This visualization shows what happens when you have zeros and poles
in the complex function plane.
Each of the colored points either corresponds to a zero or a pole
of a rational function. The exponent of the point can be changed by the slider.
Up to four such special points can be visualized. The exponents $\alpha_i$ can eb changed
by the sliders.
\[
f(z):=(z-z_1)^{\alpha_1}+(z-z_2)^{\alpha_2}+(z-z_3)^{\alpha_3}+(z-z_4)^{\alpha_4}
\]
Negative exponents correspont to poles positive exponents correspond to roots.
The image of this map applied to a grid of polar coordinates is shown.
Moving the slider on the bottom lets the polar grid rotate around the origin.