Multi-sheeted region

A region $S$ of a Riemann surface $R$, considered as a covering surface over the complex plane $\mathbf C$, such that above each point of its projection $D \subset \mathbf C$ there are at least two points of $S$; a branch point of $R$ of order $k- 1$ is regarded here as $k$ distinct points. For example, the analytic function $w = z ^ {2}$ is a one-to-one mapping of the disc $D = \{ {z \in \mathbf C } : {| z | < 1 } \}$ onto the two-sheeted region (two-sheeted disc) $S = \{ {w \in R } : {| w | < 1 } \}$ of the Riemann surface $R$ of this function; this mapping is conformal everywhere except at the origin.

For analytic functions of several complex variables there arise multi-sheeted Riemann domains (cf. Riemannian domain) over the complex space $\mathbf C ^ {n}$.

References

[a1]	C.L. Siegel, "Topics in complex functions" , 1 , Wiley (Interscience) (1988) pp. Chapt. 1, Sect. 4
[a2]	G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10