# Laplace equation in spherical coordinates

1. Nov 9, 2011

### romsofia

1. The problem statement, all variables and given/known data
Verify by direct substitution in Laplace's equation that the functions (2.19) are harmonic in in appropriate domains in ℝ2

2. Relevant equations
(2.19)= $${u_n(r, \theta)= \lbrace{1,r^{n}cos(n \theta), r^{n}sin(n \theta), n= 1, 2...; log(r), r^{-n}cos(n \theta), r^{-n}sin(n \theta); n= 1, 2,...; \rbrace}}$$

Should look like a piece-wise function (Don't know how to do that in latex).

Laplace equation in spherical coordinates
$${\Delta u = \frac{1}{r^{n-1}} \frac{\partial (r^{n-1}\frac{\partial u(r, \theta)}{\partial r})}{\partial r} + \frac{1}{r^{2}} \frac{\partial^2 u(r, \theta)}{\partial^2 \theta} = 0}$$

3. The attempt at a solution

I don't what I'm suppose to substitute, do I substitute the whole thing into the Laplace equation, each time I see u(r, θ)?