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.