Phase Diagrams of Meromorphic Functions

Abstract

The paper demonstrates the use of phase diagrams as tools for visualizing and exploring meromorphic functions. With any such function \(f:D\ \rightarrow\ \hat{\rm C}\) we associate two mappings $$P_{f}:D\rightarrow{\rm T}\cup \lbrace {0,\infty}\rbrace,z\mapsto {f(z)\over|f(z)|},\qquad V_{f}:D\rightarrow {\rm C},z\mapsto - {f(z){\overline f^\prime(z)}\over |f(z)|^{2}+|f^{\prime}(z)|^{2}},$$ with an appropriate definition at zeros and poles. Color-coding the points of \({\rm T}\cup\lbrace{0,\infty}\rbrace\) converts the function Pf to an image which visualizes the function f directly on its domain. Endowing this phase plot with the orbits of the vector field Vf yields the phase diagram of f.