Visually Identify Features
You can identify multiple features in a complex function from its plot.
Looking at the plot of , you can easily see the zeros at the third roots of unity and the four poles at .
From the plot of , you can note that there are zeros of order two at and poles of order three at when is a non-positive integer.
The plot of shows more complicated behavior. You can see that there are no zeros, but there is a pole of order 3 at . There are also four poles of order one along each angle and . There are also poles of order one along the angles 0 and that accumulate at essential singularities at and .