How to plot complex functions on the paper by your hand?

Question

I want to know the exact method of plotting complex function used by human, computer, and whatever who can do mathematics. For example how should I plot this : $w = u+iv$ , $z = x+iy$ , $w= f(z)= z^2$ I'm completely confused imagining the complex functions and I want to know how you would imagine such functions and do mathematics with it. Thanks in advance

Answer 1

You don't plot these. To plot them would require $2$ axes to plot the real and imaginary components of the inputs and they it would require another $2$ axes to plot the real and imaginary components of the outputs, totaling $4$ axes. However, we are unable to plot in $4$-dimensions in our $3$ dimensional world. So we must make a choice: plot the imaginary part of the output or the real part of the output. For example, take the function $f(z)=z^2$. Then we have $$ f(2+i)=(2+i)^2=4+4i-1=3+4i $$ We could then plot the imaginary part of the output, $4i$. So this would be the same as plotting the point $(2,1,4)$ in $\mathbb{R}^3$.

NOTE. This isn't the only thing we could plot. For example, another common choice is to plot the absolute value of the output. In our example above, we would have $f(2+i)=3+4i$. Then we know that $|3+4i|=\sqrt{25}=5$. So we would plot the point $(2,1,5)$. It all depends on the choice of the final variable to plot while the $2$ first axes are almost always the real and imaginary components of the input, respectively.

Answer 2

I just wanted to point out that I wrote a Python package, cplot, that makes plotting complex-valued functions fairly simple. It combines domain coloring and contour lines for constant arg/abs. e.g., for $\sin(z)/z$:

import cplot
import numpy as np


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


plt = cplot.plot(f, (-7.0, +7.0, 400), (-7.0, +7.0, 400))
plt.show()

Answer 3

You can plot such functions! Look at https://people.math.osu.edu/fowler.291/phase/

for instance. The color at a point tells you the phase of the image of that point. To see what the base "phase chart" is, just plot the identity function $z$.

If you wanted phase and modulus info, you could do a 3D plot colored by phase. I did that too using webGL and cannot find it now. Will update this later when I do...

Answer 4

Like Steven Gubkin says, coloring by phase can give you excellent geometric intuitions. I put together this Python code while I was taking complex analysis, let me know what you think: https://github.com/seaplant3/complex-plotting

It uses contours (like an elevation map) to show magnitude, I found that was the easiest way to keep everything visible.