### Color Graphs of Complex Functions

It is easy to set up a correspondence between the argument of a complex number and the hue of a color since both are measured as angles.  We use red to mean argument 0 (or 360) degrees;  green means 120 degrees, and blue means 240 degrees.  I have been using contour lines in saturation and value to represent the modulus of the complex value.  The contours distinguish moduli between 0 and 1, and between 1 and e, and e and e2, etc.  Moduli between 0 and 1 (or any of these ranges) are essentially wrapped around a rectangle in saturation-value space with one corner at saturation = 1 and value = 1, and the opposite corner at saturation = 0.3 and value = 0.6.

See Visual Basic code (.txt file).

The function f(z) = z  is plotted to the left below.  The contours appear as ridges with a shadow on the high modulus side.  Zeros always have a pinwheel of color, red to green to blue in a counterclockwise direction. For a simple zero there is one cycle of color; for a double zero two cycles, etc.  To the right below is a plot of a function with a simple zero, a double zero, and a triple pole.  Colors cycle in the reverse order around a pole, and there is always a set of modulus contours approaching the pole.

The plot to the right is the square root of the function above it, and shows several branch cuts.
The left plot below is of the function f(z) = (z2 + 1)/(z2 - 1).  The 'eyes' are poles, and the muzzle and forehead are zeros.  The plot on the right was carefully constructed with poles for eyes, and a zero for a nose.  The three teeth each consist of a zero just above a pole.
Where to go next:
A Gallery of G Functions     To see some very complicated graphs of complex valued functions
Alternatives                      To see an alternate method of plotting complex valued functions
Larry Crone Home Page
AU Mathematics and Statistics