# Difference between revisions of "Holomorphic envelope"

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.

#### 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)