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Covering domain

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domain over

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=49715
This article was adapted from an original article by V.V. Zharinov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article