu }\left(x\right)dx=\frac{{\left(1-x^2\right)}^{(\mu +1)/2}}{(\nu -\mu )(\nu +\mu +1)}{\mathsf{P}}_{\nu }^{\mu +1}\left(x\right),$$
$\mu \ne \nu$ or $-\nu -1$.
ⓘ Symbols: : Ferrers function of the first kind, $dx$: differential of $x$, $\int$: integral, $x$: real variable, $\mu$: general order and $\nu$: general degree Permalink: http://dlmf.nist.gov/14.17.E2 Encodings: TeX, pMML, png See also: Annotations for §14.17(i), §14.17 and Ch.14

14.17.3		$$\int x{\mathsf{P}}_{\nu }^{\mu }\left(x\right){\mathsf{Q}}_{\nu }^{\mu }\left(x\right)dx=\frac{1}{2\nu (\nu +1)}(\begin{array}{l}\begin{array}{l}({\mu }^2-(\nu +1)(\nu +x^2)){\mathsf{P}}_{\nu }^{\mu }\left(x\right){\mathsf{Q}}_{\nu }^{\mu }\left(x\right)\\ +(\nu +1)(\nu -\mu +1)x({\mathsf{P}}_{\nu }^{\mu }\left(x\right){\mathsf{Q}}_{\nu +1}^{\mu }\left(x\right)+{\mathsf{P}}_{\nu +1}^{\mu }\left(x\right){\mathsf{Q}}_{\nu }^{\mu }\left(x\right))\end{array}\\ -{(\nu -\mu +1)}^2{\mathsf{P}}_{\nu +1}^{\mu }\left(x\right){\mathsf{Q}}_{\nu +1}^{\mu }\left(x\right)),\end{array}$$
$\nu \ne 0,-1$.
ⓘ Symbols: : Ferrers function of the first kind, : Ferrers function of the second kind, $dx$: differential of $x$, $\int$: integral, $x$: 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 See also: Annotations for §14.17(i), §14.17 and Ch.14

14.17.4		$$\begin{array}{l}\int \frac{x}{{\left(1-x^2\right)}^{3/2}}{\mathsf{P}}_{\nu }^{\mu }\left(x\right){\mathsf{Q}}_{\nu }^{\mu }\left(x\right)dx\\ =\begin{array}{l}\frac{1}{\left(1-4{\mu }^2\right){\left(1-x^2\right)}^{1/2}}\\ \times (\begin{array}{l}(1-2{\mu }^2+2\nu (\nu +1)){\mathsf{P}}_{\nu }^{\mu }\left(x\right){\mathsf{Q}}_{\nu }^{\mu }\left(x\right)\\ +\begin{array}{l}(2\nu +1)(\mu -\nu -1)x\\ \times ({\mathsf{P}}_{\nu }^{\mu }\left(x\right){\mathsf{Q}}_{\nu +1}^{\mu }\left(x\right)+{\mathsf{P}}_{\nu +1}^{\mu }\left(x\right){\mathsf{Q}}_{\nu }^{\mu }\left(x\right))\end{array}\\ +2{(\mu -\nu -1)}^2{\mathsf{P}}_{\nu +1}^{\mu }\left(x\right){\mathsf{Q}}_{\nu +1}^{\mu }\left(x\right)),\end{array}\end{array}\end{array}$$
$\mu \ne \pm \frac{1}{2}$.
ⓘ Symbols: : Ferrers function of the first kind, : Ferrers function of the second kind, $dx$: differential of $x$, $\int$: integral, $x$: 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 See also: Annotations for §14.17(i), §14.17 and Ch.14

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

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).

14.17.5		$${\int }_0^1{x}^{\sigma }{\left(1-x^2\right)}^{\mu /2}{\mathsf{P}}_{\nu }^{-\mu }\left(x\right)dx=\frac{\mathrm{\Gamma }\left(\frac{1}{2}\sigma +\frac{1}{2}\right)\mathrm{\Gamma }\left(\frac{1}{2}\sigma +1\right)}{2^{\mu +1}\mathrm{\Gamma }\left(\frac{1}{2}\sigma -\frac{1}{2}\nu +\frac{1}{2}\mu +1\right)\mathrm{\Gamma }\left(\frac{1}{2}\sigma +\frac{1}{2}\nu +\frac{1}{2}\mu +\frac{3}{2}\right)},$$
$\mathrm{\Re }\sigma >-1$, $\mathrm{\Re }\mu >-1$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $dx$: differential of $x$, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable, $\mu$: general order and $\nu$: general degree Permalink: http://dlmf.nist.gov/14.17.E5 Encodings: TeX, pMML, png See also: Annotations for §14.17(ii), §14.17 and Ch.14

