Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-05T08:49:11.465Z Has data issue: false hasContentIssue false
  • English
  • Français

A theorem of continuation for functions of several complex variables

Published online by Cambridge University Press:  24 October 2008

J. G. Taylor
J. G. Taylor
Affiliation:
Institute for Advanced StudyChrist's CollegePrincetonCambridge
Get access Rights & Permissions [Opens in a new window]

Extract

In the last few years it has been found useful to apply known theorems in the theory of functions of several complex variables to solve problems arising in the quantum theory of fields (11). In particular, in order to derive the dispersion relations of quantum field theory from the general postulates of that theory it appears useful to apply known theorems on holomorphic continuation for functions of several complex variables ((2), (10)). The most important theorems are those which enable a determination to be made of the largest domain to which every function which is holomorphic in a domain D may be continued. This domain is called the envelope of holomorphy of D, and denoted by E(D). If D = E(D) then D is termed a domain of holomorphy. E(D) may be defined as the smallest domain of holomorphy containing D. Only in the special cases that D is a tube, semi-tube, Hartogs, or circular domain has it been possible to determine the envelope of holomorphy E(D) ((3), (7)). An iterative method for the computation of envelopes of holomorphy has recently been given by Bremmerman(4). It is also possible to use the continuity theorem (1) in a direct manner, though in most cases this is exceedingly difficult.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 54 , Issue 3 , July 1958 , pp. 377 - 382
Copyright
Copyright © Cambridge Philosophical Society 1958

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Behnke, H. and Thullen, P.Ergebn. Math. (3), 3 (1934).Google Scholar
(2)Bremmerman, H. J., Oehme, R. and Taylor, J. G.A proof of dispersion relations in quantized field theories. Phys. Rev. (in the Press).Google Scholar
(3)Bremmerman, H. J.Math. Ann. 127 (1954), 406.CrossRefGoogle Scholar
(4)Bremmerman, H. J.Construction of the envelopes of holomorphy of arbitrary domains. Rev. Mat. Hisp. Amer. (in the Press).Google Scholar
(5)Bremmerman, H. J.Complex convexity. Trans. Amer. Math. Soc. 82 (1956), 17.CrossRefGoogle Scholar
(6)Bremmerman, H. J.Holomorphic functionals and complex convexity in Banach spaces. Pacif. J. Math. 7 (1957), 811.CrossRefGoogle Scholar
(7)Bochner, S. and Martin, W. T.Several complex variables (Princeton, 1948).Google Scholar
(8)Lelong, P.Les fonctions plurisousharmoniques. Ann. Ecole Norm. 62 (1945), 301.Google Scholar
(9)Riesz, F.Sur les fonctions subharmoniques. Acta Math. 48 (1926), 329.CrossRefGoogle Scholar
(10)Taylor, J. G.A possible proof of dispersion relations, preprint of talk given for the Colloque sur les Problèmes Mathématiques de la Théorie Quantique des Champs (Lille, 1957).Google Scholar
(11)Wightman, A. S.Some mathematical problems of relativistic quantum theory, preprint of talk given for the Colloque sur les Problèmes Mathématiques de la Théorie Quantique des Champs (Lille, 1957).Google Scholar