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This book presents complex analysis in one variable in the
context of modern mathematics, with clear connections to several
complex variables, de Rham theory, real analysis, and other
branches of mathematics. Thus, covering spaces are used explicitly
in dealing with Cauchy's theorem, real variable methods are
illustrated in the Loman-Menchoff theorem and in the corona
theorem, and the algebraic structure of the ring of holomorphic
functions is studied.
Using the unique position of complex analysis, a field drawing on
many disciplines, the book also illustrates powerful mathematical
ideas and tools, and requires minimal background material.
Cohomological methods are introduced, both in connection with the
existence of primitives and in the study of meromorphic functionas
on a compact Riemann surface. The proof of Picard's theorem given
here illustrates the strong restrictions on holomorphic mappings
imposed by curvature conditions.
Drawn from lectures given by Raghavan Narasimhan at the University
of Geneva and the University of Chicago, this book presents the
part of the theory of several complex variables pertaining to
unramified domains over C . Topics discussed are Hartogs' theory,
domains in holomorphy, and automorphism of bounded domains.
The original edition of this book has been out of print for some
years. The appear ance of the present second edition owes much to
the initiative of Yves Nievergelt at Eastern Washington University,
and the support of Ann Kostant, Mathematics Editor at Birkhauser.
Since the book was first published, several people have remarked on
the absence of exercises and expressed the opinion that the book
would have been more useful had exercises been included. In 1997,
Yves Nievergelt informed me that, for a decade, he had regularly
taught a course at Eastern Washington based on the book, and that
he had systematically compiled exercises for his course. He kindly
put his work at my disposal. Thus, the present edition appears in
two parts. The first is essentially just a reprint of the original
edition. I have corrected the misprints of which I have become
aware (including those pointed out to me by others), and have made
a small number of other minor changes.
Bernhard Riemanns Werk hat bis heute wesentlichen Einfluss auf die
Entwicklung der Mathematik genommen. Seine Ideen sind uberraschend
modern und pragen die heutige mathematische Forschung. Die
Gesammelten Abhandlungen (1892) samt Supplement von 1902 waren seit
langer Zeit vergriffen. R. Narasimhan hat die muhevolle Edition
dieser Neuausgabe ubernommen. Es koennen nur einige Hoehepunkte
genannt werden: - H. Weils Kommentare uber Riemanns
Habilitationsschrift - C.L. Siegel uber Riemanns Nachlass zur
analytischen Zahlentheorie - W. Wirtingers beruhmter Vortrag beim
internationalen Mathematikerkongress Heidelberg 1904 uber Riemanns
Vorlesungen uber die hypergeometrische Reihe. Neben diesen
historischen Wurdigungen von Riemanns Werk gibt es aktuelle
Beitrage, insbesondere zur Mechanik und uber "shock waves" von S.
Chandrasekhar, N. Lebovitz und P. Lax. Raghavan Narasimhan gibt in
einer ausfuhrlichen Einleitung eine Wurdigung, insbesondere des
funktionentheoretischen Werks von Bernhard Riemann. Ferner sind
Fotos und zahlreiche Nachtrage zum Lebenslauf aufgenommen worden.
Eine Bibliographie mit mehr als 800 Literaturstellen erarbeitet von
E. Neuenschwander und W. Purkert rundet diese Werkausgabe ab.
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