In a previous post I explored complex functions through domain coloring. Here I would like to explore another element of complex functions – conformal mapping. The idea here is to consider some curve in the complex plane, and apply a function to it, and see what happens.
The panel on the left represents the complex plane. The intersection of the back lines is the origin. I’ve drawn a straight green line through a point. The panel on the left represents the image of the panel on the left under the complex function z = f(z) = z^2. The black curve in the right panel represents the image of the green line from the left panel under the above transformation. So as we see the function z = f(z) = z^2 takes a horizontal line to a parabola.
Technical point; When I was in high school I was exposed to an interactive geometry program called The Geometers Sketchpad. I spent countless hours building dynamic geometric constructions. Dynamic in this context means that you can drag the points around on the screen and all the lines etc.. that depend on that point adjust themselves accordingly. My goal with this project was to create that type of environment in which to explore complex functions. That little black circle in the left panel defines the green horizontal line. The above is a screenshot from a program I wrote in C++. If you click on that black circle you can drag it around on the screen. The green line will always pass through that point. The parabola in the right panel will adjust itself accordingly. I wrote this program in C++ using the SFML library, and I am pretty proud of this programming accomplishment.
Now I’ve added a vertical line. Notice how this also gets transformed to a parabola, just that the new one opens to the left, and the old one opens to the right. Also Note that the new parabola is much narrower that the old one. This is because the vertical line is closer to the origin.
I’ve dragged the vertical line to coincide with the y-axis. Note that the image in the right pane is now just a ray originating from the origin.
Now the vertical line is on the other side of the y-axis. Note how again we get the same parabola as we had when the vertical line was the same distance away from the y-axis, but on the other side of the y-axis.
Look at how those parabolas intersect. Can you guess the angle of intersection? Looks to be about 90 degrees. These are called conformal mappings because the transformation preserves the angles of intersections.
Pretty boring so far.
Now the fun begins.
All of the above have been transformations using z=f(z)=z^2. Now we can try different functions. Here is z = f(z) = sin(z).
Here is z = f(z) = tan(z). This one happens to be my favorite, especially the image of the diagonal lines.
Here is z = f(z) = ln(z)
This is z = f(z) = 1/z. This one is also pretty cool.
As you can see, the fun never has to stop. In the program I wrote you can drag around the shapes and see the transformed shape move around on the other panel.