Maximum Modulus Principle question

[bachdylan]

My lecturer gave this problem as an exercise:
http://img839.imageshack.us/img839/2405/imagewys.jpg

Uploaded with ImageShack.us



I attempted to use the Maximum Modulus Principle to solve this problem but failed. Any helps/suggestions are thankfully welcome.

 

[micromass]

OK, I was hoping that other people might reply, because my solution is fishy. But let's do this anyway.

Basically, what you need to do is showing that the function extends to \infty. So infinity is a singularity, and we want to extend our function to this singularity.

Now, because infinity is tough to work with, we are going to replace our singularity. So show that our problem is equivalent to a problem where the function is undefined in 0 (for example).

Try to show this by using either the Riemann mapping theorem or by working with the transformation z\rightarrow \frac{1}{z}.

Are you following me?