# The Complex Grapher

0+0i

Function:

Magnitude modulus:

Top left corner:

0+0i

Bottom right corner:

0+0i

Created by Tal Brenev

## What is the Complex Grapher?

The Complex Grapher creates visualizations of complex functions (i.e. functions of a complex variable). If you are unfamiliar with complex numbers and functions, you can click here to learn more about them.

This program uses the domain coloring method to graph complex functions. The graph is a representation of the complex plane: each pixel corresponds to a complex number $z$, with the pixel's horizontal and vertical position representing the real and imaginary components of $z$, respectively. The color of the pixel is determined by the value of $f(z)$, where $f$ is the function that is being graphed. Click here to read more about domain coloring.

## Using the Complex Grapher

You can enter any mathematical expression in the "Function" textbox. The expression must use $z$ as the input variable. The imaginary unit $i$ is also valid, as well as the constants $\pi$ and $e$, which can be expressed as "p" and "e" in the textbox. The Complex Grapher currently supports the following mathematical operations: addition (+), subtraction (-), multiplication (*), division (/), and exponentiation (^). Additionally, the following built-in functions are supported: sine (sin), cosine (cos), tangent (tan), and the natural logarithm (log).

The "Magnitude modulus" textbox must contain a number greater than 0. To find out what this value does, please read the section on domain coloring.

The "Top left corner" label specifies the complex number which is represented by the top-left pixel on the graph. The "Bottom right corner" label does the same for the opposite corner.

The arrow buttons shift the graph along the complex plane, while the plus/minus buttons zoom the graph in and out.

## Domain Coloring: How it Works

In domain coloring, each pixel on the graph corresponds to a complex number $z$. The pixel's horizontal and vertical positions represent the real and imaginary components of $z$, respectively. The value of $f(z)$ is then calculated, and the pixel's color is based on this value.

The color does not depend on the real/imaginary components of $f(z)$: the argument and magnitude of $f(z)$ are used instead. The argument determines the hue, while the magnitude determines the brightness.

We come across an issue here: the pixel's brightness has a maximum, while the magnitude of $f(z)$ does not. So, the magnitude is computed modulo some number $n$ in order to keep it within a certain range. The "Modulo magnitude" setting controls the value of this number $n$. If $m$ is the magnitude of $f(z)$, then the exact formula for calculating the brightness is:

$$b = \begin{cases} (m\% n)/n, & \mbox{if } (m\% 2n) \leq n \\ 1- (m\% n)/n, & \mbox{if } (m\% 2n) > n \end{cases}$$

where $b$ is the brightness, in the range 0 to 1 inclusive.

## An Introduction to Complex Numbers

The following is a brief explanation of complex numbers for those who are unfamiliar with the concept.

You've probably been taught in high school that you can't calculate the square root of a negative number. It's certainly true that the result won't be a real number, but what if we create an entirely new set of numbers to account for square roots of negatives? Let's start by creating a number which we will call $i$:

$$i = \sqrt{-1}$$

The square root of any negative number can now be written as a multiple of $i$. For example:

$$\sqrt{-9}$$ $$=\sqrt{9}\sqrt{-1}$$ $$=3i$$

Numbers that are a multiple of $i$ are not real numbers: hence, they are named imaginary numbers. $3i$, $5i$, $6.24i$, and $\pi i$ are all examples of imaginary numbers. The constant $i$ is known as the imaginary unit.

But what happens if we add a real number and an imaginary number together? The result is called a complex number. Any number of the form $a+bi$, where $a$ and $b$ are real numbers, is a complex number. $a$ is called the real part, while $b$ is called the imaginary part. Note that $a$ and/or $b$ can be equal to 0, and therefore all real numbers and all imaginary numbers are also complex numbers.

Arithmetic with complex numbers is very similar to arithmetic with real numbers. Basic rules of arithmetic still apply. For example, let's simplify the expression $(3i+1)(5i-7)$:

$$(3i+1)(5i-7)$$ $$=(3i)(5i)-(3i)(7)+(1)(5i)-(1)(7)$$ $$=15i^2-21i+5i-7$$ $$=-15-16i-7$$ $$=-16i-22$$

How can we visualize complex numbers? While real numbers are visualized as part of a number line, complex numbers are visualized as part of the complex plane: this is simply a Cartesian plane, with the $x$-axis representing the real part of a complex number, and the $y$-axis representing the imaginary part. For example, the number $3+2i$ is drawn on the complex plane below:

There is an alternative way to represent complex numbers. Instead of expressing a complex number in terms of real and imaginary parts, it can be expressed in terms of argument and magnitude. Consider the number $z=a+bi$, plotted on the complex plane below:

The angle $\theta$ between the arrow and the $x$-axis is called the argument of $z$. The length of the arrow, $m$, is called the magnitude of $z$. With some basic trigonometry, and the Pythagorean theorem, it is possible to express the argument and magnitude in terms of $a$ and $b$:

$$\theta=\text{tan}^{-1}(b/a)$$ $$m=\sqrt{a^2+b^2}$$