On the local behaviour of a holomorphic function

[Tsakanikas Nickos]

Let \( f \) be a non-constant holomorphic function defined on an open neighborhood \( U \) of \( 0 \) with \( f(0)=0 \). Show that

• if \( f^{\prime}(0) \neq 0 \), then \( f \) is biholomorphic on a smaller neighborhood of \( 0 \).
• if \( f^{\prime}(0) = 0 \), then there is a unique integer \( k \geq 2 \) such that \( f = g^{k} \) on some smaller neighborhood of \( 0 \) for some biholomorphic function \( g \) defined there.