# Holomorphic envelope

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2010 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.

How to Cite This Entry:
Holomorphic envelope. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Holomorphic_envelope&oldid=35287
This article was adapted from an original article by V.S. Vladimirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article