## Color Graphs of Complex Functions |
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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 e 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. |
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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)
= (z^{2} + 1)/(z^{2} - 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. |
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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 |