# Complex variable power series

1. Oct 8, 2008

### student85

Hi all, I'm having trouble understading this problem I got for homework. They're actually two problems in one, (a) and (b)... Any comment you can give me for understaing this will REALLY help. This is due Thursday, so please if anyone knows anything about this, can you share it with me? Thank you.

1. The problem statement, all variables and given/known data
a) If f(z) = $\sum$ an(z-z0)n has a radius of convergence R>0 and if f(z)=0 for all z,|z-z0| < r $\leq$ R, show that a0=a1=...=0.

b) If F(z) = $\sum$ an(z-z0)n and G(z) = $\sum$ bn(z-z0)n are equal on some disc |z-z0|< r, show that an = bn for all n.

NOTE: All sums go from n=0 to infinity.

2. Relevant equations

3. The attempt at a solution

Ok first of all I don't really get (a). It says f(z) is equal to 0 for all z, so obviously all a's must be equal to 0, no? Can someone please tell me what I am missing here? I typed the wording for the problem exactly as it is in the textbook.
Then for (b), again I don't see the point in the problem I mean if F and G are equal in the disc, then obviously all a's and b's are equal because each is the coefficient to a specific power of z, so a0 must be equal to b0, a1 = b1, and so on. But there must be more to these problems, there must be some mathematical proof or something I must do. Can someone shed some light on me? Thanks again.

2. May 23, 2009

### 22/7

Hints
(a) Use the same Taylor's Theorem from real analysis (works the same way for analytic functions)

(b) Subtract F and G from each other and compose the series into one series. Use the result from (a) so that a-b=0 for all n.