Kindly help me with the following integral :
$ I_l(a) = \int_{-1}^{+1} dx\, e^{a x} P_l(x) \quad $ ($a$ is real and positive).
I thought to use the following relation given in Gradshteiyn and also in Prudnikov,
$\int_{-1}^{+1}\, dx\, e^{ipx} P_l(x) = \sqrt{\frac{2 \pi}{p}} \, e^{\frac{i l \pi}{2}} J_{l + \frac{1}{2}} (p) $.
But it seems that the formula cited there holds for real $p$. Can someone please enlighten me if $p$ becomes complex? I am unable to trace the original source.