# 2-D Poisson's equation - Green's function

In the x-y plane, we have the equation

$$\nabla^{2} \Psi = - 4\pi \delta(x- x_{0}) \delta (y- y_{0})$$

with $$\Psi = 0$$ at the rectangular boundaries, of size L.

A paper I'm looking at says that for

$$R^{2}$$ = $$(x-x_{0})^{2}$$ + $$(y-y_{0})^{2}$$ << $$L^{2}$$ ,

that is, for points very close to the source, the solution must behave as if the boundaries were at infinity, and

$$\Psi \approx -2 ln R.$$

I see that -2 ln R will satisfy the equation, but why should it be the only solution valid at R<<L? And how does it satisfy the boundary condition?

- 2 ln R goes to infinity as R goes to zero, does that have anything to do with it?

Morse and Feshbach have the same thing on Pg 798 of vol 1 of their Methods ... , but I cant see an explanation there either.

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