Slight generalization of the Local Normal Form
The Local Normal Form states that if is a holomorphic map at , which is not constant, then there is a unique integer which satisfies the following property: for every chart on centered at , there exists a chart on centered at such that .
The way I understand the proof, I think we can extend it to say that for any chart on centered at , then there exists a chart on centered at such that . The only added condition is that is non-zero except at .
Proving this would be quite easy, and is left as an exercise.