Holomorphic Root Of A Holomorphic Function

Complex Analysis
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Tsakanikas Nickos
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Holomorphic Root Of A Holomorphic Function

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Post by Tsakanikas Nickos »

Let \( f \) be a non-constant holomorphic function defined on an open neighborhood \( U \subset \mathbb{C} \) of \( 0 \) with \( f(0)=0 \). Show that there is a unique integer \( k \geq 1 \) such that \( f(z) = g(z)^{k} , \ z \in V, \) for some holomorphic function \( g \) defined on an open neighborhood \( V \subset U \) of \( 0 \) with \( g^{\prime}(0) \neq 0 \).
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