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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 $|z| < \epsilon$.

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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: math.stackexchange.com/questions/35304/…. Duplicate? –  Jonas Meyer Jan 23 '12 at 5:34

1 Answer 1

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

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