# §14.17 Integrals

## §14.17(i) Indefinite Integrals

 14.17.1 ${\int\left(1-x^{2}\right)^{-\mu/2}\mathop{\mathsf{P}^{\mu}_{\nu}\/}\nolimits\!% \left(x\right)\mathrm{d}x}={-\left(1-x^{2}\right)^{-(\mu-1)/2}\mathop{\mathsf{% P}^{\mu-1}_{\nu}\/}\nolimits\!\left(x\right)}.$
 14.17.2 $\int\left(1-x^{2}\right)^{\mu/2}\mathop{\mathsf{P}^{\mu}_{\nu}\/}\nolimits\!% \left(x\right)\mathrm{d}x=\frac{\left(1-x^{2}\right)^{(\mu+1)/2}}{(\nu-\mu)(% \nu+\mu+1)}\mathop{\mathsf{P}^{\mu+1}_{\nu}\/}\nolimits\!\left(x\right),$ $\mu\neq\nu$ or $-\nu-1$.
 14.17.3 $\int x\mathop{\mathsf{P}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)\mathop{% \mathsf{Q}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)\mathrm{d}x=\frac{1}{2\nu(% \nu+1)}\left((\mu^{2}-(\nu+1)(\nu+x^{2}))\mathop{\mathsf{P}^{\mu}_{\nu}\/}% \nolimits\!\left(x\right)\mathop{\mathsf{Q}^{\mu}_{\nu}\/}\nolimits\!\left(x% \right)+(\nu+1)(\nu-\mu+1)x(\mathop{\mathsf{P}^{\mu}_{\nu}\/}\nolimits\!\left(% x\right)\mathop{\mathsf{Q}^{\mu}_{\nu+1}\/}\nolimits\!\left(x\right)+\mathop{% \mathsf{P}^{\mu}_{\nu+1}\/}\nolimits\!\left(x\right)\mathop{\mathsf{Q}^{\mu}_{% \nu}\/}\nolimits\!\left(x\right))-(\nu-\mu+1)^{2}\mathop{\mathsf{P}^{\mu}_{\nu% +1}\/}\nolimits\!\left(x\right)\mathop{\mathsf{Q}^{\mu}_{\nu+1}\/}\nolimits\!% \left(x\right)\right),$ $\nu\neq 0,-1$.
 14.17.4 $\int\frac{x}{\left(1-x^{2}\right)^{3/2}}\mathop{\mathsf{P}^{\mu}_{\nu}\/}% \nolimits\!\left(x\right)\mathop{\mathsf{Q}^{\mu}_{\nu}\/}\nolimits\!\left(x% \right)\mathrm{d}x=\frac{1}{\left(1-4\mu^{2}\right)\left(1-x^{2}\right)^{1/2}}% \left((1-2\mu^{2}+2\nu(\nu+1))\mathop{\mathsf{P}^{\mu}_{\nu}\/}\nolimits\!% \left(x\right)\mathop{\mathsf{Q}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)+(2\nu% +1)(\mu-\nu-1)x(\mathop{\mathsf{P}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)% \mathop{\mathsf{Q}^{\mu}_{\nu+1}\/}\nolimits\!\left(x\right)+\mathop{\mathsf{P% }^{\mu}_{\nu+1}\/}\nolimits\!\left(x\right)\mathop{\mathsf{Q}^{\mu}_{\nu}\/}% \nolimits\!\left(x\right))+2(\mu-\nu-1)^{2}\mathop{\mathsf{P}^{\mu}_{\nu+1}\/}% \nolimits\!\left(x\right)\mathop{\mathsf{Q}^{\mu}_{\nu+1}\/}\nolimits\!\left(x% \right)\right),$ $\mu\neq\pm\frac{1}{2}$.

