03
This image shows the third power of 1/z. The singularity at z=0 is a pole of third order. A pole of third order appears as a white spot which represents the value "complex infinity". On a small circle around the pole all colors appear exactly three times.
The absolute value of the function decreases very rapidly with increasing absolute value of z. If |z|=3, the absolute value of f(z) is 1/27 which appears as black in the standard color map. The Mathematica package VQM`ComplexPlot allows one to adjust the standard color map if one wants to investigate functions in regions where the values are very small or very large.
Note that the absolute value of this function is spherically symmetric. While the computer defined "lightness" of the colors depends only on the distance to the origin, the visually perceived brightness depends on the color. This gives the pole a star-like appearance. While this property of the colormap is a disadvantage here, it is an advantage in other situations, because it allows to identify the lines with constant argument Pi/3, Pi, 4 Pi/3 with good accuracy.