Integrals with Legendre end bessel functions

Question

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!!

Answer 1

Hint:

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 $$P_n(-x)=(-1)^nP_n(x)$$