TITLE: How to plot a complex function?
QUESTION [6 upvotes]: We cannot plot graph of a complex function $f:\mathbb {C\to C}$ as it requires $4$ dimensions.But we can show how the mapping transforms the domain plane into image plane.We can draw grid lines parallel and perpendicular to $x$-axis and see how the grid lines are modified.But often it becomes tedious task to plot these kind of diagrams.Is there any systematic procedure to draw such figures without help of any software?
For example , $z^2,z^3,\sin(z),\log(z),\exp(z)$ etc.
I want a method to visualize any given function.Is there a way out?

REPLY [3 votes]: Another popular way for plotting complex functions is domain coloring, which associates brightness with the modules of $f(z)$ and hue with its argument.
I recently write cplot, a Python package which makes creating these plots easy. For example, for the natural log:

import cplot
import numpy as np

plt = cplot.plot(np.log, (-2.0, +2.0, 400), (-2.0, +2.0, 400))
plt.show()

cplot's gallery has many more cool examples.