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Commonly it's believed that one cannot fully visualize a complex function $f:\mathbb{C}\rightarrow \mathbb{C}$ because the full plot would have to be 4-dimensional. But is this true? And why would the following surface (embedded in $\mathbb{R}^3$) not count as a full plot of $f$?

Consider complex polynomials $f(z)$ (which are prototypes of entire functions). For each $r$ you can plot the circle $C_r = \{r e^{i\varphi}\ |\ 0 \leq \varphi < 2\pi\}$ which is a closed curve in $\mathbb{R}^2$. Adding $r$ as a third dimension you get an intricated surface in $\mathbb{R}^3$ which can be considered a plot of $f$:

enter image description here source

For a given point $(u,v,r)$ on the surface – which indicates a pair $(z,f(z))$ – you can tell $f(z) = (u,v)$ and $z = re^{i\varphi}$ for some $0 \leq \varphi < 2\pi$. But you cannot tell $\varphi$, that's the missing information (dimension). But in the two branched plot of the real square root you cannot tell what the square root of $4$ is, neither: $+2$ or $-2$?

Nevertheless the plot gives you a unique picture of the function – I guess there are no two polynomials with the same plot, are there?

My questions is:

  • What's the name of these surfaces? (Are they some kind of "Riemannian surfaces"?)

  • If they are not Riemannian surfaces: How are they related to those?

  • Why do they seem to be not so prominent as they seem to deserve (at least in my opinion)?

They reveal a lot about a complex function, and they are rather easy to grasp - at least easier than domain-colored plots, at least for the beginner. But you rarely find them, even in the visualization of complex functions literature, do you?

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