Mean Value Theorem

[Hypnotoad]

Prove that for charge-free two-dimensional space the value of the electrostatic potential at any point is equal to the average of the potential over the surface of any circle centered on that point. Do this by considering the electrostatic potential as the real part of an analytic function.

I have no idea how to start this problem and am not sure what to do with the analytic function information. Any hints on how to start this would be really appreciated.

[Wong]

In fact what you said is a general theorem regarding harmonic functions in any dimensions. By definition a harmonic function is a function f satisfying \sum_{i} {\partial_{i}^{2} f}= 0 .

To prove the assertion in two dimension, you may like to recall that in a source free region, the potential satisfies the laplace equation and may be regarded as the real part of an analytic function (because of the riemann condition on analyticity). Then you may like to recall which theorem in complex analysis allows you to express the value of an analytic function at a point as an integral over a contour?