# §14.17 Integrals

## §14.17(i) Indefinite Integrals

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

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(ii) Barnes’ Integral

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

## §14.17(iii) Orthogonality Properties

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

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

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

## §14.17(iv) Definite Integrals of Products

With $\psi\left(x\right)=\Gamma'\left(x\right)/\Gamma\left(x\right)$5.2(i)),

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

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

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

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).