I am interested in evaluating the integral:
$$\small\iint\limits_{[0,2\pi) \times [0,2\pi)} (x^2 + y^2)^{-1/2}((b-x)^2 + (c-y)^2)^{-1/2} J_{1}\left(\rho \sqrt{x^2 + y^2}\right)J_{1}\left(\rho \sqrt{(b-x)^2 + (c - y)^2}\right)\mathrm{d}x\mathrm{d}y,$$
where $b, c, \rho > 0$ are constants, and $J_{\nu}$ denotes the Bessel function of the first kind.
I've tried integrating this with Mathematica, using an input such as:
Integrate[((x^2 + y^2)^(-1/2))*(((b-x)^2 + (c-y)^2)^(-1/2))*BesselJ[1, k*Sqrt[x^2 + y^2]]*
BesselJ[1, k*Sqrt[(b - x)^2 + (y - c)^2]], {x, 0, 2*Pi}, {y, 0,
2*Pi}]
but this just returns the input. We can bound the integral using the bound $|J_{\nu}(z)| \leqslant C_{\nu}|z|^{-1/2},$ but I can't seem to be able to compute this either. Can anyone help me evaluate this integral?