Visualizing Complex Functions with Phase Plots

This is the complex plane. The center of the image is zero, the real numbers extend to the left and right, and the complex axis up and down. The distinctive part of the graph, compared to common representations of the number line, is the color: every value is colored according to its angle around zero, also called its phase. The specific color assignment doesn't matter, just the progression of colors around the circle.

The dark lines overlay the grid of cartesian coordinates (x and y coordinates in the plane) and the light lines overlay the grid of polar coordinates (absolute value and phase).

With this association, we can graph more complicated functions:

This image plots the function $f(z) = z^2$. The square of any real number is positive, and in the original plot, the positive reals were colored blue, which is why the center line in this diagram is blue. On the other hand, the complex axis is now purple -- corresponding to the negative reals -- because of the relationship $i^2 = -1$.

$z^2 - 1$. Drag the zeroes to move them.

Here we have $z^2 - 1$, and we can start to see what information comes from these plots: the most important properties of a complex function are its zeroes and its poles, and anywhere that a complex function is zero, there will be a swirl of color in its phase plot, because zero is the only value that is next to every other color in the complex plane. $z^2 - 1$ is zero at $1$ and $-1$, which is visible in the plot -- a full color spiral comes from each point, pushing away from each other to either side.

All plots in this post are interactive -- drag the zeroes around to plot other nearby quadratic functions.

Try dragging one zero on top of the other. You'll end up with the same graph as $z^2$, with a double color swirl. As you pull them apart, the two color swirls point away from each other. But from a large distance, the two zeroes still look pretty much the same as $z^2$, with two copies of the full color spectrum coming out of the pair.

Poles

But zeroes are only half of the story. Here's a plot of $1/z$:

Now the cartesian grid projects onto a sequence of nested circles moving away to infinity as the function value increases near the origin. But the phase lines are almost unchanged -- there's still a swirl of color at the center, all that has changed is that the order of colors is reversed. In many ways, zeroes and infinities behave identically in complex functions, and phase plots help show that connection.

Try another demo, with two zeroes and one pole near each other:

$(z^2 - 4)/(z - 2i)$

With a little experimenting, you'll probably get the intuition that the zeroes are emitting phase lines and the pole is absorbing it, which is pretty much accurate. You can prove this formally, and most complex analysis classes will do this, but it's a lot easier to make it through those proofs with this kind of visualization in mind.

Generally, the intuition is: phase lines cannot cross each other, and phase is repelled from zeroes and attracted to poles (or vice versa, since the two end up being equivalent). This applies to any function that has a complex derivative, and it lets us think about complex functions in a more physical geometric way.

Here's a couple more demos to play with:

Five zeroes
Four zeroes and four poles