# Exponential Sums

Posted on Aug 4th, 2019

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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