visualize complex functions with
domain coloring
First, we associate a color with each point of the complex plane.
The hue cycles once around the rainbow as we go around the origin
(red for the point 1); we approach white as we approach the origin,
and we approach black as we go to infinity. I included the axes
and the unit circle to help with identification of points.
Now, to illustrate a complex function f(z), we draw a plane. At each point
z of the plane, we use the color for the value f(z).
Below is the domain color diagram for the function f(z)=z2.
It is zero at the origin, shown by the white color. Note that as we move
around the origin, the colors cycle through the rainbow twice.
This is the sign of a double zero.
Next is a sixth-degree polynomial, with one triple zero and three simple
zeros. See that the colors cycle through the rainbow three times
as we move around the point z = -1.
Below is another presentation of the same function. On the left is
the z-plane, and on the right is the w-plane. If they are related by
the equation w=f(z), then corresponding points have the same color.
Note that more than one z-value may correspond to the same w-value,
and thus colors may be repeated in the left picture.
Next is the function 1/z. It gets black as we approach the origin,
and gets white as we go far away from the origin. The black spot
at the origin illustrates a pole in this function. The colors
cycle once as we go around it, so it is a simple pole.
Now consider the function f(z) = (z2-i)/(z2+i).
There are two simple zeros and two simple poles. Can you recognize them
from the colors?
The derivative of this function is 0 at the origin.
Next, we show f(z)-f(0) to illustrate this.
Next is the complex cosine funtion. If you move the whole thing
to the right by the appropriate amount, it repeats itself. This
shows the periodicity of the function. Note all the simple zeros
on the real axis.
Next is the complex exponential function exp(z). If you move upward
by a certain distance, the picture repeats...the function is periodic with
purely imaginary period. The function approaches 0 as you move to
the left, and approaches infinity as you move to the right.
Here is the complex logarithm. Note that there is an abrupt color-change
when you cross the negative real axis. This function (the principal
value of the logarithm) is discontinuous on the negative real axis.
Can you recognize the simple zero at z = 1 ?
Below is an interesting one... f(z) = exp(1/z) . What happens at the origin
here? It is not a zero, and not a pole. We call this an essential
singularity.
Below is the same function exp(1/z) but magnified by a factor of 100.
In every neighborhood of the origin, this function takes all values (except
zero).
An alternate coloring scheme: instead of continuous shading, only a certain
limited number of colors...
Others, using this coloring scheme...
More domain coloring pictures by Jan Hlavacek
...
Martin Pergler
...
François Labelle
...
Hans Lundmark
...
more Frank Farris
email: edgar@math.ohio-state.edu