Help with Greens function/Fourier transformation to solve screened Poisson equation

Question

I am having trouble getting from one line to the next from this wiki page. I am referring to the text line

Green's function in $r$ is therefore given by the inverse Fourier transform,

where

$$G(r) ~=~ \frac{1}{(2\pi)^3} \iiint d^3k \frac{e^{i{\bf k}\cdot{\bf r}}}{k^2+\lambda^2}$$

goes to

$$G(r) ~=~ \frac{1}{2\pi^2r} \int^{\infty}_0 \!dk_r \frac{k_r \sin(k_r r)}{k_r^2+\lambda^2}.$$

Where does the $\frac{1}{r}$ term come from and what is $k_r$? How did they simplify the triple integral? Divergence theorem? Stokes? Detailed steps would be much appreciated.

Answer 1

You convert the integral to spherical coordinates and the $k \cdot r$ term becomes $kr\cos\theta$. Integrating over $\theta$ and $\phi$ gets you the single dimensional integral expression.