I am trying to understand how to do the inverse Fourier transform for the Coulomb/Yukawa potential in 2D. So $$V(r) = \frac{e^2}{r}e^{-\lambda r}.$$ The Fourier transform says $$V(q) = \frac{2\pi e^2}{\sqrt{q^2+\lambda^2}}.$$
I learned how to do this from this question. I also understand how to go back in the case when $\lambda = 0$. But for positive $\lambda$, I run into $$f(\lambda) = \int_0^{\infty}dq J_0(q)\frac{q}{\sqrt{q^2+\lambda^2}}.$$ From the Fourier transform I know $f(\lambda) = \exp(-\lambda)$. But I am not sure how to get there.