Introduction

Not too many years ago fractals became very popular because they looked very pretty when generated by computer graphics even though the vast majority of the people generating them had no idea what was their significance.

Graphs of functions of a complex variable are also pretty but the concept is more basic and easier to understand. It’s my hope that these graphs will be, not only aesthetically appreciated, but also fundamentally understood. For this purpose I would like to compare them with graphs of functions of a real variable, which are understood by most people who studied them in a pre-calculus course. I will start by reviewing these real functions.

We start with two real axes, one is called the domain, or x-axis, and the other, the range, or y-axis. We first calibrate these axes. See Figure 1.

Figure 1

Figure 2

Now, let’s review plotting by using an example. Let’s plot the function y = x2. Using a "string" connect 0 in the x-axis to 0 in the y, 1 to 1, 2 to 4, -1 to 1, -2 to 4, etc. See Figure 2.

You say you never did it this way? You’re right! That’s because someone by the name of René Descartes thought about placing the y-axis vertically, and then plotting as you already know how. See Figure 3.

Figure 3

Figure 4

By doing it this way, one does not need to "tie strings" from one number in the x-axis to the corresponding one in the y-axis. It’s easy to tell what goes into what simply by starting at the value in the x-axis, travelling vertically up (or down) to the graph and then moving horizontally to touch the y-axis. Wherever it touches the y-axis is the corresponding y value. See Figure 4.

A complex number, z, has the form x+iy, where x and y are real and i is . In complex plotting, instead of two real axes there are two so-called complex planes, the domain is called the z-plane and the range, the w-plane. Just like every real number can be represented by a point in the real line and every point in the real line represents a real number, every complex number can be represented by a point in the complex plane and every point in the complex plane represents a complex number. A complex plane has two perpendicular axes, the real and the imaginary ones. The point z = x+iy is located the same way that (x,y) is located in real variables. See Figure 5 for examples.

Figure 5

There is no method comparable to Descartes’ procedure for plotting complex functions. Instead, plotting in this case is done analogously to the way we plotted y = x² using two horizontal axes and pieces of "string" to show the correspondence. However, our clever "trick" here consists in using different colors to show the correspondence between domain and range points instead of "strings". So that, if a point in the w-plane is red it’s because "it came" from the red point in the z-plane. Therefore, except for the striped pattern maps, every pixel in the domain and its corresponding pixel in the range have their own unique red, green, and blue components. For the striped pattern maps, this scheme is used for each pair of corresponding stripes.

To give the reader an idea how the range points are calculated, consider the function w = z², where z = x+iy and w = u+iv, where x, y, u, and v are real. Then,

Now, by equating real and imaginary parts, we have

These two equations give us a way to calculate the value of w for each value of z. For example, if z = 2 - i, then x = 2 and y = -1; u = 2² – (-1)² = 3; v = 2(2)(-1) = -4; and w = 3 – 4i. See Figure 6 for the plot of this point.

Figure 6

Many textbooks show plots of functions of a complex variable but since they are printed in black and white only the few points that are labeled exhibit their correspondence. Hopefully, the day will come when color printing will be cheap enough so that these plots, also called maps, will be printed in color in order to use color as another dimension.

Here’s a collection of such plots for your enjoyment. Just click on the function in the menu of functions whose plots you would like to see, and then, on the coloring scheme you desire. Please allow plenty of time to download the animations. A typical MATLAB program to generate the maps is available for your perusal.