Integration of a generic radial function in polar coordinates

Question

I need to perform the following integral

$\int{P(k) e^{i \vec{k}\cdot \vec{\Delta r}} \frac{d^2k}{(2 \pi) ^2}}$

using polar coordinates. I think the result should depend on some Bessel function, but I don't know how to proceed to show that. The form of $P(k)$ is, in principle, a generic function dependent only on the modulus $k$.

EDIT:

I think I was able to arrive to the following form, using polar coordinates where the axis is oriented along the vector $\vec{\Delta r}$.

$ \int{ P(\rho) e^{-i \rho \Delta r cos \theta }\rho d\rho d\theta}$

But then I don't know how to proceed.

Answer 1

Try to use method, which is used in the calculation of Green function of wave equation.