Is this a thing? Visualizing complex functions with 3D animation

Question

I have read about various ways to visual complex functions, e.g. colored graphs, vector fields, conformal maps, etc.

Here is another way: 3D animation. Picture a 3D plot with three axes for $\text{Re}(z), \text{Im}(z), \text{Re}(f(z))$. Press "play" and the plot starts moving, with time $t=\text{Im}(f(z))$. Alternatively, the three axes could be $|z|, \operatorname{arg}(z), |f(z)|$, with time $t=\operatorname{arg}(f(z))$.

Questions: Is this method of visualizing complex functions known? Could it be useful?

(Needham's book Visual Complex Analysis does not seem to mention this method.)

Answer 1

I haven't seen before a visualization like this, but I think it could be very interesting. Just like any other visualization technique:

• You need several interesting examples to "get used" to it. You need to learn how features of a known visualization, or of the theory itself, fit int this new technique. What happens at poles or at zeros of meromorphic functions?, what happen in zeros with higher degree.
• Surely it won't be complete, in the sense that there will be features that cannot be seen. But hopefully some results of the theory could be better understood in this way.

I am really curious about what animations would arise with $z^2$, $\overline{z}$, $\frac{1}{z}$,... Please, if you finally develop this kind of representation, please paste links to the videos at the end of your question.