Geodesic
Normal forms for singularities of one dimensional holomorphic vector fields
Electronic journal of differential equations, Tome 2004 (2004)
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.
Classification : 34C20, 34A34, 32A10, 37C10
Keywords: meromorphic vector field, holomorphic vector field, normal form 
@article{EJDE_2004__2004__a133,
     author = {Garijo,  Antonio and Gasull,  Armengol and Jarque,  Xavier},
     title = {Normal forms for singularities of one dimensional holomorphic vector fields},
     journal = {Electronic journal of differential equations},
     year = {2004},
     volume = {2004},
     zbl = {1075.34089},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2004__2004__a133/}
}
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AU  - Jarque,  Xavier
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%F EJDE_2004__2004__a133
Garijo,  Antonio; Gasull,  Armengol; Jarque,  Xavier. Normal forms for singularities of one dimensional holomorphic vector fields. Electronic journal of differential equations, Tome 2004 (2004). http://geodesic.mathdoc.fr/item/EJDE_2004__2004__a133/