Exegesis. Review the geometry of $e^z$ and $n^{th}$ [rmsf-1500][edit]

$\gdef\CC{\mathbb{C}}$ $\gdef\spaces#1{~ #1 ~}$

To finish off this lecture, we’ll review the geometry of some of the more basic holomorphic maps, namely the exponential function and the $n^{th}$ power map for $n$ a positive integer.

The exponential function $z \to e^z$ is a holomorphic map map $\CC \to \CC$. To picture it, it’s easiest to use Cartesian real coordinates on the source $\CC$. Namely

$$ e^{a+bi} \spaces= e^a (\cos b + i \cdot \sin b)$$

In other words, $e^{a+bi}$ maps to the complex number with polar coordinates $(e^a, b)$ where the angle $b$ is given in radians. What does this look like? Well, let’s first imagine the field of vertical lines in the source $\CC$. Each of these wraps around a circle centered at the origin, with periodicity of $2\pi i$. If our line is far to the left this circle is of small radius, and if our line is far to the right it is of large radius. One the other hand the horizontal lines simply map to rays emanating from the origin. Note, however, that the origin itself is not in the image of the map. As you move far to the left your values approach the origin, but they never get there.

All in all there is a periodicity in this map: adding a multiple of $2\pi i$ does not change the value under the exponential map, thus it’s convenient to imagine taking the horizontal strip ranging between the lines $b=0$ and $b=2\pi$, and making an infinite cylinder out of it by gluing these boundary this cylinder and the open subset $\CC\backslash\{0\}$, thus the exponential map “straightens out” $\CC\backslash\{0\}$ into a more manifestly symmetric geometry model having the same conformal geometry (angles are preserved by the exponential function). E.g., now the operation of scaling by a non-zero complex number on $\CC\backslash\{0\}$ turns into just translating along the cylinder.

What about the $n^{th}$ power map? Since it’s a question of multiplication, polar coordinates are the more useful: it sends $0$ to $0$, and it sends $(r, \theta)$ to $(r^n, \theta^n)$. The picture is that on the unit circle, the $n^{th}$ power maps wraps around $n$ times evenly. That is, if you travel around the unit circle $n$ times at constant speed $ns$. If you move away from the unit circle along a radial line, the argument of your image doesn’t change, but your radius will change according to the real-valued $n^{th}$ power map applied to your origin radius. We’ll have occasion to point out some features of this example later.