# Holomorphic envelope

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=35287