The Cauchy-Riemann Complex: Integral Formulae and Neumann Problem (Aspects of Mathematics) - Softcover

Lieb, Ingo; Michel, Joachim

Softcover

ISBN 10: 3322916103 ISBN 13: 9783322916105

Publisher: Vieweg+Teubner Verlag, 2012

Synopsis

This book presents complex analysis of several variables from the point of view of the Cauchy-Riemann equations and integral representations. A more detailed description of our methods and main results can be found in the introduction. Here we only make some remarks on our aims and on the required background knowledge. Integral representation methods serve a twofold purpose: 1° they yield regularity results not easily obtained by other methods and 2°, along the way, they lead to a fairly simple development of parts of the classical theory of several complex variables. We try to reach both aims. Thus, the first three to four chapters, if complemented by an elementary chapter on holomorphic functions, can be used by a lecturer as an introductory course to com­ plex analysis. They contain standard applications of the Bochner-Martinelli-Koppelman integral representation, a complete presentation of Cauchy-Fantappie forms giving also the numerical constants of the theory, and a direct study of the Cauchy-Riemann com­ plex on strictly pseudoconvex domains leading, among other things, to a rather elementary solution of Levi's problem in complex number space en. Chapter IV carries the theory from domains in en to strictly pseudoconvex subdomains of arbitrary - not necessarily Stein - manifolds. We develop this theory taking as a model classical Hodge theory on compact Riemannian manifolds; the relation between a parametrix for the real Laplacian and the generalised Bochner-Martinelli-Koppelman formula is crucial for the success of the method.

About this title

About the Author

Prof. Dr. Ingo Lieb ist Professor für Mathematik an der Universität Bonn. Er ist Autor der beiden Bücher "Funktionentheorie" und "Ausgewählte Kapitel aus der Funktionentheorie" in der Reihe vieweg studium/Aufbaukurs Mathematik.
Prof. Dr. Joachim Michel ist Professor für Mathematik am "Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville" (L.M.P.A.) in Calais, Frankreich.

From the Back Cover

The method of integral representations is developed in order to establish 1. classical fundamental results of complex analysis both elementary and advanced, 2. subtle existence and regularity theorems for the Cauchy-Riemann equations on complex manifolds. These results are then applied to important function theoretic questions. The book can be used for advanced courses and seminars at the graduate level; it contains to a large extent material which has not yet been covered in text books.

Edition details

• Publisher: Vieweg+Teubner Verlag
• Publication date: 2012
• Language: English
• ISBN 10: 3322916103
• ISBN 13: 9783322916105
• Binding: Paperback
• Number of pages: 372