[Math] Plotting in the Complex Plane

Question

I just wonder how do you plot a function on the complex plane? For example,$$f(z)=\left|\dfrac{1}{z}\right|$$
What is the difference plotting this function in the complex plane or real plane?

Answer 1

First, the graph of a complex function is by definition $\{(z,w)\in{\mathbb C}^2: w=f(z)\}$, which lives in 4D. Thus, there's not a "simple" way to visualize this.

Nonetheless, there are a number of standard tricks to help us visualize the function. Perhaps the most standard is to illustrate the effect that the function has on some standard domain or set of domains. If you type "plot 1/z" into WolframAlpha, for example, you'll see a number of images related to the function, one of which is labeled "Complex map" and looks like so:

This illustrates how a grid of lines throughout the square $[-1,1]\times[-1,1]$ is affected by the map $z\rightarrow 1/z$. This is a bit like trying to understand the map $f(x)=x^2$ by examining the following picture:

The top picture represents a set of points evenly distributed throughout the interval $[0,2]$; the bottom illustrates the image of the points under $f(z)=z^2$.

I reiterate, though, this is just one of a number of techniques for visualizing these types of functions. Other possibilities include: plotting the real and imaginary part as 3D graphs, doing the same with the magnitude and argument, or creating contour plots of these. This works because for example, writing $w=|f(x+i*y)|$, we see that $w$ is a real-valued function of the two real variables $x$ and $y$, which means we can make a 3D plot. Here's the result of the WolframAlpha query "plot |1/z|", for example:

It can get really crazy if you move to the Riemann sphere. Here's WolframAlpha's "Riemann sphere map":

To really understand this, you should check out the video Möbius Transformations Revealed on YouTube!