Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Where can I find a proof/reference for the following fact?

Let $f$ be a holomorphic function with a zero of order $n$ at $z = 0$. Then for sufficiently small $\epsilon > 0$, there exists $\delta > 0$ such that for all $a$ with $0 < |a| < \delta$, $f(z) = a$ has exactly $n$ roots in the disc $|z| < \epsilon$.

share|cite|improve this question
First you have $\delta$, and then you have $a$... anyway the series for your holomorphic function will look like $c_1 z^n+c_2 z^{n+1}+\cdots$, no? – J. M. Nov 19 '11 at 8:07
A reference is IV.7.4 in Conway's Functions of one complex variable. I think there was a question about this on this site, but I do not know the link. – Jonas Meyer Nov 19 '11 at 8:12
Great, found the theorem in Conway, thanks – Joe Nov 19 '11 at 8:14
I just found the question I was thinking of:…. Duplicate? – Jonas Meyer Jan 23 '12 at 5:34

A reference is IV.7.4 in J.B. Conway's Functions of one complex variable.

$_{\text{This was copied from a comment in an attempt to get the question off of the Unanswered list.}}$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.