Interpreting a complex valued function as a vector field over its domain provides a wealth of opportunities for producing visually appealing images. Starting with a plot of the vector field over a rectangular grid and assigning colors as functions of length and/or direction we easily discern the multiplicities of the zeros and poles of the function. Plotting the vector field of the function only along certain paths, inventing functions for assigning length and color to each vector produces some beautiful images.

For our purposes we will interpret a complex valued function of a complex number, w = f(z) as a vector in the complex plane emanating from z = x+iy and we will restrict the domain to a small region that contains the “interesting” points, such as zeros or poles of f