Just to get a bit of a picture of what taking a function of a complex number means, we can play a bit of a game (I use this term in the loosest sense). Normally we think of functions as going from a real number to another real number. $\sin(x)$ takes a real number $x$ and gives you another real number. We can plot this on a graph by plotting a two dimensional set of data which tells you about the value that $\sin(x)$ takes for every $x$ along the real line. We are very used to this idea of a function. However, a function of a complex number is more difficult to visualise.

Complex numbers themselves live in 2 dimensions (they have a real part and an imaginary part) and when you apply a function to them, very often the result is another complex number which also lives in a 2 dimensional space. So trying to plot this function would require us to draw a three dimensional surface in four dimensions (cf. a one dimensional line in 2 dimensions for purely real functions). This we can’t do, but what we can do is to ask what a set of complex numbers looks like when we apply a function to them, by having two complex planes next to each other.

Let’s think about the complex numbers which lie along a point of fixed real part but varying imaginary part. Let’s pick the real part to be 2 for now. These numbers are thus of the form: $z=2+ib$. As we vary $b$ we trace out a vertical line which cuts through the real axis at $2$. What happens to these points when we apply to them the function $f(z)=\frac{1}{z}$. We know that they will give: $\frac{2-ib}{2^2+b^2}$. As we vary the value $b$ these will trace out not a straight line, but in fact a circle of radius $2$. This is not immediately obvious until you’ve played around quite a bit with complex numbers, so don’t be put off if you don’t see it. However, we can plot this using Mathematica and find that vertical lines are mapped to circles of the form shown here: Vertical lines in the complex plan get mapped to circles when we apply the function $f(z)=\frac{1}{z}$ to them. Think of the line as being made up of a load of points in the complex plane, each of those points get mapped to another set of points but if we were to plot both the original line and the mapped image in the same graph it would look very confusing, so instead we draw the image in a plane next to the original. The arrow shows two example points on lines which get mapped to two points on circles.

while horizontal lines are mapped to circles of a slightly different form as can be seen here And in the following we have both, for completeness. The colours of each line are matched, and you can see four points which are mapped to their correspondingly coloured points in the complex plane. The mapping of $z\rightarrow\frac{1}{z}$ from one complex plane to its image.

How about a more complicated function like the exponential of a complex number ( $e^z$)? Well, it turns out that in this case vertical lines get mapped to circles, and horizontal lines get mapped to radial spikes as can be seen from the mapping of red and blue lines here: The complex plane mapped under the exponential function. You see that horizontal lines get mapped to radial lines, while vertical lines get mapped to circles where the radius is the exponential of the real value of the points on the line.

Actually this particular mapping is a special type of mapping which preserves angles. You can see that all the intersections of the lines in the left plot are at 90 degrees, as are all the intersections of lines in the right hand plot. This class of mappings is known as a conformal mapping.

 How clear is this post?