Complex analysis - something really confusing

[sweetvirgogirl]

I think I have misunderstood one of the theorems in complex analysis

(k reperesents the order of the derivative)


Theorem: Suppose f is analytic on a domain D and, further, at some point z0 subset of D, f (k) (z0) = 0. Then f(z) = 0 for all z subset of D ...

Is the theorem basically saying is that if f(z) equals 0 at any z0, then it will equal zero for all of the points?? That doesnt' sound right at all ...

any help with be greatly appreciated

 

[leach]

That theorem is obviously wrong. The function $$f(z) = z^{k+1}$$ is a counterexample. It verifies $$f^{k}(0) = 0$$, is holomorphic in any domain, yet it is nonzero for every $$z\ne 0$$.

Most probably your theorem is one of the following:

1) If $$f^k(z_0) = 0$$ for all $$k\ge 0$$, then f=0 in D.

2) If $$f=0$$ in some open subset of D, then f=0 in D.