A Hankel Transform Integral: $\int_0^\infty\frac{1}{k^2-k_p^2}J_0\left(k\rho\right)\;k\,dk$

Question

$$ \int_0^\infty\frac{1}{k^2-k_p^2}J_0\left(k\rho\right)\;k\,dk $$

Suppose that $k_p$ is in the first quadrant in the complex plane, and that $\rho$ is purely real. $J_0$ is the Bessel function of the first kind and order zero.

What is the right set of moves to calculate this integral? What are the complex analysis arguments you have to use (closing contours around the two poles, residues, etc.; are these applications of the Cauchy integral formula, or the residue theorem)? Do you have extend the integral to the whole real line and replace the Bessel function with a Hankel function? What's the final result? Mathematica is happy to tell me the answer is a specific Hankel function, but I would like to know how these things work for THIS particular problem.

Answer 1

We may exploit the following facts: $$\mathcal{L}(J_0(k\rho)) = \frac{1}{\sqrt{\rho^2+s^2}},\qquad \mathcal{L}^{-1}\left(\frac{k}{k^2-k_p^2}\right) = \cosh(k_p s). $$ If we temporarily assume that $k_p = ip$ and $p\in\mathbb{R}^+$, the original integral equals: $$ \int_{0}^{+\infty}\frac{\cos(ps)}{\sqrt{\rho^2+s^2}}\,ds = \int_{0}^{+\infty}\frac{\cos(\rho p s)}{\sqrt{s^2+1}} = K_0(\rho p)$$ by the integral definition of $K_0$. By analytic continuation, it is expected that the original integral, if converging, equals $\color{red}{K_0(-i\rho k_p)}$.