How can a complex plot be projected onto a real plot?

Question

The plot of $f(z)=e^{iz}$ looks like a unit circle on the complex plane, and likewise the plot of $x^2+y^2=1$ looks like a unit circle on the real plane. Is there any way to draw a complex plot on the real plane and then derive a relationship between $x$ and $y$ that describes the real plot? For instance, is there any way to get from $f(z)=e^{iz}$ to $x^2+y^2=1$? What about more complicated examples, like coming up with a relation that plots the shape of $\zeta(s)$ on the real plane?

Answer 1

Think at $f(z)$ as $u+iv=f(x+iy)=\phi(x,y)+i\psi(x,y)$ .

Each $f(z)$ has the associated functions $\phi(x,y) =Re(f(z))\quad \psi(x,y)=Im(f(z))$ which, if $f(z)$ is analytic shall respect the Cauchy-Rieman conditions.

Then a map from the plane $(x,y)$ to the plane $(u,v)$ is defined, which for an analytic $f(z)$ is a Conformal Map

This is for example a sketch of the conformal map induced by $\zeta(z)$