Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question

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).

share|cite|improve this answer
Thank you. This is very helpful. – user61928 Feb 12 '13 at 2:27

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.