A gallery of holomorphic and meromorphic functions

My first course in complex analysis involved almost no pictures. This is the first post in a series dedicated to distilling a few neat properties of analytic and meromorphic functions by letting pictures and videos do as much heavy lifting as possible.

A survey of complex functions

We begin with the identity function $f(z) = z$.

This form of domain coloring is essentially a kind of polar coordinates where the “base color/shade” is an angular coordinate and the “darkness” is the distance from the origin. This is why when we zoom out, most of the picture is white (large magnitude).

A key here is that near the origin (the only zero of the identity function), a simple closed counterclockwise curve traces out red, green, blue, violet, then back to red. In some sense, we “climbed up from red up to blue and fell back down to red, doing no net work.” I will call this “positively oriented” to mean “seeing colors in the order of ROYGBIV” as one travels around a curve counterclockwise around a point.

In some sense, a complex analytic function is one where the “color changes sufficiently smoothly and is always positively oriented around a given point.”

Simple Zeros and Poles

A zero and pole can have a “multiplicity” which basically means that there is a constant factor of “extra angular velocity” when traveling around the zero.

There is an analogous image whereby rather than a zero where the orientation is “red->green->blue”, we have a pole where the orientation is “blue->green->red.”

Let’s next admire even the simplest kind of polynomial/series.

$$ f_n(z) = \sum_{k=0}^{n} z^k $$

Note that we essentially have an increasing number of “positively oriented” zeroes around the unit circle. We get one zero for each added term of the power series, all along the unit circle. As we add more terms, the vanishing locus will be dense in the unit circle. As it turns out, this is an obstruction to anayticity. The grug reason is that in this imaginary “limit” of the power series, when we trace out a simpled closed curve around a point on the unit circle, we will not be able to go “red green blue.”

Triginometric Functions

We can interpolate between the rotated sine functions.

$$ f_0(z) = \sin(z) \quad f_1(z) = \sin(iz) $$

And similarly for cosine functions.

These functions have large magnitude/absolute value outside of an axis. We can see that these functions are analytic because even though at every zero/black point, the phase alternates between a ‘starting state’, the color orientation when traveling counterclockwise around a zero is always “red->green->blue.”

The tangent function is especially neat because it alternates between a zero and a pole on its ‘critical axis.’

We can quickly internalize the fact that $\sin(z)$, $\cos(z)$, $\sinh(z)$ and $\cosh(z)$ are related via translation and rotation.

Namely, we have the following.

$$ \cosh(z) = i\cos(iz) \quad \sinh(z) = -i \sin(iz) $$

In particular, in the graphs of $\cos(z)$, $\sin(z)$ and $\cosh(z)$, we observe red and blue along the “critical” axis, whereas the “critical axis” of the $\sinh(z)$ graph alternates between blue and green. The green basically comes from multiplying $\sin(z)$ by $-i$. In the very first graph we analyzed for $f(z) = z$, notice that multiplying by $-i$ is rotating clockwise by $\pi/2$. This brings red to blue, and blue to green. This reflects the fact that $\sinh(z)$ and $\cosh(z)$ might be defined (though I wouldn’t do so) as the odd and even parts of $\exp(z)$, respectively.

I will write more about these four functions in another post because they are seriously OP.

A brief aside on how to plot complex valued functions

TLDR: you freeload on the work of computer engineers who give us tremendous hardware. Samuel Li’s WebGL Complex Function Plotter ¹, under the hood, involves directly putting $r$ and $\theta$ in the cylindrical HSV color space , some sampling of the grid, and lots of scalar additions and multiplications.