# §14.12 Integral Representations

## §14.12(i) $-1

### Mehler–Dirichlet Formula

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

compare (14.6.6).

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

## §14.12(ii) $1

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

where

 14.12.10 $u=\frac{1}{2}\mathop{\ln\/}\nolimits\!\left(\frac{x+1}{x-1}\right).$ Symbols: $\mathop{\ln\/}\nolimits\NVar{z}$: 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)
 14.12.11 $\mathop{\boldsymbol{Q}^{m}_{n}\/}\nolimits\!\left(x\right)=\frac{\left(x^{2}-1% \right)^{m/2}}{2^{n+1}n!}\int_{-1}^{1}\frac{\left(1-t^{2}\right)^{n}}{(x-t)^{n% +m+1}}\mathrm{d}t,$
 14.12.12 $\mathop{\boldsymbol{Q}^{m}_{n}\/}\nolimits\!\left(x\right)=\frac{1}{(n-m)!}% \mathop{P^{m}_{n}\/}\nolimits\!\left(x\right)\int_{x}^{\infty}\frac{\mathrm{d}% t}{\left(t^{2}-1\right)\left(\displaystyle\mathop{P^{m}_{n}\/}\nolimits\!\left% (t\right)\right)^{2}},$ $n\geq m$.

### Neumann’s Integral

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

### Heine’s Integral

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

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.