# Proof of Liouville's Theorem +

1. Jun 30, 2011

### MurraySt

I'm asked to use Rouche's Theorem to prove Liouville's - I really don't have much of a clue as proofs are not my strong suit.

Next up: Find the max and min of abs(f(z)) over the unit disk where f(z) = z^2 - 2

Do I use the maximum modulus theorem?

Lastly I'm given epsilon>0 and the set e^(1/z) where 0<abs(z)<epsilon. This set is equal to the entire complex plane minus 0 as e^(1/z) cannot take on that value. The question is: What can I say about the set? Besides the fact that it cannot be 0 I'm out of ideas.

Thanks as always

Edit: One final question
I'm give that f(1) = 1, f(-1) = i and f(-i) = 1. I need to find a Mobius transformation.

I believe I need to use the cross ratio - but the problem is that Mobius transformations should send something to 0, 1 and infinity (which this one does not) how can I get around this issue?

Last edited: Jun 30, 2011
2. Jul 1, 2011

### hunt_mat

Is this homework? As if it is it's in the wrong forum.