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''envelope of holomorphy, of a (Riemann) domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047530/h0475301.png" />''
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The largest domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047530/h0475302.png" /> with the following property: Any holomorphic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047530/h0475303.png" /> can be holomorphically continued to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047530/h0475304.png" />. The problem of constructing the envelope of holomorphy for a given domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047530/h0475305.png" /> arises in connection with the fact that in a complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047530/h0475306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047530/h0475307.png" />, not all domains are domains of holomorphy (cf. [[Domain of holomorphy|Domain of holomorphy]]), i.e. there exist domains such that any function that is holomorphic in this domain has a holomorphic continuation to a larger (usually not single-layered) domain. The envelope of holomorphy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047530/h0475308.png" /> is a domain of holomorphy; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047530/h0475309.png" /> is a domain of holomorphy, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047530/h04753010.png" />.
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''envelope of holomorphy, of a (Riemann) domain $D$''
  
In applications in axiomatic quantum field theory there arises the non-trivial problem of constructing envelopes of holomorphy of a special kind, which reflect the physical requirements of spectrality, local commutativity and Lorentz covariance. The [[Bogolyubov theorem|Bogolyubov theorem]] on the edge-of-the-wedge and continuity theorems (cf. [[Continuity theorem|Continuity theorem]]) are especially useful in this connection.
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The largest domain $H(D)$ with the following property: Any [[holomorphic function]] in $H(D)$ can be holomorphically continued to $D$. The problem of constructing the envelope of holomorphy for a given domain $D$ arises in connection with the fact that in a complex space $\mathbb{C}^n$, $n \ge 2$, not all domains are domains of holomorphy (cf. [[Domain of holomorphy]]), i.e. there exist domains such that any function that is holomorphic in this domain has a holomorphic continuation to a larger (usually not single-layered) domain. The envelope of holomorphy $H(D)$ is a domain of holomorphy; if $D$ is a domain of holomorphy, then $H(D) = D$.
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In applications in axiomatic quantum field theory there arises the non-trivial problem of constructing envelopes of holomorphy of a special kind, which reflect the physical requirements of spectrality, local commutativity and Lorentz covariance. The [[Bogolyubov theorem]] on the edge-of-the-wedge and continuity theorems (cf. [[Continuity theorem]]) are especially useful in this connection.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.S. Vladimirov,  "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , 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">  V.S. Vladimirov,  "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  (In Russian)</TD></TR>
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</table>
  
  

Latest revision as of 21:13, 1 December 2014

2020 Mathematics Subject Classification: Primary: 32D10 [MSN][ZBL]

envelope of holomorphy, of a (Riemann) domain $D$

The largest domain $H(D)$ with the following property: Any holomorphic function in $H(D)$ can be holomorphically continued to $D$. The problem of constructing the envelope of holomorphy for a given domain $D$ arises in connection with the fact that in a complex space $\mathbb{C}^n$, $n \ge 2$, not all domains are domains of holomorphy (cf. Domain of holomorphy), i.e. there exist domains such that any function that is holomorphic in this domain has a holomorphic continuation to a larger (usually not single-layered) domain. The envelope of holomorphy $H(D)$ is a domain of holomorphy; if $D$ is a domain of holomorphy, then $H(D) = D$.

In applications in axiomatic quantum field theory there arises the non-trivial problem of constructing envelopes of holomorphy of a special kind, which reflect the physical requirements of spectrality, local commutativity and Lorentz covariance. The Bogolyubov theorem on the edge-of-the-wedge and continuity theorems (cf. Continuity theorem) are especially useful in this connection.

References

[1] V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)
[2] B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian)


Comments

References

[a1] R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965) pp. Chapt. 1, Sect. G
How to Cite This Entry:
Holomorphic envelope. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Holomorphic_envelope&oldid=13411
This article was adapted from an original article by V.S. Vladimirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article