On the local and global phase portrait of the 1-dimensional complex equation $z{\dot}= f (z)$

Abstract

This work consists of studying the complex first order differential equation $z{\dot} = \dfrac{dz}{dt}=f(z),\hspace{2cm} z \in\mathbb{C},t\in\mathbb{R}$ where $f$ is an analytic function of $C$ except, possibly, at isolated singularities. This is a rather general family of complex functions that includes polynomial, rational, holomorphic and entire functions, and functions with isolated essential singularities.