How to interpret the plot of a function of a complex variable?

Question

I know what a complex number is: $a+bi$. But I have seen these functions that make no sense to me, something such as this: $$f(z)=z^2+1$$ where $z$ is a complex number.

Does this have to do with that plane where the "y"-axis is Real numbers and the "x"-axis is Imaginary numbers? I typed it into this complex grapher http://davidbau.com/conformal/#z%5E2%2B1 and I am utterly confused. What's with all the colors? What's with that weird 8 shape?

Answer 1

One way to imagine a map $f : D \rightarrow \mathbb{C}$ with $D \subseteq \mathbb{C}$ is to think of it as a 2-dimensional vector field. Remember that $\mathbb{C}$ is just the vector space $\mathbb{R}^{2}$ equipped with a special multiplication

$$ * : \ \mathbb{R}^2 \times \mathbb{R}^2 \rightarrow \mathbb{R}^2 $$ $$(v_1,v_2)*(w_1,w_2) = (v_1w_1-v_2w_2,v_1w_2+v_2w_1)$$

that makes $(\mathbb{R}^2,+,*)$ a field. The imaginary unit $i$ is not some magical object, but just a notational symbol to simplifly the practical use of the above multiplication. Therefore $f$ just maps a set $D\subseteq \mathbb{R}^2$ into the $\mathbb{R}^2$. For every vector $v \in D$ you can visualize $f$ by mentally attaching $f(v)$ to $v$. If you would plot this, you would get a cohort of arrows in the plane, all starting somewhere in $D$.

Answer 2

I like @Joker123's suggestion of visualizing complex maps as vector fields, but I want to point out that this is not what is done in the tool you linked at davidbau.com.

The plot at davidbau.com is drawing the preimage of a flat grid and unit circle under the input map. If you try the identity map ($z$), you'll see that flat grid and unit circle. If you try a translation (e.g. $z+1$) you'll see that the grid is still composed of straight lines, but the circle in the middle has been shifted. Note that for $z\to z+1$ the circle is shifted to the left not the right because this is a preimage.

If you mapped the points of the big grey figure eight in your plot by $z\to z^2+1$ they would form the unit circle. Similarly, if you applied $z\to z^2+1$ to all the points in the curved grid of your plot, the result would be the same flat grid that's drawn for the identity map.

These sorts of plots are frequently used to visualize conformal maps.

Answer 3

In complex functions, every complex number in $\mathbb{C}$ is assigned another number in $\mathbb{C}$. The resulting curve cannot be represented in a 2D or 3D plot, since both domain and image have two real dimensions.

One way around that is to use colors. In the polar representation of $f(z)$ $$ f(z) = r \exp(i \phi) $$ associate "brightness" with the absolute value of $r$ of $f(z)$ and "hue" with the argument $\phi$. This idea is called domain coloring.

I recently wrote cplot, a Python library for domain coloring, which should hopefully make creating and understanding such plots easier. For your function:

import cplot

plt = cplot.plot(lambda z: z ** 2 + 1, (-2.0, 2.0, 400), (-2.0, 2.0,400))
plt.savefig("out.png", bbox_inches="tight")
plt.show()

The dark areas are zeros of your function ($\pm i$, makes sense).