View source for Multi-sheeted region

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A region  $  S $
of a [[Riemann surface|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|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|Riemannian domain]]) over the complex space  $  \mathbf C  ^ {n} $.

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.L. Siegel,   "Topics in complex functions" , '''1''' , Wiley (Interscience)  (1988)  pp. Chapt. 1, Sect. 4</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Springer,   "Introduction to Riemann surfaces" , Addison-Wesley  (1957)  pp. Chapt.10</TD></TR></table>