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

14.17.6		$${\int }_{-1}^1{\mathsf{P}}_l^m\left(x\right){\mathsf{P}}_n^m\left(x\right)dx=\frac{(n+m)!}{(n-m)!\left(n+\frac{1}{2}\right)}{\delta }_{l,n},$$
ⓘ Symbols: : Ferrers function of the first kind, ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$, $!$: factorial (as in $n!$), $\int$: integral, $x$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $l$: nonnegative integer A&S Ref: 8.14.11 8.14.13 Referenced by: §14.17(iii), Erratum (V1.0.17) for Equations (10.22.37), (10.22.38), (14.17.6)–(14.17.9) Permalink: http://dlmf.nist.gov/14.17.E6 Encodings: TeX, pMML, png Clarification (effective with 1.0.17): The Kronecker delta symbol has been moved furthest to the right. See also: Annotations for §14.17(iii), §14.17 and Ch.14

14.17.7		${\displaystyle {\int }_{-1}^1}{\mathsf{P}}_l^m\left(x\right){\mathsf{P}}_n^{-m}\left(x\right)dx$	$={\displaystyle \frac{{(-1)}^m}{l+\frac{1}{2}}}{\delta }_{l,n},$
ⓘ Symbols: : Ferrers function of the first kind, ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$, $\int$: integral, $x$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $l$: nonnegative integer Referenced by: §14.17(iii) Permalink: http://dlmf.nist.gov/14.17.E7 Encodings: TeX, pMML, png Clarification (effective with 1.0.17): The Kronecker delta symbol has been moved furthest to the right and ${(-1)}^m$ has been moved to the numerator of the fraction. See also: Annotations for §14.17(iii), §14.17 and Ch.14
14.17.8		${\displaystyle {\int }_{-1}^1}{\displaystyle \frac{{\mathsf{P}}_n^l\left(x\right){\mathsf{P}}_n^m\left(x\right)}{1-x^2}}dx$	$={\displaystyle \frac{(n+m)!}{(n-m)!m}}{\delta }_{l,m},$
$m>0$,
ⓘ Symbols: : Ferrers function of the first kind, ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$, $!$: factorial (as in $n!$), $\int$: integral, $x$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $l$: 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 Clarification (effective with 1.0.17): The Kronecker delta symbol has been moved furthest to the right. See also: Annotations for §14.17(iii), §14.17 and Ch.14
14.17.9		${\displaystyle {\int }_{-1}^1}{\displaystyle \frac{{\mathsf{P}}_n^l\left(x\right){\mathsf{P}}_n^{-m}\left(x\right)}{1-x^2}}dx$	$={\displaystyle \frac{{(-1)}^l}{l}}{\delta }_{l,m},$
$l>0$.
ⓘ Symbols: : Ferrers function of the first kind, ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$, $\int$: integral, $x$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $l$: nonnegative integer Referenced by: §14.17(iii), Erratum (V1.0.17) for Equations (10.22.37), (10.22.38), (14.17.6)–(14.17.9) Permalink: http://dlmf.nist.gov/14.17.E9 Encodings: TeX, pMML, png Clarification (effective with 1.0.17): The Kronecker delta symbol has been moved furthest to the right and ${(-1)}^l$ has been moved to the numerator of the fraction. See also: Annotations for §14.17(iii), §14.17 and Ch.14

Orthogonality relations for the associated Legendre functions of imaginary order are given in Bielski (2013).

