I'm trying to prove the follwing: 

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> For a holomorphic function $f$ with a zero $z\_0$ of multiplicity $n$ the $k$-th root of $f$ exists near $z\_0$ if and only if $k\mid n$.
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I thought it might be a good idea to prove the fact for $f=z^n$ first. If $k|n$ I can define $h(z):=z^{n/k}$ which is clearly holomorphic and satisfies $h(z)^k=f(z)$. Converserly if $k$ does not divide $n$ I tried to compute $(z^n)^{1/k}=\exp(\frac{1}{k}\log|z^n|+\frac{1}{k}\arg(z^n))=\exp(\frac{n}{k}(\log |z|+\arg z))$. How can I continue from here? How to prove the result for general $f$?