To be clear from the beginning, I am looking to develop my visual intuition for complex functions. I can explain algebraically why the plot below for $q(z) = e^{2\pi i z}$ looks the way it does, but I want to develop better intuitive reasoning with these complex contour plots.
With that out of the way... I have the following three complex plots, all generated in SageMath:
- $e^z$
- $z^{2\pi i}$
- $e^{2\pi i z}$
Visually, it seems that the transformation from the first plot $(e^z)$, to the third $(e^{2\pi i z})$ is clear: the chart rotates 90 degrees clockwise, the period of the color strips moves from $2\pi i$ to $1$, and the transitions to both light and dark as are sped up as we move away from the origin.
Again, I can explain each of these pieces algebraically, but I'm not able to see why that happens from combining the first and second graphs. Of course the third is the plot of the composition $(e^{z})^{2\pi i}$, so somehow it should be applying the second transformation to the first plot.
Does anyone have good enough intuition with complex plots to explain pictorially how we get the third from composing the first two?
In the complex plane, multiplication by a unit modulus complex number encodes rotation about the origin through an angle equal to its argument. In particular, since $\arg i = \frac\pi2$, the map $z \mapsto iz$ rotates counterclockwise through a right angle. Now, multiplication by a real number scales the modulus of a complex number (stretches/shrinks it from the origin). These two operations commute, so it doesn't matter which you think of as happening first.
Putting this together, $$ z \mapsto iz \mapsto 2\pi i z $$ (1) rotates CCW by a right angle and (2) stretches by a factor of $2\pi$, so if $w = 2\pi i z$, then $w$ is a rotated and stretched version of $z$. Now, since this is happening in the input of the function $w \mapsto \exp w$, the effect on the graph is the inverse of this relationship, as usual. To wit, if $v = \exp w$ (and $w = 2\pi iz$), then for a particular $w$, the $z$ value that is fed into the exponential function is $$ z = \frac{w}{2\pi i} = -i \cdot \frac{1}{2\pi} \cdot w, $$ which is shrunk by a factor of $2\pi$ and rotated back (CW) by a right angle, so the colorful bands now stretch downward and are squeezed together. This is the value of $z$ that produces the particular output $v$, hence the pixel at this $z$ is colored according to $v$.
As an experiment, use some different scaling factors, e.g. $z \mapsto \exp(cz)$ with $$ c = \frac{1 - \sqrt{3}}{4} = \frac12 \cdot \exp(-\tfrac\pi3), $$ should have colorful bands extending in $+60^\circ$ direction with double spacing.