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

14.17.10		$${\int }_{-1}^1{\mathsf{P}}_{\nu }\left(x\right){\mathsf{P}}_{\lambda }\left(x\right)dx=\frac{2\left(2\sin \left(\nu \pi \right)\sin \left(\lambda \pi \right)\left(\psi \left(\nu +1\right)-\psi \left(\lambda +1\right)\right)+\pi \sin \left((\lambda -\nu )\pi \right)\right)}{{\pi }^2(\lambda -\nu )(\lambda +\nu +1)},$$
$\lambda \ne \nu$ or $-\nu -1$.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, ${\mathsf{P}}_{\nu }\left(x\right)={\mathsf{P}}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $\sin z$: sine function, $x$: real variable and $\nu$: general degree A&S Ref: 8.14.4 Permalink: http://dlmf.nist.gov/14.17.E10 Encodings: TeX, pMML, png See also: Annotations for §14.17(iv), §14.17 and Ch.14

14.17.11		$${\int }_{-1}^1{\left({\mathsf{P}}_{\nu }\left(x\right)\right)}^{2}dx=\frac{{\pi }^2-2{\sin }^2\left(\nu \pi \right){\psi }^{\prime }\left(\nu +1\right)}{{\pi }^2\left(\nu +\frac{1}{2}\right)},$$
$\nu \ne -\frac{1}{2}$.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, ${\mathsf{P}}_{\nu }\left(x\right)={\mathsf{P}}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $\sin z$: sine function, $x$: 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 See also: Annotations for §14.17(iv), §14.17 and Ch.14

14.17.12		$${\int }_{-1}^1{\mathsf{Q}}_{\nu }\left(x\right){\mathsf{Q}}_{\lambda }\left(x\right)dx=\frac{\left((\psi \left(\nu +1\right)-\psi \left(\lambda +1\right))(1+\cos \left(\nu \pi \right)\cos \left(\lambda \pi \right))+\frac{1}{2}\pi \sin \left((\lambda -\nu )\pi \right)\right)}{(\lambda -\nu )(\lambda +\nu +1)},$$
$\lambda \ne \nu$ or $-\nu -1$, $\lambda \text{ and }\nu \ne -1,-2,-3,\mathrm{\dots }$.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, ${\mathsf{Q}}_{\nu }\left(x\right)={\mathsf{Q}}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $\sin z$: sine function, $x$: real variable and $\nu$: general degree A&S Ref: 8.14.6 Permalink: http://dlmf.nist.gov/14.17.E12 Encodings: TeX, pMML, png See also: Annotations for §14.17(iv), §14.17 and Ch.14

14.17.13		$${\int }_{-1}^1{\left({\mathsf{Q}}_{\nu }\left(x\right)\right)}^{2}dx=\frac{{\pi }^2-2\left(1+{\cos }^2\left(\nu \pi \right)\right){\psi }^{\prime }\left(\nu +1\right)}{2(2\nu +1)},$$
$\nu \ne -\frac{1}{2}$ or $-1,-2,-3,\mathrm{\dots }$.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, ${\mathsf{Q}}_{\nu }\left(x\right)={\mathsf{Q}}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $x$: real variable and $\nu$: general degree A&S Ref: 8.14.7 Permalink: http://dlmf.nist.gov/14.17.E13 Encodings: TeX, pMML, png See also: Annotations for §14.17(iv), §14.17 and Ch.14

14.17.14		$${\int }_{-1}^1{\mathsf{P}}_{\nu }\left(x\right){\mathsf{Q}}_{\lambda }\left(x\right)dx=\frac{2\sin \left(\nu \pi \right)\cos \left(\lambda \pi \right)\left(\psi \left(\nu +1\right)-\psi \left(\lambda +1\right)\right)+\pi \cos \left((\lambda -\nu )\pi \right)-\pi }{\pi (\lambda -\nu )(\lambda +\nu +1)},$$
$\mathrm{\Re }\lambda >0$, $\mathrm{\Re }\nu >0$, $\lambda \ne \nu$.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, $\mathrm{\Re }$: real part, ${\mathsf{P}}_{\nu }\left(x\right)={\mathsf{P}}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, ${\mathsf{Q}}_{\nu }\left(x\right)={\mathsf{Q}}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $\sin z$: sine function, $x$: real variable and $\nu$: general degree A&S Ref: 8.14.8 Permalink: http://dlmf.nist.gov/14.17.E14 Encodings: TeX, pMML, png See also: Annotations for §14.17(iv), §14.17 and Ch.14

