How do I see geometrically complex exponentiation? e.g $(a +bi)^{c+di}$ $\forall(a, b, c, d)$ reals

Question

Geometric interpretation of exponentiation with complex numbers

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How do I see geometrically complex exponentiation? e.g $(a+bi)^{c+di}$ $\forall(a, b, c, d)$ reals

Answer 1

I am familiar with at least two ways of visualizing complex functions.

Use Colors

We can assign a color to each complex number, then "graph" a function by coloring each pixel in some piece of the complex plane by the color of the image point. For example, we can let the argument of a number represent the hue, and the modulus represent the brightness (some combination of saturation and value). In that case, the complex plane would look something like this:

After application of the map $z \mapsto z^3 + 1$, you would have something like this:

Note that the dark spots are near the 3rd roots of $-1$, which makes sense, as these are zeros of the given polynomial. More complicated functions can be visualized similarly.

The downside is that you kind of have to have software to help you out---until you get some intuition, these visualizations are hard to do in your head.

Use a Grid

One of the nice things about complex analytic functions is that they are conformal. That is, they preserve angles. Thus if two lines meet at a right angle, the image lines will also meet at a right angle. Thus by considering where lines go, you have a chance at understanding how a function behaves. Using the same example as above (and the glory that is Wolfram|alpha), we get something like

This has the advantage of being more-or-less doable by hand---if you can figure out where lines and/or circles go, you are golden. This is also a technique that is supported by a lot of software, including the above-mentioned Wolfram|alpha.

Visualizing the Complex Exponential

I realize that this doesn't really answer your question. Indeed, I am intentionally avoiding your questions as it is just a bit too big to try to address---the graphs of complex functions are four dimensional already (two dimensions for the domain, two for the codomain), and you want a family graphs with a complex parameter (fix the exponent, graph the function $z^a$; do this for each possible exponent (or vice-versa)).

A couple of things to consider:

1. If $w = e^{i\theta} \in \mathbb{C}$ (with $\theta\in\mathbb{R}$, then $w^x = e^{ix\theta}$ is a rotation for any real $x$. As $x$ varies from $0$ to $2\pi$, the complex plane spins around the origin.
2. If $r > 0$, then $r^x$ is simply the usual exponential. Combine this with (1) to explore $(re^{\theta})^x$ (where $re^{i\theta}$ is a complex number in polar form, and $x$ is real).
3. Complex exponents get a little more complicated. You are going to have start dealing with things like branch cuts.

As I said in a comment above, this is a rather difficult problem. I would suggest playing around with software that can graph complex functions. Vary the parameters a bit and see what happens.

Good luck!