Electron. J. Diff. Eqns., Vol. 2004(2004), No. 122, pp. 1-7.

Normal forms for singularities of one dimensional holomorphic vector fields

Antonio Garijo, Armengol Gasull, Xavier Jarque

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.

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Antonio Garijo
Dep. d'Eng. Informática i Matemátiques
Universitat Rovira i Virgili
Av. Pa&iunl;sos Catalans, 26
43007 Tarragona, Spain
email: agarijo@etse.urv.es
Armengol Gasull
Dept. de Matemátiques
Universitat Autónoma de Barcelona
Edifici C, 08193 Bellaterra, Barcelona, Spain
email: gasull@mat.uab.es
Xavier Jarque
Dep. de Matemáatica Aplicada i Análisi
Universitat de Barcelona
Gran Via 585
08007 Barcelona, Spain
email: xavier.jarque@ub.edu

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