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''domain over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026920/c0269202.png" />''
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''domain over $\mathbf{C}^n$''
  
A pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026920/c0269203.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026920/c0269204.png" /> is an arcwise-connected Hausdorff space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026920/c0269205.png" /> is a local homeomorphism, called a projection. Covering domains are encountered in the analytic continuation of holomorphic functions. For every analytic (possibly multivalent) function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026920/c0269206.png" /> in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026920/c0269207.png" /> there is a corresponding covering domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026920/c0269208.png" /> with a projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026920/c0269209.png" />, just as for every analytic function of one complex variable there is a corresponding Riemann surface; the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026920/c02692010.png" /> is single-valued on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026920/c02692011.png" />. Covering domains are also called Riemann domains.
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A pair $(X,\pi)$, where $X$ is an [[Arcwise connected space|arcwise-connected]] Hausdorff space and $\pi$ is a [[local homeomorphism]], called a projection. Covering domains are encountered in the [[analytic continuation]] of holomorphic functions. For every analytic (possibly multivalent) function $f$ in a domain $D \subset \mathbf{C}^n$ there is a corresponding covering domain $\tilde D$ with a projection $\pi : \tilde D \rightarrow D$, just as for every analytic function of one complex variable there is a corresponding [[Riemann surface]]; the function $f$ is single-valued on $\tilde D$. Covering domains are also called Riemann domains.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''2''' , Moscow  (1976)  (In Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''2''' , Moscow  (1976)  (In Russian)</TD></TR>
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</table>
  
  
  
 
====Comments====
 
====Comments====
A covering domain is sometimes called a manifold spread over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026920/c02692013.png" />. See also [[Domain of holomorphy|Domain of holomorphy]]; [[Riemannian domain|Riemannian domain]]; [[Holomorphic envelope|Holomorphic envelope]].
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A covering domain is sometimes called a manifold spread over $\mathbf{C}^n$. See also [[Domain of holomorphy]]; [[Riemannian domain]]; [[Holomorphic envelope]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.C. Gunning,  H. Rossi,  "Analytic functions of several complex variables" , Prentice-Hall  (1965)  pp. Chapt. 1, Section G</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Grauert,  R. Remmert,  "Theory of Stein spaces" , Springer  (1979)  (Translated from German)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  R.C. Gunning,  H. Rossi,  "Analytic functions of several complex variables" , Prentice-Hall  (1965)  pp. Chapt. 1, Section G</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Grauert,  R. Remmert,  "Theory of Stein spaces" , Springer  (1979)  (Translated from German)</TD></TR>
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</table>
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Latest revision as of 17:02, 13 June 2020

domain over $\mathbf{C}^n$

A pair $(X,\pi)$, where $X$ is an arcwise-connected Hausdorff space and $\pi$ is a local homeomorphism, called a projection. Covering domains are encountered in the analytic continuation of holomorphic functions. For every analytic (possibly multivalent) function $f$ in a domain $D \subset \mathbf{C}^n$ there is a corresponding covering domain $\tilde D$ with a projection $\pi : \tilde D \rightarrow D$, just as for every analytic function of one complex variable there is a corresponding Riemann surface; the function $f$ is single-valued on $\tilde D$. Covering domains are also called Riemann domains.

References

[1] B.V. Shabat, "Introduction of complex analysis" , 2 , Moscow (1976) (In Russian)


Comments

A covering domain is sometimes called a manifold spread over $\mathbf{C}^n$. See also Domain of holomorphy; Riemannian domain; Holomorphic envelope.

References

[a1] R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965) pp. Chapt. 1, Section G
[a2] H. Grauert, R. Remmert, "Theory of Stein spaces" , Springer (1979) (Translated from German)
How to Cite This Entry:
Covering domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Covering_domain&oldid=14568
This article was adapted from an original article by V.V. Zharinov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article