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Hans Lundmark's complex analysis pages

1. Introduction

Example. Below are two plots of the function f (z) = sin(z3−1) / z :

Virtually all information in the real plot can be read off from just the middle horizontal line z = x+0i in the complex plot (once you learn how to interpret it). But the complex plot also reveals new phenomena that can't be seen in the real plot.

Real plot of f f(z)=sin(z^3-1)/z Complex plot of f

2. My stuff

The HTML files are produced with the TtH TeX-to-HTML translator, which uses symbol fonts to render mathematical characters. The versions differ in how the math symbols are encoded. See the TtH manual for details.

You can also get the LaTeX source and a typeset DVI version (without the images).

3. References


There are lots and lots of books about complex analysis. There is one which should be of particular interest for visitors to this site:

Here are a few other books that I can recommend:

Internet sources

The first web page I saw on domain coloring was the one by Frank Farris, who also coined the term. There he shows the idea, but his pictures are a little primitive. Other people, including me of course, have tried to make better pictures. Here are some that I am aware of (in no particular order):

4. Technical details

Software used

Most pictures were made on a PC running GNU/Linux, using the GNU Image Manipulation Program (GIMP). For the actual domain coloring stuff, and for the y = f (x) plot above, I used the MathMap GIMP plugin by Mark Probst, slightly hacked by me to allow the elementary functions to take complex arguments. These changes were then incorporated into MathMap v0.12 (a long time ago), and the plugin now also comes with example files for making domaing coloring plots.

Some of the more recent pictures were made with a buggy and ugly Python program that I wrote, since I had problems with MathMap for a while (old versions of MathMap not working with new versions of GIMP). If I can pull myself together some day and tidy up this Python program (not likely to happen very soon) I will make it available here, but right now I'd like to keep it to myself. But you don't need to worry about that, since nowadays MathMap development is active again, so you should be able to use that if you want to make your own plots. You can also try Michael J. Gruber's Python-Fu GIMP plugin.

The Julia set was drawn with the Fractal Explorer GIMP plug-in. The formula in the example above was generated by the TeX input for GIMP plug-in written by Dov Grobgeld.

The HTML in the main document "Visualizing complex analytic functions using domain coloring" was generated by Ian Hutchinson's TtH TeX-to-HTML translator. The LaTeX source for that, like the HTML on this start page, was written using Emacs.

(Obsolete) Scheme script for making domain coloring plots

Here is a script that used to work once upon a time, with older versions of GIMP and MathMap. If you want to try it out, good luck, but I can't give any warranties or support.

You need GIMP with the MathMap plug-in installed. Download the Scheme file and put it in the GIMP scripts directory. The next time you start GIMP, a menu entry Script-Fu/MathMap/Domain Coloring should appear in the Xtns menu in the toolbox.

Original version 2000-08-10. Last modified 2022-02-05. Hans Lundmark (