Riemann's legacy

"Among a number of wonderful ideas we owe Riemann, the idea of Riemann surface is, without doubt, the most beautiful, everlasting, intensively developing, unifying and fertilizing a number of other ideas, penetrating the whole body of mathematics, and, in turn, many branches of physics."
K. Maurin. The Riemann Legacy, Kluwer, 1997

Main idea of Riemann surfaces:

Given a function [Graphics:../Images/TalkGD99_gr_1.gif] of a complex variable [Graphics:../Images/TalkGD99_gr_2.gif], make [Graphics:../Images/TalkGD99_gr_3.gif] single valued by using various copies of [Graphics:../Images/TalkGD99_gr_4.gif] (or parts of it in case of functions with natural boundaries of analyticity) and glue these copies properly together.

Older visualizations

old sketches and plaster models

Why visualize Riemann surfaces today?

It's intrumental for teaching.

"We see every day how difficult it is for the newcomer to grasp the idea of a Riemann surface and how he owns the whole theory once he has understood this fundamental concept."
("Wir sehen noch täglich, wie hart es dem Neuling ankommt, das Wesen der Riemannschen Fläche zu begreifen, und wie er auf einmal die ganze Theorie besitzt, wenn er die fundamentale Vorstellungsweise erfasst hat.")
F. Klein Jahresber. DMV 1895

"The Riemann surface is a very valuable, very suggestive thing to materialize and visualize the multi--valuedness of functions."
("Die Riemannsche Fläche [ist ein] sehr wertvolles, sehr suggestives Mittel zur Vergegenwärtigung und Veranschaulichung der Vieldeutigkeit von Funktionen.")
H. Weyl. Die Idee der Riemannschen Fläche, Teubner 1913

It's a popular subject.

  • Field theories on Riemann surfaces
  • Altavista search for "Riemann Surface*" --> > 3000 matches

Computer algebra meets analytic functions.

  • All analytic functions are based on Taylor--, Laurent--, and Puisseux series
  • Puisseux series have fractional powers or logarithms
         --> branch cuts of all functions follow uniquely from branch cuts of Log and Power

Example: [Graphics:../Images/TalkGD99_gr_5.gif]                  textbook branch cut: [Graphics:../Images/TalkGD99_gr_6.gif]



goal          :   Riemann surfaces for all special functions
this talk   :   compositions of elementary functions

Poster Photograph

Interactive 3D graphics and contour integral representations of special functions.

The beta function B[Graphics:../Images/TalkGD99_gr_9.gif] can be represented by the following integral


Here C is a contour which encloses the point [Graphics:../Images/TalkGD99_gr_11.gif] and [Graphics:../Images/TalkGD99_gr_12.gif] in a way which is topologically equivalent to the following:


An example showing this path on a Riemann surface:


A simple example: [Graphics:../Images/TalkGD99_gr_15.gif]

Find a parametrization.



Choose the imaginary part of [Graphics:../Images/TalkGD99_gr_18.gif]as a representation of the Riemann surface.
Rewriting [Graphics:../Images/TalkGD99_gr_19.gif] in polar form as [Graphics:../Images/TalkGD99_gr_20.gif] --> parametric representation [Graphics:../Images/TalkGD99_gr_21.gif]:



A more complicated example: [Graphics:../Images/TalkGD99_gr_24.gif] (or [Graphics:../Images/TalkGD99_gr_25.gif])

Find all branch cuts.



Using ContourPlot one can see the branch cuts nicely--they form clusters of contour lines.







Further pictures with simple branch cuts

One more example (accidentally easily doable): [Graphics:../Images/TalkGD99_gr_34.gif]

Use natural parametrization.




The inverse trigonometric functions


A hopeless example: [Graphics:../Images/TalkGD99_gr_41.gif]



Introduction | Construction Method | Implementation | Outlook |