Constant temperature distribution across the surface of a disk

[ps1]

Homework Statement

First of all can i say that my question is part of a bigger problem that I'm trying to solve, but for the moment I'm stuck at this bit!

I'm trying to obtain a function that will return the temperature at a chosen point $(r,\theta)$ on a disk of radius $r_{0}$. The temperature of the disk is constant across its area.

Homework Equations

So far what I have done is use a general solution for the polar form of the 2d steady state laplace equation;

$f(r,\theta) = (1/2)a_0 + \sum\limits_{n} (r/r_0)^n (a_n cos(n\theta) + b_n sin(n\theta))$

and solved for the Fourier coefficients using $g(\theta) = T$ where $T$ is the temperature distribution around the disk's perimeter.

The Attempt at a Solution

I had hoped to get an equation that would give the same temperature for all points on the disk but instead i have this;

$f(r,\theta) = T + \sum\limits_{n} (r/r_0)^n (T/\pi) cos(n\theta)$

which returns varying temperatures depending on which point i choose. some points are warmer than the disk itself! something is obviously wrong. i believe that I'm thinking about the problem the wrong way or I've not fully understood what I'm trying to do. the idea is that i can eventually choose a point outside the disk and find the temperature there, then plot a heat flow graph with other boundaries at different temperatures.

 

[haruspex]

It's trivial to find a function that takes a constant value. You appear to be assuming that it is possible to find a single function in Fourier expansion form that returns a constant value over an entire disc but other values elsewhere. That is not going to happen. Your end result will at least consist of two equations, one for points in the disc and one for elsewhere. There share a boundary, so just ensure the two equations satisfy that boundary condition.