A tool to create images and video of complex functions as they approach 0, useful for visualizing singularities.
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.
Instead of domain coloring, the program just transforms the following reference circle with the given function:
Pole of Order 6:
ConclusionIt 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):
you will notice it is not approaching infinity like (1/x) would, but instead oscillating as it gets close to x = 0. 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).