I'm not an expert with this type of functions and its transformations, but I would like to know how to work with them when they are in an integral. First of all I know the definition of Bessel functions and Legendre functions. And the relation between them, with the generating function:
$$e^{tx}J_0\left ( t\sqrt{1-x^2} \right )=\sum \frac {P_n(x)}{n!}t^n $$
The integrals that I don't know how to start to solve are:
$$\int_{0}^{\pi }e^{r\cos}~J_0(r\sin\theta )\sin\theta ~d\theta $$ And $$\int_{-1}^{1 }e^{ax}~J_0\left ( a\sqrt{1-x^2} \right )~P_m(x)~dx $$
If someone can explain how to work with integrals and summations in this cases I'd be really grateful. Thanks!!