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Let $K$ be a open connected subset of complex numbers and $f$ holomorphic on $K$. If $f=0$ on some open disc $D$ in $K$, then is it true that

  1. $n$th derivative of $f$ is $0$ for all points in $D$

  2. And this implies $f=0$ on the whole of $K$.

Progress

I know that holomorphic implies power series expansion exists about the point and was trying to see if it can be done using just this. My main question is, if $ f$ is holomorphic and zero in some disc then is it zero on any connected set containing that disc?

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  • $\begingroup$ Please add some context or thoughts of your own. For example: do you know the identity theorem for holomorphic functions? $\endgroup$ – mrf Dec 16 '14 at 11:44
  • $\begingroup$ well i know that holomorphic implies power series expansion exists about the point and was trying to see if it can be done using just this. $\endgroup$ – mac Dec 16 '14 at 11:48
  • $\begingroup$ My main question is, If f is holomorpic and zero in some disc then is it zero on any connected set containing that disc. $\endgroup$ – mac Dec 16 '14 at 11:52
  • $\begingroup$ That is basically the identity principle. One way to proceed is to prove that the set where $f=0$ is both open and closed. $\endgroup$ – mrf Dec 16 '14 at 11:54
  • $\begingroup$ Can you show me please how you show that the set where f=0 is both open and closed $\endgroup$ – mac Dec 16 '14 at 11:57
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By the identity theorem both claims follow trivially.

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