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.
Long time.