I want to do this integral $$U(k)=\int_{k'=0}^{\infty} F(k')\int_{r=0}^{c}J_0(k'r)J_0(kr)rdr k' dk'$$ where $J_0(x)$ is the 0th order Bessel function of the first kind, $F(x)$ is a function that I can compute numerically but don't know the explicit form.
I know $$\mathrm{if} \quad c\rightarrow \infty,\qquad \mathrm{then} \quad U(k) = F(k)$$ by using the bessel function orthogonality $$\int_{0}^{\infty}J_{m}(kr)J_{m}(k'r)rdr=\frac{1}{k}\delta(k-k') $$ What for a positive finite value $c$?
A maybe useful hint:Integral of product of Bessel functions of the first kind