I'm reading through a book on Riemann surfaces, and I tend to get stuck on some of the proofs (maybe because of my lacking background in complex analysis). Anyway, here it goes:(adsbygoogle = window.adsbygoogle || []).push({});

It's a rather lengthy proof, and I'd prefer not to give it in full detail. The question itself is pretty self-contained.

Let h(z) be a holomorphic function from some open subset of [itex]\mathbb{C}[/itex], with h(0) non-zero.

Then why can one find an open disc around 0 such that is [itex]z \mapsto zh(z)[/itex] is biholomorphic onto its image? I.e., why does it have an inverse near 0?

The theorem is about the local behavour of holomorphic functions, basically that one can biholomorphically transform any non-constant holomorphic function to [itex]z \mapsto z^k[/itex] for some integer k > 0 locally around a point.

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# Question on holomorphic functions

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