In my review of Tristan Needham's Visual Complex Analysis, I introduce a new way to use color in visualizing complex-valued functions in the plane.
Here is a brief description.
To visualize a complex-valued function in the plane we use what we call a domain coloring diagram: for each pixel in the domain of the function compute the color associated with that input value and use that color for that pixel.
Next, observe the function f(z)=(z^{2}-i)/(2z^{2}+2i). This rational function has two zeroes (the white points) and two poles (the black ones). Each is simple. |
Notice that in a neighborhood of the origin, the cyan color appears constant. This leads us to guess that the derivative of this function is zero there. We make a domain coloring diagram of f(z)-f(0) and observe a double zero: indeed the derivative is zero at the origin.
If you want to learn more about geometric approaches to complex function theory, I highly recommend Tristan Needham's wonderful new book Visual Complex Analysis. You can read portions of the text and get information about ordering from the web site linked to the title.
Link to homepage for Frank Farris.