## A tool to create images and video of complex functions as they approach 0, useful for visualizing singularities. |

### 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.

Instead of domain coloring, the program just transforms the following reference circle with the given function:

### Movies

#### Pole of Order 6:

#### e^(1/z):

**tan(1/z)**

### Source Code

https://github.com/plurSKI/singularityViewer### 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):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).

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