$|z| \leq \pi$, complex function $f(z)=e^z$ 's graph

Question

When the problem has these conditions, what graph shape does this function have?

In complex, $|z| \leq \pi$, $\rightarrow$ what is the exponential graph of complex function $f(z)=e^z$?

Answer 1

This is a function of complex numbers into complex numbers, $\Bbb Z \to \Bbb Z$, so we have the initial handicap of being able to visualize directly four variables ($z=a+bi \to f(z)=a'+b'i$).

$f(z)=e^z$ is single valued, so you can try to visualize it using a complex domain color map.

Here is a complex domain color map of $f(z)=e^z$ at the interval defined by $[-8,8]+[-6,6]i$ (disregard the left and bottom numbers, they are just automatically plotted by Python because it is a $800$x$600$ pixel image):

And it is possible to convert it into a depth map autostereogram, so you can see the map as a "magic eye" object:

Since $|z| \leq \pi$ is the same as $(Re(z))^2+(Im(z))^2 \leq \pi^2$, and that is the circle equation for a radius $r=\pi$ circle. It means that you just want to see the result of the complex exponential applied to those complex numbers $z$ whose absolute value is inside or exactly equal to the radius of the circle centered in the origin of coordinates of the complex plain and radius $\pi$. To be able to distinguish, I will apply $f(z)=0$ for any other value outside the circle. The color map looks like this:

And the autostereogram looks like this:

Basically, if my calculations are not wrong, the values $f(z)$ what you want to visualize are inside the circle and at the circle itself.