Visualizing a complex valued function of one real parameter

Question

I'm looking for a way to capture/graph or visualize it in my head, but I can't find how..

a 2-dimensional path won't do, because it doesn't reveal the rate-of-change..
2 1-dimensional graphs on top of each other doesn't help much either..
3-dimensional space with 2-dimensional slices for values of the parameter is the best I could come up with..

Is there a better way?

Answer 1

If you want a picture that represents the complete information inherent in a function $$f:\quad {\mathbb R}\to{\mathbb C},\qquad t\mapsto z(t)=x(t)+iy(t)$$ you need three real dimensions, because informationwise a function is equivalent to its graph, which is a subset of domain$\times$range, and this cartesian product has real dimension $3$ in your case.

When the given function $f$ describes a phenomenon in time you should draw the $t$-axis more or less horizontally extending to the right, the $x$-axis would have to go to the rear, and the $y$-axis points vertically upwards. The "worldline" of your particle moving around in ${\mathbb C}$ then looks as follows:

If you are much more interested in the geometry of the $2$D-curve obtained by projection to ${\mathbb C}$, and want to see self-intersections, etc., instead of the exact time-table, you'd rather lie ${\mathbb C}$ horizontally by drawing the $x$-axis going to the right and the $y$-axis going to the rear; finally the $t$-axis goes vertically upwards. This would give rise to a picture of the following kind:

Answer 2

[> with(plots):
  f := z-> exp(I*z):
  complexplot3d(f, -2-2*I .. 2+2*I);

Or

[> complexplot(exp(I*x), x = -Pi .. Pi);