# 3D-plots of complex functions

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 $$f$$:

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?

• – lhf Jan 8 at 13:43