I'm trying to find the Green's function for the screened Poisson equation in two dimensions, i.e. the $G(\mathbf{r})$ that solves
$$(\nabla^2-1/\rho^2) G(\mathbf{r}) = \delta(\mathbf{r}), \qquad \mathbf{r}\in\mathbb{R}^2,$$
via Fourier transform methods. Everything is fine except for the final integral in the inverse transform, which is over the radius $k$ in polar coordinates.
Essentially my question is: "how do you do the last step on the last line of the relevant wikipedia page?" which is:
$$\int_0^\infty dk\frac{k}{k^2+1/\rho^2}J_0(kr) = K_0(r/\rho).$$
I can't find anywhere such a relation between the 0th-order Bessel function of the first kind $J_0(x)$ and the 0th order modified Bessel function of the second kind $K_0(x)$, except on that particular wikipedia page.
Could someone point me to a reputable reference for this integral relation, or otherwise indicate how I might show this result? Thanks!
Best Answer
For the integral, $$ I:=\int_0^\infty \frac{x}{x^2+q^2} \text{J}_0(x\,r)\ dx $$ insert the well-known integral $$ \frac{x}{x^2+q^2} = \int_0^\infty e^{-x\,u}\cos{(q\ u)} \ du.$$ Switch integrations. Many integral tables have $$ \int_0^\infty e^{-x\,u} \text{J}_0(x\,r)\ dx = \frac{1}{\sqrt{r^2+u^2}}. $$ Scale and use Digital Library of Mathematical Functions 10.32.6, $$ \text{K}_0(w)=\int_0^\infty \frac{\cos{(w \ t)}}{\sqrt{1+t^2}} dt.$$ You'll find $I=\text{K}_0(r\ q), $ equivalent to your question with $q=1/\rho.$ For more rigor, justify the interchange of integrations.