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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?

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    $\begingroup$ This might be helpful. $\endgroup$ Commented Feb 13, 2013 at 6:32
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    $\begingroup$ Welcome to MSE! As an editorial suggestion, if you put your equations between double dollar signs, they center nicely. $\endgroup$ Commented Feb 13, 2013 at 6:34
  • $\begingroup$ See Domain coloring. $\endgroup$ Commented Jul 6, 2016 at 22:39

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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.

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  • $\begingroup$ For the most part. This particular text references w and z planes. From what I've stated above, what variables make up which plane? $\endgroup$ Commented Feb 13, 2013 at 6:30
  • $\begingroup$ The $z$ plane is usually referring to the domain of $f$ and the $w$ plane to the codomain. $\endgroup$ Commented Feb 13, 2013 at 6:35
  • $\begingroup$ Ah. So the z plane plane has axes (x,iy), where the w plane has axes (f(z),z) then? $\endgroup$ Commented Feb 13, 2013 at 6:40
  • $\begingroup$ not quite. The $w$ plane has coordinates (t,s). The graph of the function is the set $\{((x,iy),f(x,iy))\}$, which consists of four coordinates in total since $f(x,iy)$ is in the $w$ plane. $\endgroup$ Commented Feb 13, 2013 at 7:03
  • $\begingroup$ (t,s)? Are those just arbitrary? $\endgroup$ Commented Feb 14, 2013 at 1:52
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You can think of complex functions as maps from one plane (say, $z$) to another ($w$). Both of these planes have a real axis (usually shown horizontal) and an imaginary axis (usually vertical). Maps from 2D space to 2D space are hard to picture but it can be done in a number of ways. One of the most helpful (for me) is using color to represent the output of the function (called domain coloring). You can try it here, and if you want more functionality (like showing the magnitude as contours) I put together some Python code here.

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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)

enter image description here

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