Fundamental Solution to Laplace equation using inverse Fourier

Question

I'm trying to do the following integral (which is an inverse Fourier transform):

$$\int_{-\infty}^\infty \int_{-\infty}^\infty \frac{e^{i (x_1\xi_1 + x_2\xi_2) }} {4\pi^2 (\xi_1^2 +\xi_2^2)}d\xi_1d\xi_2$$

Any help with this? I tried converting to polar but that didn't make my life that much easier.

Answer 1

In polars, the integral is equal to

$$\frac1{4 \pi^2} \int_0^{\infty} dk \, \frac1{k} \, \int_0^{2 \pi} d\theta \, e^{i k r \cos{\theta}} = \frac1{2 \pi} \int_0^{\infty} dk \, \frac{J_0(k r)}{k}$$

You should be able to see that the integral does not converge.