# §14.12 Integral Representations

## §14.12(i) $-1

### Mehler–Dirichlet Formula

 14.12.1 $\displaystyle\mathsf{P}^{\mu}_{\nu}\left(\cos\theta\right)$ $\displaystyle=\frac{2^{1/2}(\sin\theta)^{\mu}}{\pi^{1/2}\Gamma\left(\frac{1}{2% }-\mu\right)}\int_{0}^{\theta}\frac{\cos\left(\left(\nu+\frac{1}{2}\right)t% \right)}{(\cos t-\cos\theta)^{\mu+(1/2)}}\,\mathrm{d}t,$ $0<\theta<\pi$, $\Re\mu<\tfrac{1}{2}$. 14.12.2 $\displaystyle\mathsf{P}^{-\mu}_{\nu}\left(x\right)$ $\displaystyle=\frac{\left(1-x^{2}\right)^{-\mu/2}}{\Gamma\left(\mu\right)}\int% _{x}^{1}\mathsf{P}_{\nu}\left(t\right)(t-x)^{\mu-1}\,\mathrm{d}t,$ $\Re\mu>0$;

compare (14.6.6).

 14.12.3 $\mathsf{Q}^{\mu}_{\nu}\left(\cos\theta\right)=\frac{\pi^{1/2}\Gamma\left(\nu+% \mu+1\right)(\sin\theta)^{\mu}}{2^{\mu+1}\Gamma\left(\mu+\frac{1}{2}\right)% \Gamma\left(\nu-\mu+1\right)}\*\left(\int_{0}^{\infty}\frac{(\sinh t)^{2\mu}}{% (\cos\theta+i\sin\theta\cosh t)^{\nu+\mu+1}}\,\mathrm{d}t+\int_{0}^{\infty}% \frac{(\sinh t)^{2\mu}}{(\cos\theta-i\sin\theta\cosh t)^{\nu+\mu+1}}\,\mathrm{% d}t\right),$ $0<\theta<\pi$, $\Re\mu>-\tfrac{1}{2}$, $\Re\nu\pm\mu>-1$.

## §14.12(ii) $1

 14.12.4 $\displaystyle P^{-\mu}_{\nu}\left(x\right)$ $\displaystyle=\frac{2^{1/2}\Gamma\left(\mu+\frac{1}{2}\right)\left(x^{2}-1% \right)^{\mu/2}}{\pi^{1/2}\Gamma\left(\nu+\mu+1\right)\Gamma\left(\mu-\nu% \right)}\*\int_{0}^{\infty}\frac{\cosh\left(\left(\nu+\frac{1}{2}\right)t% \right)}{(x+\cosh t)^{\mu+(1/2)}}\,\mathrm{d}t,$ $\nu+\mu\neq-1,-2,-3,\dots$, $\Re\left(\mu-\nu\right)>0$. 14.12.5 $\displaystyle P^{-\mu}_{\nu}\left(x\right)$ $\displaystyle=\frac{\left(x^{2}-1\right)^{-\mu/2}}{\Gamma\left(\mu\right)}\int% _{1}^{x}P_{\nu}\left(t\right)(x-t)^{\mu-1}\,\mathrm{d}t,$ $\Re\mu>0$. 14.12.6 $\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)$ $\displaystyle=\frac{\pi^{1/2}\left(x^{2}-1\right)^{\mu/2}}{2^{\mu}\Gamma\left(% \mu+\frac{1}{2}\right)\Gamma\left(\nu-\mu+1\right)}\*\int_{0}^{\infty}\frac{(% \sinh t)^{2\mu}}{\left(x+(x^{2}-1)^{1/2}\cosh t\right)^{\nu+\mu+1}}\,\mathrm{d% }t,$ $\Re\left(\nu+1\right)>\Re\mu>-\tfrac{1}{2}$. 14.12.7 $\displaystyle P^{m}_{\nu}\left(x\right)$ $\displaystyle=\frac{{\left(\nu+1\right)_{m}}}{\pi}\*\int_{0}^{\pi}\left(x+% \left(x^{2}-1\right)^{1/2}\cos\phi\right)^{\nu}\cos\left(m\phi\right)\,\mathrm% {d}\phi,$ 14.12.8 $\displaystyle P^{m}_{n}\left(x\right)$ $\displaystyle=\frac{2^{m}m!(n+m)!\left(x^{2}-1\right)^{m/2}}{(2m)!(n-m)!\pi}% \int_{0}^{\pi}\left(x+\left(x^{2}-1\right)^{1/2}\cos\phi\right)^{n-m}(\sin\phi% )^{2m}\,\mathrm{d}\phi,$ $n\geq m$.
 14.12.9 $\boldsymbol{Q}^{m}_{n}\left(x\right)=\frac{1}{n!}\int_{0}^{u}\left(x-\left(x^{% 2}-1\right)^{1/2}\cosh t\right)^{n}\cosh\left(mt\right)\,\mathrm{d}t,$

where

 14.12.10 $u=\frac{1}{2}\ln\left(\frac{x+1}{x-1}\right).$ ⓘ Symbols: $\ln\NVar{z}$: 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 See also: Annotations for §14.12(ii), §14.12 and Ch.14
 14.12.11 $\boldsymbol{Q}^{m}_{n}\left(x\right)=\frac{\left(x^{2}-1\right)^{m/2}}{2^{n+1}% n!}\int_{-1}^{1}\frac{\left(1-t^{2}\right)^{n}}{(x-t)^{n+m+1}}\,\mathrm{d}t,$
 14.12.12 $\boldsymbol{Q}^{m}_{n}\left(x\right)=\frac{1}{(n-m)!}P^{m}_{n}\left(x\right)% \int_{x}^{\infty}\frac{\,\mathrm{d}t}{\left(t^{2}-1\right)\left(\displaystyle P% ^{m}_{n}\left(t\right)\right)^{2}},$ $n\geq m$.

### Neumann’s Integral

 14.12.13 $\boldsymbol{Q}_{n}\left(x\right)=\frac{1}{2(n!)}\int_{-1}^{1}\frac{P_{n}\left(% t\right)}{x-t}\,\mathrm{d}t.$

### Heine’s Integral

 14.12.14 $\boldsymbol{Q}_{n}\left(x\right)=\frac{1}{n!}\int_{0}^{\infty}\frac{\,\mathrm{% d}t}{\left(x+(x^{2}-1)^{1/2}\cosh t\right)^{n+1}}.$

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.