Normal forms for singularities of one dimensional holomorphic vector fields

Abstract

We study the normal form of the ordinary differential equation ż = ƒ(z), z ∈ ℂ, in a neighbourhood of a point p ∈ ℂ, where ƒ is a one-dimensional holomorphic function in a punctured neighbourhood of p. Our results include all cases except when p is an essential singularity. We treat all the other situations, namely when p is a regular point, a pole or a zero of order n. Our approach is based on a formula that uses the flow associated with the differential equation to search for the change of variables that gives the normal form.