# Proof of Liouville's Theorem

1. Oct 31, 2013

### eaglesmath15

1. The problem statement, all variables and given/known data
Prove Liouville's theorem directly using the Cauchy Integral formula by showing that f(z)-f(0)=0.

2. Relevant equations
f(a) = $\frac{1}{2πi}$$\oint\frac{f(z)}{z-a}dz$

3. The attempt at a solution
So the thing is, I know how to prove Liouville's theorem using CIF, but it doesn't show f(z)-f(0)=0, or at least not directly, and I've tried looking up other methods of proving it this way, but can't find any.

Last edited: Oct 31, 2013
2. Oct 31, 2013

### brmath

The proof I know expands f into a Taylor's series at zero , and shows that each $a_k$ has to be zero except for k = 0. We know $a_0$ = f(0). Are you familiar with this approach?