Electronic Journal of Differential Equations,
Vol. 2004(2004), No. 122, pp. 1-7.

Title: Normal forms for singularities of one dimensional 
       holomorphic vector fields

Authors: Antonio Garijo (Univ. Rovira i Virgili,Tarragona, Spain)
         Armengol Gasull (Univ. Autonoma de Barcelona, Spain)
	 Xavier Jarque (Univ. de Barcelona, Spain)

Abstract: 
 We study the normal form of the ordinary differential equation
 $\dot z=f(z)$, $z\in\mathbb{C}$, in a neighbourhood of a point
 $p\in\mathbb{C}$, where $f$ 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.
 
Submitted June 23, 2004. Published October 15, 2004.
Math Subject Classifications: 34C20, 34A34, 32A10, 37C10.
Key Words: Meromorphic vector field; holomorphic vector field; normal form.