14.17.15		$${\int }_{-1}^1{\mathsf{P}}_{\nu }\left(x\right){\mathsf{Q}}_{\nu }\left(x\right)dx=-\frac{\sin \left(2\nu \pi \right){\psi }^{\prime }\left(\nu +1\right)}{\pi (2\nu +1)},$$
$\mathrm{\Re }\nu >0$.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, $\mathrm{\Re }$: real part, ${\mathsf{P}}_{\nu }\left(x\right)={\mathsf{P}}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, ${\mathsf{Q}}_{\nu }\left(x\right)={\mathsf{Q}}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $\sin z$: sine function, $x$: real variable and $\nu$: general degree A&S Ref: 8.14.9 Permalink: http://dlmf.nist.gov/14.17.E15 Encodings: TeX, pMML, png See also: Annotations for §14.17(iv), §14.17 and Ch.14

14.17.16		$${\int }_{-1}^1{\mathsf{P}}_l^m\left(x\right){\mathsf{Q}}_n^m\left(x\right)dx=\frac{\left(1-{(-1)}^{l+n}\right)(l+m)!}{(l-n)(l+n+1)(l-m)!},$$
$l,m,n=0,1,2,\mathrm{\dots }$, $l\ne n$.
ⓘ Symbols: : Ferrers function of the first kind, : Ferrers function of the second kind, $dx$: differential of $x$, $!$: factorial (as in $n!$), $\int$: integral, $x$: real variable, $m$: nonnegative integer and $n$: 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 See also: Annotations for §14.17(iv), §14.17 and Ch.14

14.17.17		$${\int }_0^{\pi }{\mathsf{Q}}_l\left(\cos \theta \right){\mathsf{P}}_m\left(\cos \theta \right){\mathsf{P}}_n\left(\cos \theta \right)\sin \theta d\theta =0,$$
$l,m,n=1,2,3,\mathrm{\dots }$, .
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral, ${\mathsf{P}}_{\nu }\left(x\right)={\mathsf{P}}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, ${\mathsf{Q}}_{\nu }\left(x\right)={\mathsf{Q}}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $\sin z$: sine function, $m$: nonnegative integer and $n$: nonnegative integer Referenced by: §14.17(iv) Permalink: http://dlmf.nist.gov/14.17.E17 Encodings: TeX, pMML, png See also: Annotations for §14.17(iv), §14.17 and Ch.14

(When $l+m+n$ is even the condition is not needed.) Next,

14.17.18		$${\int }_1^{\mathrm{\infty }}{P}_{\nu }\left(x\right)Q_{\lambda }\left(x\right)dx=\frac{1}{(\lambda -\nu )(\nu +\lambda +1)},$$
$\mathrm{\Re }\lambda >\mathrm{\Re }\nu >0$.
ⓘ Symbols: $dx$: differential of $x$, $\int$: integral, $\mathrm{\Re }$: real part, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $x$: real variable and $\nu$: general degree A&S Ref: 8.14.1 Permalink: http://dlmf.nist.gov/14.17.E18 Encodings: TeX, pMML, png See also: Annotations for §14.17(iv), §14.17 and Ch.14

14.17.19		$${\int }_1^{\mathrm{\infty }}{Q}_{\nu }\left(x\right)Q_{\lambda }\left(x\right)dx=\frac{\psi \left(\lambda +1\right)-\psi \left(\nu +1\right)}{(\lambda -\nu )(\lambda +\nu +1)},$$
$\mathrm{\Re }\left(\lambda +\nu \right)>-1$, $\lambda \ne \nu$, $\lambda$ and $\nu \ne -1,-2,-3,\mathrm{\dots }$.
ⓘ Symbols: $dx$: differential of $x$, $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, $\mathrm{\Re }$: real part, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $x$: real variable and $\nu$: general degree A&S Ref: 8.14.2 Permalink: http://dlmf.nist.gov/14.17.E19 Encodings: TeX, pMML, png See also: Annotations for §14.17(iv), §14.17 and Ch.14

14.17.20		$${\int }_1^{\mathrm{\infty }}{(Q_{\nu }\left(x\right))}^{2}dx=\frac{{\psi }^{\prime }\left(\nu +1\right)}{2\nu +1},$$
$\mathrm{\Re }\nu >-\frac{1}{2}$.
ⓘ Symbols: $dx$: differential of $x$, $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, $\mathrm{\Re }$: real part, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $x$: real variable and $\nu$: general degree A&S Ref: 8.14.3 Permalink: http://dlmf.nist.gov/14.17.E20 Encodings: TeX, pMML, png See also: Annotations for §14.17(iv), §14.17 and Ch.14

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

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).

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).