Plotting a Function of a Complex Variable

Question

I've been teaching myself complex analysis using "Introductory Complex Analysis" by Richard A. Silverman, and I'm a few chapters in and still have no clue how to plot anything. I know that

$$ z = x + iy $$

corresponds to an imaginary plane and a real plane, ie

$$ z = x^{2} + iy $$

Gives a parabolic plane for the real part, and a flat plane for the imaginary part. The issue I'm having is plotting

$$ w = f(z) = u+vi $$

where

$$ u\rightarrow u(x,y) \wedge v\rightarrow v(x,y) .$$

I get the idea that 1 function of a complex variable produces two graphs that can be regarded as real, so would I be correct in assuming that:

$$ f(z) = w = z^{2} = (x +iy)(x +iy) = x^2 + i2xy -y^{2} $$

thus

$$ u=x^{2} - y^{2} \wedge v=2xy $$

as applied to the definition above, giving me the real plot "u" and an imaginary plot "v", are the only two graphs? Also, which -- if either -- is to be regarded as the w plane and the z plane?

Answer 1

Firstly, the collection of points $x+iy$ gives you a single plane. It has in it the real axis and the imaginary axis, but it's just one plane.

The graph of a function $f:\mathbb C \to \mathbb C$ is a four dimensional object (since $\mathbb C$ is a two-dimensional object). So, if you are having trouble visualizing the graph it ok - it's just because most people are not very good at visualizing in four dimensions.

As you say, every function $f:\mathbb C \to \mathbb C$ gives rise to two real valued functions $u,v:\mathbb C \to \mathbb R$. The graph of each of these is a three dimensional object that can be plotted. The two graphs together determine the given function $f$ but the the visual information given by the geometry of these component real-valued functions' graphs is quite limited if you want for the study of $f$ as an analytic function. I hope this helps clearing some of the confusion.

Answer 2

There are (at least) two approaches to this problem:

• domain coloring
• plotting contour lines at equal absolute value and/or equal phase

With cplot (a Python package that I wrote) you can use both of these; for example for $$ f(z) = \frac{\sin(z^3)}{z} $$

import cplot
import numpy as np


def f(z):
    return np.sin(z ** 3) / z


cplot.show(f, (-2.0, +2.0), (-2.0, +2.0), 1000)