# The identity theroem complex analysis

1. Apr 1, 2008

### Scousergirl

1. The problem statement, all variables and given/known data
Prove that there is no holomorphic function f in the open unit disk such that f(1/n)=((-1)^n)/(n^2) for n=2,3,4....

2. Relevant equations
The identity theorem: Let f and g be holomorphic functions in the connected open subset of C, G. If f(z)=g(z) for all z in a subset of G that has a limit point in G, then f=g.

3. The attempt at a solution

We proved in an earlier example that there is not holomorphic function f in the open unit disk such that f(1/n)=2^(-n) n=2,3,4... Is the above function identical to this? I thought this was the route to go but i am not sure anymore. I know we have to somehow use the identity theorem.

2. Apr 1, 2008

### Pere Callahan

Maybe it helps to consider the function $g(z)=(-1)^{\frac{1}{z}}z^2$