Visualising functions from complex numbers to complex numbers
I think that complex analysis is hard because graphs of even basic functions are 4 dimensional. Does anyone have any good visual representations of basic complex functions or know of any tools for generating them?
Solution 1:
One way that functions from C to C can be represented is to show the image of a grid. That is, plot the images of the lines x = constant and y = constant under your function, where z = x + yi.
Another is what Wikipedia calls domain coloring (see this article by Hans Lundmark for a more detailed exposition. The idea here is to color the range of the complex function -- since color space is three-dimensional there are multiple ways to do this -- and then color each point in the domain by the color of the corresponding point in the range.
Solution 2:
For Moebius transformations, check out this nice YouTube video.
Solution 3:
Open two browser windows side-by-side and use wolfram alpha. Recast your function f(z) as f(x+iy) and plot the real part in one window and the imaginary part in another. My example links provide plots for z3.