# Exponential Sums

August 04, 2019

Exponential sums are an area of math represented by a series with terms that are complex exponentials. These types of sums are related to Fourier analysis and number theory. Plotting exponential sums produces various designs based on the type of input function.

An exponential sum is represented by the following equation

where the exponential is a complex number. The function $f(n)$ is a real-valued function defined for a sequence of positive integers. The exponential sum can be plotted in the complex plane as a progression of partial sums. The x-axis on the plot is the real part and the y-axis is the imaginary part.

## Python package

The Python package `expsum`

is a command line tool to plot and optionally animate the exponential sum for a specific function. See the exponential-sums repository on GitHub for installation and usage instructions. Examples of using the `expsum`

tool for various functions are shown below. An animated plot can be displayed using the optional `--anim`

argument. See the GitHub repo for more details.

## Function 1

```
>>> python expsum func1 2000 10 7 17
```

```
>>> python expsum func1 8000 11 21 31
```

## Function 2

```
>>> python expsum func2 1200 100
```

```
>>> python expsum func2 4000 800
```

## Function 3

```
>>> python expsum func3 1000
```

```
>>> python expsum func3 4000
```

## Function 4

```
>>> python expsum func4 4000 4
```

## Function 5

```
>>> python expsum func5 4000 50 100
```

## Function 6

```
>>> python expsum func6 2000 4
```

## Function 7

```
>>> python expsum func7 8000 4
```

## References

This article was inspired by John Cook’s blog post “Exponential sums make pretty pictures”.

- John D. Cook. Exponential sums make pretty pictures. October 7, 2017.
- David Angell. Exponential sums. School of Mathematics and Statistics, UNSW. Accessed July 7, 2019.
- Wikipedia contributors. Exponential sum. In Wikipedia, The Free Encyclopedia. Accessed July 20, 2019.

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