# Functions of a Complex Variable

My precalculus classes just finished a unit on polar coordinates and complex numbers. When I teach about complex numbers,  I mention functions of a complex variable in passing, but we don’t really give it much thought. We do complex arithmetic and that’s all; that is, problems like these:

Evaluate.

$\frac{i^5(2-i)}{1+3i}$

$\left(2+2i\sqrt{3}\right)^{10}$

$\sqrt[4]{40.5+40.5i\sqrt{3}}$

In our precalculus class, we also understand how to plot complex numbers. Complex numbers must be plotted on a two-dimensional plane because complex numbers are…well…two dimensional! The real number line has no place for them. For instance, we represent the complex number $w=2-3i$ as the point $(2,-3)$

But we don’t ask questions about complex functions. This is sad! Because functions of a complex variable are fairly accessible.  That is, we want to consider functions like

$f(z)=z+1$

$f(z)=z+4i$

$f(z)=3z$

$f(z)=iz$

$f(z)=z^2$

The first thing you’ll notice is that I’ve written these functions in terms of $z$, to indicate that they take complex arguments and (possibly) return complex values. Here’s where the problem comes. Take for instance, $f(x)=x+1$. We’re used to visualizing it this way:

Notice we’re wired (because of schooling, perhaps) to understand the $x$ coordinate as being the “input” to the function and the $y$ coordinate as being the “output” from the function. Now, think about $f(z)=z+1$ where $z$ is complex. Do you see the problem? Remember, complex numbers are two-dimensional.  A function $f(z)$ that operates on complex numbers has two-dimensional input and two-dimensional output. For example, take the function $f(z)=z+1$. If we try putting a few complex numbers into the function for $z$, what happens? If $z=-2+4i$, then $f(z)=-1+4i$. Geometrically, what is happening to a complex number on the complex plane when we apply $f(z)$? If you thought “the function translates complex numbers to the right by one unit in the complex plane,” you’re right!

I’ve built another Geogebra applet to help you visualize this kind of function. Make sure you use it with purpose, rather than just dragging things around randomly. Try making predictions about what will happen before revealing the result. Read the directions.

Have fun, and I hope you learn something about complex functions! I’m sure to post more on them someday. There’s a lot more to say.