# Mean value of a harmonic function on a square

1. Apr 28, 2010

### bndnchrs

1. The problem statement, all variables and given/known data

The idea is to prove that the average of a harmonic function over a square is the same as the average over its diagonals.

2. Relevant equations

Really, none, other than the mean value theorem, that is the value of the function at a point is the same as the average of the function over the boundary of a ball at any radius from it.

3. The attempt at a solution

I'm not really sure how to get here. I've thought about using the compactness of the square to my advantage but that doesn't guarantee anything about the collection of balls and wouldn't do my much good anyways, at least I can't see how. I know of a proof using complex analytic functions but this isn't applicable here, I'm looking strictly at real-defined functions. An important thing in that proof is that the integral over a closed contour is zero, but I can't assume that for a real function... right?

Steps in the right direction are appreciated, I wan't to work this one out but I need a nudge.