# Integrals with Legendre end bessel functions

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!!

Try using that $$\int_{-1}^{1}P_m(x)P_n(x) dx=\frac{2}{2n+1}\delta_{mn}$$
Where $$\delta_{mn}$$ represents the Kronecker delta alongwith the Generating functions you have at hand.
Also with the generating function you have, try substituting $$x=\cos \theta$$ in the first integral, and then use that $$\int_{-1}^{1}P_n(x)dx=0$$ $$\forall$$ $$n\ge 1$$ which is pretty evident from the property of Legendre polynomials that $$P_n(-x)=(-1)^nP_n(x)$$