Back in June of 2009, I wrote about the issues involved in graphing complex functions. Since then, I’ve been showing this material to my students and discussing the relationship between graphing real-valued functions and graphing complex-valued functions.
As part of these talks, I’ve had to make explicit the difference between the Cartesian Plane and the Argand Diagram. I’ve been doing this by showing the mapping of the points from the x-real number line to the y-real number line. George Abdo and Paul Godfrey have a nice website that shows this process.
OK, then if we cross the x and y number lines to create the Cartesian Plane, we’ll see the picture of the graph that we’re used to.
Then, we have to consider the complex mapping. In this situation, each x value is two-dimensional and is mapped to a two-dimensional y coordinate, like so:
(Click the picture for a better view, or check out the link to Prof. Abdo’s website)
But to show these together, as the Cartesian Plane does for real-valued functions would require 4 dimensions, which creates difficulty for human beings who normally have enough trouble with 3 dimensions. What people have done instead is to take the x values from the complex plane and color them based on their corresponding y values.
Lawrence Crone at American University has some nice pictures showing this effect.
Andrew Bennett at Kansas State University has a nifty online complex graphing calculator on his website. You can type in a formula and the software will show you a representation of the mapping. I prefer the top view to see the roots, but the side view is interesting as well.
The graphing utility window is limited to complex x values a+bi in which a and b are both between 2 and -2, but this can be adjusted by right clicking on the window.
Something that shows up nicely on the complex grapher at the previous link are roots of unity – typing in z³-1 will show the cube roots of 1, both real and complex…