How can I solve this equality?

Question

I need prove this equality $$ K_O(r/\xi) = \int_{0}^{\infty} \frac{k}{{k^2}+\xi^{-2}}J_o(kr) dk $$

Where $$K_0$$ is a modified bessel function of zero order.

Answer 1

Using Laplace Transform-Evaluating integrals over the positive real axis: $\int_0^{\infty } \frac{k J_0(k r)}{k^2+\frac{1}{t^2}} \, dk=\int_0^{\infty } \left(\mathcal{L}_k^{-1}\left[\frac{k}{k^2+\frac{1}{t^2}}\right](s)\right) \left(\mathcal{L}_k[J_0(k r)](s)\right) \, ds=\int_0^{\infty } \frac{\cos \left(\frac{s}{t}\right)}{\sqrt{r^2+s^2}} \, ds=K_0\left(\frac{r}{t}\right)$