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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Analytic continuation, envelopes of holomorphy, and projective and direct limit spaces
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by Robert Carmignani PDF
Trans. Amer. Math. Soc. 209 (1975), 237-258 Request permission

Abstract:

For a Riemann domain $\Omega$, a connected complex manifold where $n(n = dimension)$ globally defined functions form a local system of coordinates at every point, and an arbitrary holomorphic function $f$ in $\Omega$, the “Riemann surface” ${\Omega _f}$, a maximal holomorphic extension Riemann domain for $f$, is formed from the direct limit of a sequence of Riemann domains. Projective limits are used to construct an envelope of holomorphy for $\Omega$, a maximal holomorphic extension Riemann domain for all holomorphic functions in $\Omega$, which is shown to be the projective limit space of the “Riemann surfaces” ${\Omega _f}$. Then it is shown that the generalized notion of envelope of holomorphy of an arbitrary subset of a Riemann domain can also be characterized in a natural way as the projective limit space of a family of “Riemann surfaces".
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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 209 (1975), 237-258
  • MSC: Primary 32D10
  • DOI: https://doi.org/10.1090/S0002-9947-1975-0385165-2
  • MathSciNet review: 0385165