Given a multi-valued complex function $f: z = x+\mathrm{i}\,y\rightarrow w=u+\mathrm{i}\,v$ with $x,y,u,v\in\mathbb{R}$, we know the image $\{f(z)\,|\,z\in\mathbb{C}\}$ is a Riemannian surface. How to visualise this surface?

What I saw on the web is usually a surface corresponding to the graph of the map i.e. $\{(x,y,u)\,|\,x,y\in\mathbb{R},u=\mathrm{Re}\,f(x+\mathrm{i}y)\}$, which is not the Riemanninan surface.


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