f is a holomorphic function at every z in C. We know that lim f(z)=0 when z->+00. Show that f(z)=0 for every Z in C.
Can you please give me some kind of hint ?? i would be sooo grateful...
Hi Eleni,
Since is holomorphic at each point of is an entire function. The idea here is to use Liouville's Theorem. Try finding a way to use the information that to prove that is bounded. Once you have bounded and entire Liouville's Theorem will tell you that is constant.
Does this get things on the right track? Good luck!
GJA - I don't see a way to easily prove that f is bounded. You can't just use the fact that f(z) goes to zero - there are functions that go to zero but are not bounded.
I think you can expand f(z) in a power series and use Cauchy's Integral Formula to prove all the coefficients are zero except for the constant coefficient. So f(z) is a constant which must be zero. That's the way the proof of Liouville's Theorem goes.
I think it might be possible to argue directly from Cauchy's Integral Formula. If so, that would be a much better solution.
- Hollywood
If as then there is a compact disc centered at the origin such that for outside of this disc. Since this disc is compact and is entire (so continuous), is bounded on the disc. The bound for is then the maximum of 1 and the bound on the disc.