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

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

## §14.17(iii) Orthogonality Properties

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

 14.17.6 $\int_{-1}^{1}\mathop{\mathsf{P}^{m}_{l}\/}\nolimits\!\left(x\right)\mathop{% \mathsf{P}^{m}_{n}\/}\nolimits\!\left(x\right)\mathrm{d}x=\delta_{l,n}\frac{(n% +m)!}{(n-m)!\left(n+\frac{1}{2}\right)},$
 14.17.7 $\displaystyle\int_{-1}^{1}\mathop{\mathsf{P}^{m}_{l}\/}\nolimits\!\left(x% \right)\mathop{\mathsf{P}^{-m}_{n}\/}\nolimits\!\left(x\right)\mathrm{d}x$ $\displaystyle=(-1)^{m}\delta_{l,n}\frac{1}{l+\frac{1}{2}},$ 14.17.8 $\displaystyle\int_{-1}^{1}\frac{\mathop{\mathsf{P}^{l}_{n}\/}\nolimits\!\left(% x\right)\mathop{\mathsf{P}^{m}_{n}\/}\nolimits\!\left(x\right)}{1-x^{2}}% \mathrm{d}x$ $\displaystyle=\delta_{l,m}\frac{(n+m)!}{(n-m)!m},$ $m>0$, 14.17.9 $\displaystyle\int_{-1}^{1}\frac{\mathop{\mathsf{P}^{l}_{n}\/}\nolimits\!\left(% x\right)\mathop{\mathsf{P}^{-m}_{n}\/}\nolimits\!\left(x\right)}{1-x^{2}}% \mathrm{d}x$ $\displaystyle=(-1)^{l}\delta_{l,m}\frac{1}{l},$ $l>0$.

