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How does one compute the Green's function of the laplacian on $ \mathbb{R}^2 $? Can someone point me to a reference? In particular, the fourier transform: $$ \int_0^\infty \int_0^\infty \frac{e^{i m x} e^{i m y}}{m^2+n^2} $$ does not converge. Is there some standard way to regulate this?

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The integral diverges for a good reason: $\mathbb R^2$ does not have Green's function. In potential-theory-speak, $\mathbb R^2$ is not a Greenian domain. More precisely, a domain $\Omega\subset \mathbb R^2$ is Greenian if and only if its complement has positive logarithmic capacity. This is discussed in all potential theory books, such as Armitage-Gardiner.

The closest thing to Green's function for the plane is $\log\sqrt{x^2+y^2}$, which inverts the Laplacian but does not have zero boundary values (or constant sign).

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Thank you. This is very helpful. – user61928 Feb 12 '13 at 2:27

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