The beauty of complex functions
In math, a complex function is a function that takes a complex number a+bi and maps it to another complex number c+di. A single complex number x+iy can be visualized as the point (x, y) in a 2D plane. One way to visualize a complex function is to use a plane to represent the input space and the color of a point to represent the value of the function at that point. Let the distance between the output point (c, d) (let’s call it P) and the point (0, 0) (let’s call it O) be r, and let the angle between the line OP and the x-axis be θ. We use brightness to represent r (the brighter the color, the larger r is) and the hue to represent θ. Doing so, we get beautiful images, as well as insight into the behavior of the plotted functions.
Identity function
This plot is of the identity function, f(z)=z. The origin is dark, and the colors become lighter as you move away from the origin. The colors rotate once around the origin.
Sine function
This graph is of the sine function evaluated on complex numbers. The sine function is one examples of a periodic real function, and indeed we see the periodicity going left-right. However, the graph also clearly shows that the sine function is not periodic going up-down. The magnitude increases the further you go from the x-axis.
Another interesting feature of this graph are the points for which the sine is 0, visible as the black dots.If we look carefully, we see that the colors rotate exactly once around the black spot, corresponding to the fact that these zeros are of order 1.
Exponential function
This graph is of the exponential function. Going from left to right, we see the magnitudes increasing, as we expect. Going vertically, we observe that the exponential function is periodic, an important discovery by Euler and perhaps a surprise for someone not familiar with complex variables.
Elliptic functions
Both the sine and exponential functions are periodic, but they are periodic in only one direction. Are there differentiable functions that are periodic both vertically and horizontally?
Yes! Elliptic functions, first constructed in the nineteenth century, are doubly-periodic, and they are special in that they are the only doubly-periodic differentiable functions. In the graph to the left, you can clearly see the double period. There’s a black dot (a place where the function is 0) and a white dot (a place where the function is infinite) in every lattice cell, which is in fact a consequence of a theorem in complex analysis that states that a bounded differentiable function must be constant.
Gamma function
One way of getting interesting complex functions is asking whether some function on another domain generalizes to a differentiable complex function. For instance, we might wonder if the function f(x)=x! generalizes beyond the positive integers, and in fact, it does. The gamma function Γ(z) is equal to (z-1)! for z larger than 1. On the graph, we see interesting behavior on the negative x-axis. We observe white dots spaced evenly on every nonpositive integer; these correspond to the function evaluating to infinity at those points. Around each of these points, the colors wrap around once, because these poles have order 1.
Riemann zeta function
Another interesting function formed by generalizing from the positive integers is the zeta function. On the positive real numbers larger than 1, the zeta function is defined to be ζ(s)=1+1/2^s+1/3^s+…, and it turns out that you can find a generalization of this function to all complex numbers. This function has interesting connections to number theory.
Looking at the plot, we see dark spots (zeros) at regular intervals on the negative x-axis. It turns out the zeta function is zero at all negative even integers, and these are called the trivial zeros. We also see zeros on a vertical line, but spaced irregularly. The Riemann Hypothesis, one of the greatest unsolved problems of mathematics, hypothesizes that all the nontrivial zeros lie on this line. Proving this will answer deep questions in number theory and earn the prover a million dollars as prize money.