Riemann Surface of $z^3$

Question

I'm new to the topic of Riemann surfaces and complex analysis, and I tried to compute the Riemann surface of $z^3$ on Wolfram Alpha but couldn't because its not a multivalued function. How can I compute the riemann surface of $z^3$ or I got something really wrong?

Answer 1

When thinking of the Riemann surface of an expression with one complex variable, you should really think of converting that into an equation with two complex variables, like this: $$w = z^3 $$ The resulting Riemann surface is then the solution set of this equation: $$S = \{(w,z) \in \mathbb C^2 \mid w=z^3\} $$ However, as you have observed, $w=z^3$ is single valued as a function of the variable $z$. So that equation defines a function with independent variable $z$ and dependent variable $w$. Also $S$ is simply the graph of this function (with domain--codomain variables switched from their usual "first--last" positions). So as is usual with functions, we obtain an isomorphism between the domain of the function and the graph of the function, namely $\mathbb C \mapsto S$ given by $z \mapsto (w,z^3)$. In this context "isomorphism" means "biholomorphism" and so the Riemann surface is (biholomorphic to) $\mathbb C$.