Computational Methods and Function Theory

, Volume 10, Issue 2, pp 639–661 | Cite as

Phase Diagrams of Meromorphic Functions

  • Elias Wegert


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 P f to an image which visualizes the function f directly on its domain. Endowing this phase plot with the orbits of the vector field V f yields the phase diagram of f.

We describe the local normal forms of phase diagrams, study properties of their orbits, and investigate the basins of attraction of zeros. Special attention is paid to the interplay between zeros, poles and critical points. In particular we derive formulas which relate the numbers of these points in a Jordan domain G to the winding numbers of P f and V f along the boundary of G. A short proof of Walsh’s theorem on the critical points of Blaschke products serves as an illustration.


meromorphic function phase plot phase diagram basin of attraction visualization of complex functions Gauss-Lucas theorem Walsh theorem 

2000 MSC

30D30 30A99 


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  1. 1.
    T. Arens, F. Hettlich, Ch. Karpfinger, U. Kockelkorn, K. Lichtenegger and H. Stachel, Mathematik (in German), ktrum Akademischer Verlag 2008.MATHGoogle Scholar
  2. 2.
    F. A. Farris, Review of Visual Complex Analysis by Tristan Needham, American Mathematical Monthly 105 (1998), 570–576.MathSciNetCrossRefGoogle Scholar
  3. 3.
    H. Lundmark, Visualizing complex analytic functions using domain coloring,∼halun/complex/domain_coloring-unicode.html (as seen on Sep 16, 2010).
  4. 4.
    L. Perko, Differential Equations and Dynamical Systems (2nd edition), Springer, 1993.Google Scholar
  5. 5.
    Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer, 1992.Google Scholar
  6. 6.
    J. L. Walsh, The Location of Critical Points of Analytic and Harmonic Functions, American Mathematical Society Colloquium Publ. 34, New York, 1950, 386 pp.MATHGoogle Scholar
  7. 7.
    J. L. Walsh, Note on the location of zeros of extremal polynomials in the non-Euclidean plane, Acad. Serbe Sci., Publ. Inst. Math. 4 (1952), 157–160.MathSciNetMATHGoogle Scholar
  8. 8.
    E. Wegert, Phase plots of complex functions: a journey in illustration, to appear in Notices Amer. Math. Soc. Google Scholar
  9. 9.
    —, Visual exploration of complex functions using phase plots, in preparation. Elias Wegert Google Scholar

Copyright information

© Heldermann  Verlag 2010

Authors and Affiliations

  1. 1.Institute of Applied AnalysisTech. Univ. Bergakademie FreibergFreibergGermany

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