Wegert, Elias : Visualization of Complex Functions: Plea for the Phase Plot

Author(s):

Wegert, Elias

Title:

Visualization of Complex Functions: Plea for the Phase Plot

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 12, 2009

Mathematics Subject Classification:

30-01			[ Instructional exposition ]
30A99			[ None of the above, but in this section ]
30C99			[ None of the above, but in this section ]
30E25			[ Boundary value problems ]
11M06			[ $\zeta $ ]
33-01			[ Instructional exposition ]

Abstract:

The paper proposes and investigates a method for visualization of complex functions. The ``phase plot'' is a color encoded representation which depicts a function in its entity. Analytic and meromorphic functions are determined by their phase plot up to a positive factor. Location and degree of zeros and poles of a function and its derivative can easily be seen. The logarithmic derivative has an interpretation as density of the isochromatic lines. It is shown that phase plots admit a class of well-posed boundary value problems. Their solvability depends on the color index of the (given) boundary coloring. Based on these results, a theorem on the universality of the phase plot of the Riemann Zeta function is derived. Since the formulation is intuitive and needs (almost) no mathematical prerequisites it is easy to communicate also to non-experts. The converse of the theorem is equivalent to Riemann's hypothesis. Several examples illustrate the usage of phase plots as tools.

Keywords:

visualization, analytic functions, special functions, universal functions, Riemann zeta function

Language:

English

Publication time:

5 / 2009