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

## §14.17(iv) Definite Integrals of Products

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

 14.17.10 $\int_{-1}^{1}\mathop{\mathsf{P}_{\nu}\/}\nolimits\!\left(x\right)\mathop{% \mathsf{P}_{\lambda}\/}\nolimits\!\left(x\right)\mathrm{d}x=\frac{2\left(2% \mathop{\sin\/}\nolimits\!\left(\nu\pi\right)\mathop{\sin\/}\nolimits\!\left(% \lambda\pi\right)\left(\mathop{\psi\/}\nolimits\!\left(\nu+1\right)-\mathop{% \psi\/}\nolimits\!\left(\lambda+1\right)\right)+\pi\mathop{\sin\/}\nolimits\!% \left((\lambda-\nu)\pi\right)\right)}{\pi^{2}(\lambda-\nu)(\lambda+\nu+1)},$ $\lambda\neq\nu$ or $-\nu-1$.
 14.17.11 $\int_{-1}^{1}\left(\mathop{\mathsf{P}_{\nu}\/}\nolimits\!\left(x\right)\right)% ^{2}\mathrm{d}x=\frac{\pi^{2}-2{\mathop{\sin\/}\nolimits^{2}}\!\left(\nu\pi% \right)\mathop{\psi\/}\nolimits'\!\left(\nu+1\right)}{\pi^{2}\left(\nu+\frac{1% }{2}\right)},$ $\nu\neq-\frac{1}{2}$.
 14.17.12 $\int_{-1}^{1}\mathop{\mathsf{Q}_{\nu}\/}\nolimits\!\left(x\right)\mathop{% \mathsf{Q}_{\lambda}\/}\nolimits\!\left(x\right)\mathrm{d}x=\frac{\left((% \mathop{\psi\/}\nolimits\!\left(\nu+1\right)-\mathop{\psi\/}\nolimits\!\left(% \lambda+1\right))(1+\mathop{\cos\/}\nolimits\!\left(\nu\pi\right)\mathop{\cos% \/}\nolimits\!\left(\lambda\pi\right))+\frac{1}{2}\pi\mathop{\sin\/}\nolimits% \!\left((\lambda-\nu)\pi\right)\right)}{(\lambda-\nu)(\lambda+\nu+1)},$ $\lambda\neq\nu$ or $-\nu-1$, $\lambda\text{ and }\nu\neq-1,-2,-3,\dots$.
 14.17.13 $\int_{-1}^{1}\left(\mathop{\mathsf{Q}_{\nu}\/}\nolimits\!\left(x\right)\right)% ^{2}\mathrm{d}x=\frac{\pi^{2}-2\left(1+{\mathop{\cos\/}\nolimits^{2}}\!\left(% \nu\pi\right)\right)\mathop{\psi\/}\nolimits'\!\left(\nu+1\right)}{2(2\nu+1)},$ $\nu\neq-\frac{1}{2}$ or $-1,-2,-3,\dots$.
 14.17.14 $\int_{-1}^{1}\mathop{\mathsf{P}_{\nu}\/}\nolimits\!\left(x\right)\mathop{% \mathsf{Q}_{\lambda}\/}\nolimits\!\left(x\right)\mathrm{d}x=\frac{2\mathop{% \sin\/}\nolimits\!\left(\nu\pi\right)\mathop{\cos\/}\nolimits\!\left(\lambda% \pi\right)\left(\mathop{\psi\/}\nolimits\!\left(\nu+1\right)-\mathop{\psi\/}% \nolimits\!\left(\lambda+1\right)\right)+\pi\mathop{\cos\/}\nolimits\!\left((% \lambda-\nu)\pi\right)-\pi}{\pi(\lambda-\nu)(\lambda+\nu+1)},$ $\Re{\lambda}>0$, $\Re{\nu}>0$, $\lambda\neq\nu$.
 14.17.15 $\int_{-1}^{1}\mathop{\mathsf{P}_{\nu}\/}\nolimits\!\left(x\right)\mathop{% \mathsf{Q}_{\nu}\/}\nolimits\!\left(x\right)\mathrm{d}x=-\frac{\mathop{\sin\/}% \nolimits\!\left(2\nu\pi\right)\mathop{\psi\/}\nolimits'\!\left(\nu+1\right)}{% \pi(2\nu+1)},$ $\Re{\nu}>0$.
 14.17.16 $\int_{-1}^{1}\mathop{\mathsf{P}^{m}_{l}\/}\nolimits\!\left(x\right)\mathop{% \mathsf{Q}^{m}_{n}\/}\nolimits\!\left(x\right)\mathrm{d}x=\frac{\left(1-(-1)^{% l+n}\right)(l+m)!}{(l-n)(l+n+1)(l-m)!},$ $l,m,n=0,1,2,\dots$, $l\neq n$.
 14.17.17 $\int_{0}^{\pi}\mathop{\mathsf{Q}_{l}\/}\nolimits\!\left(\mathop{\cos\/}% \nolimits\theta\right)\mathop{\mathsf{P}_{m}\/}\nolimits\!\left(\mathop{\cos\/% }\nolimits\theta\right)\mathop{\mathsf{P}_{n}\/}\nolimits\!\left(\mathop{\cos% \/}\nolimits\theta\right)\mathop{\sin\/}\nolimits\theta\mathrm{d}\theta=0,$ $l,m,n=1,2,3,\dots$, $|m-n|.

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

 14.17.18 $\int_{1}^{\infty}\mathop{P_{\nu}\/}\nolimits\!\left(x\right)\mathop{Q_{\lambda% }\/}\nolimits\!\left(x\right)\mathrm{d}x=\frac{1}{(\lambda-\nu)(\nu+\lambda+1)},$ $\Re{\lambda}>\Re{\nu}>0$.
 14.17.19 $\int_{1}^{\infty}\mathop{Q_{\nu}\/}\nolimits\!\left(x\right)\mathop{Q_{\lambda% }\/}\nolimits\!\left(x\right)\mathrm{d}x=\frac{\mathop{\psi\/}\nolimits\!\left% (\lambda+1\right)-\mathop{\psi\/}\nolimits\!\left(\nu+1\right)}{(\lambda-\nu)(% \lambda+\nu+1)},$ $\Re{(\lambda+\nu)}>-1$, $\lambda\neq\nu$, $\lambda$ and $\nu\neq-1,-2,-3,\dots$.
 14.17.20 $\int_{1}^{\infty}(\mathop{Q_{\nu}\/}\nolimits\!\left(x\right))^{2}\mathrm{d}x=% \frac{\mathop{\psi\/}\nolimits'\!\left(\nu+1\right)}{2\nu+1},$ $\Re{\nu}>-\tfrac{1}{2}$.

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

## §14.17(v) Laplace Transforms

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

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).