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Q:
Integral involving Bessel’s function of the first kind
I’m trying to evaluate an integral:
$$I(m,n)=\int_{0}^{\infty}J_m(x)J_n(x)\exp(-\pi x/2)dx,$$
with $m$ and $n$ positive integers. I already checked the related problems in the forum, but I could not find a solution.
$J_n$ denotes the $n$-th Bessel’s function of the first kind. Could you suggest me how to evaluate it, or give me a link where this kind of integrals are discussed?
A:
These are the formulae for the Bessel’s functions of the first kind
$$J_{m+1}(x)=J_m(x)+\frac{2m}{x}J_m(x)$$
$$J_{ -m}(x)=J_m(x)-\frac{2m}{x}J_m(x)$$
and
$$J_n(x)=\frac{i^n}{2}\left(H_n^{(1)}(x)-H_n^{(2)}(x)\right)$$
where $H_n^{(1,2)}$ are the Hankel functions of the first and second kind. The integral is a double Hankel transform (I assume that the exponant can be removed). Then using the generating function of the Bessel’s functions
$$e^{i\pi z/2}J_n(z)=J_n(z)$$
you have
$$\left[e^{i\pi z/2}J_m(z)\right]^2=J_m(z)^2e^{i\pi z/2}=e^{i\pi z/2}J_m(z)J_m(z)=\left[e^{i\pi z/2}\frac{i^m}{2}\left(H_m^{(1)}(z)-H_m^{(2)}(z)\right)\right]^2$$
and thus
$$2\int_0^\infty J_m(x)J_n(x)e^{ -\pi x/2}dx=\frac{i^m}{2