View source for Holomorphic envelope

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''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====
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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>
<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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====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.C. Gunning,   H. Rossi,   "Analytic functions of several complex variables" , Prentice-Hall  (1965)  pp. Chapt. 1, Sect. G</TD></TR></table>