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A region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065220/m0652201.png" /> of a [[Riemann surface|Riemann surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065220/m0652202.png" />, considered as a covering surface over the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065220/m0652203.png" />, such that above each point of its projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065220/m0652204.png" /> there are at least two points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065220/m0652205.png" />; a [[Branch point|branch point]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065220/m0652206.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065220/m0652207.png" /> is regarded here as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065220/m0652208.png" /> distinct points. For example, the analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065220/m0652209.png" /> is a one-to-one mapping of the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065220/m06522010.png" /> onto the two-sheeted region (two-sheeted disc) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065220/m06522011.png" /> of the Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065220/m06522012.png" /> of this function; this mapping is conformal everywhere except at the origin.
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For analytic functions of several complex variables there arise multi-sheeted Riemann domains (cf. [[Riemannian domain|Riemannian domain]]) over the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065220/m06522013.png" />.
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A region  $  S $
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of a [[Riemann surface|Riemann surface]]  $  R $,
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considered as a covering surface over the complex plane  $  \mathbf C $,
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such that above each point of its projection  $  D \subset  \mathbf C $
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there are at least two points of  $  S $;
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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 } \} $
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onto the two-sheeted region (two-sheeted disc)  $  S = \{ {w \in R } : {| w | < 1 } \} $
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of the Riemann surface  $  R $
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of this function; this mapping is conformal everywhere except at the origin.
  
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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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 08:01, 6 June 2020


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} $.

Comments

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
How to Cite This Entry:
Multi-sheeted region. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multi-sheeted_region&oldid=47922
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article