# Laplace u => Harmonic function

1. Sep 5, 2008

### Somefantastik

In R2, I am to find all homog. polys (deg 2) that are harmonic.

the earlier homework included something like u = xy, show it's harmonic. EASY as pi. But I'm not really sure how to set this problem up. I understand the concept that a harmonic function will look like $$\nabla^{2} u = 0$$, but I'm not sure how to find all polys of degree 2.

I started out doing something like

$$u = a_{2}x^{2}_{1} + a_{1}x_{1} + a_{0} + b_{2}x^{2}_{2} + b_{1}x_{2} + b_{0}+...$$

and taking the partial with respect to each xi but that's not getting me very far.

Any suggestions?

2. Sep 5, 2008

### Ben Niehoff

If you are in R^2, then you only need x and y. The general 2nd-degree polynomial is simply

$$Ax^2 + Bxy + Cy^2 + Dx + Ey + F$$

3. Sep 5, 2008

### Somefantastik

Thank you. I ended up with f(x,y) = Ax2 + Bx + C - Ay2 + Ey + F covers harmonic polys in R2.