Singularity Viewer

Overview

In Complex Analysis you are able to work with mathematical singularities. They are classified into three main categories:

• Removable: The function can be rearranged to remove it
• Pole: Has a finite number of negative terms in the Laurent series. The simple case of these type of functions are polynomials with negative degrees.
• Essential: The Laurent series has an infinite number of negative terms. More-or-less, no matter how many times you integrate the singularity remains.

This program attempts to examine these functions by taking a circle centered around the origin and making the radius closer and closer to 0 while keeping the image size the same. The 'zoom' also tries to keep values visible by scaling up the pixel mappings. In short it's a visualizer for very small values in the complex plane.

Instead of Domain Coloring, the program just transforms the following reference circle with the given function:

Movies

Source Code

Conclusion

It seems that most poles 'repel' things from the location of the singularity (ie. Pushes them towards infinity). On the other hand, essential singularities don't have a predictable behavior. This is easy to see on the non-complex plane if you look at the graph for sin ( 1 / x):

This is also to be expected since essential singularities take on all values (except possibly one) of the complex plane infinitely many times (via Picard's Great Theorem) This means if I was able to plot infinite points of exp(1/z) on the computer, the entire complex plane would have every color in my circle except the origin (0 + 0i). But since I'm only doing 2 million points per image I am limited to the few I actually compute (which is what the images and movies are showing).