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