Determine the Green function of the two dimensional Laplace operator
Recall that the Green function has to satisfy
As an ansatz, we assume that depends only on the magnitude of , i.e. . Then let us use Gauss theorem with a “volume” being a disk with radius .
Since the boundary of the disk is the circle and , one obtains
from which follows that is given by (up to functions that lie in the kernel of the two-dimensional Laplace operator)
where is some length scale in order to make the logarithm well-defined.