Complex Mappings

What is a complex mapping?

A complex mapping is a function that takes a complex number as input and returns another complex number as output. Visualizing such a function is a bit more complicated than visualizing a real-valued function of a real variable, whose graph is simply a curve in the plane. The graph of a complex mapping would be a surface in 4-dimensional space, which is not so easy to see! Therefore it makes more sense to viualize a complex mapping another way, namely by observing how it acts on a region in the plane. For example, the picture below illustrates how a complex mapping (called f ) deforms a square in the plane. Complex mappings are the basic objects of study in complex analysis.

The Program

I wrote a Java applet to visualize such mappings. There are three types of mappings to choose from: polynomials, Mobius transforms, and exponentials (more on these later). Each type has 4 complex number parameters that can be modified to further alter the mapping. These mappings act on regions of the complex plane, which are represented in the program by rectantular grids, as in the above image. For the sake of reasonable computation time, the mappings in the program only act on the vertices of the grid. After that, the program simply "connects the dots" to create the final image. There are controls to adjust the resolution of the grid, i.e., the number of vertical and horizontal lines. There are also controls to determine which region in the plane this grid represents, and which region in the plane is actually visible. The coordinate axes can be turned on or off, the size of the output image is adjustable, and the output image can be saved as a png file.

You can download the files for the applet here:

• Mesh.class (the main class file)
• Mesh$1.class (an auxillary class file)
• Mesh$2.class (an auxillary class file)
• Mesh$3.class (an auxillary class file)
• Mesh$4.class (an auxillary class file)
• ComplexNumber.class (an auxillary class file)

You'll need the Java Runtime Environment to run it. The applet is started with the command

java Mesh

from the command line.

If you'd rather look at and/or compile the source code yourself, the two files needed are:

• Mesh.java (the applet is in this file)
• ComplexNumber.java (this defines a complex number class)

You'll need the Java SE Development Kit. Compile it with the command

javac Mesh.java

from the command line.

Polynomials

Complex polynomials are the same as regular polynomials over the real numbers, except the coefficients and variables are now complex numbers. The generic form in the program is a degree 3, i.e., a cubic:

Here are a few examples of how complex polynomials behave.

Mobius Transforms

A Mobius transformation (also called fractional linear transformations) have real analogs, but they are not as interesting. The complex versions, however, constitute a very important and interesting class of functions. An amazing video describing their nature can be found here.

Here are a few examples of how Mobius transformtions behave.

Exponentials

The exponential function is perhaps the more important function in mathematics. The complex version combines the "exponential growth" familiar from the real version with rotation in the plane. In fact, it is periodic, something one might never guess from looking at the real version.

Here are a few examples of how complex exponentials behave.