Temperature and Harmonic Functions

[jgens]

Homework Statement

Find the temperature of the semi-circular plate of radius 1 with the following boundary temperatures: T=0 along the semi-circle, T=0 along (-1,0), and T=1 along (0,1).

Homework Equations

N/A

The Attempt at a Solution

Well, the author of the text has already noted that temperature functions are harmonic, so I'm looking for a harmonic function here. I've already tried a number of stock examples (like taking the real and imaginary parts of log(z)), but I can't seem to get anything to work. Can someone give me a hint to get me working in the right direction here? Thanks!

 

[hunt_mat]

This puzzles me as when dealing with temperature you use the diffusion equations which is parabolic, and looks like:
$$\frac{\partial T}{\partial t}=\frac{1}{\alpha}\frac{\partial^{2}T}{\partial x^{2}}$$
These don't define harmonic functions in general. What equation is he using here?

 

[jgens]

The text we're using (Lang's Complex Analysis text) just says that temperature functions are harmonic. So, I'm guessing we're supposed to find a harmonic equation modeling the temperature of the plate, but I can't think of any trivial examples that make it work out. It's not a physics text, so I wouldn't be surprised if it grossly simplified the physics behind temperature.

 

[hunt_mat]

Conformal transformations in the answer. Transform a semi-circle into a region where you know you can solve the equation and then it should be easy from there.