Application of Holomorphic Functions in Two and Higher Dimensions
This book presents applications of hypercomplex analysis to boundary value and initial-boundary value problems from various areas of mathematical physics. Given that quaternion and Clifford analysis offer natural and intelligent ways to enter into higher
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pplication of Holomorphic Functions in Two and Higher Dimensions
Klaus Gürlebeck • Klaus Habetha • Wolfgang Sprößig
Application of Holomorphic Functions in Two and Higher Dimensions
Klaus Gürlebeck Bauhaus-Universität Weimar Weimar, Germany
Klaus Habetha RWTH Aachen Aachen, Germany
Wolfgang Sprößig TU Bergakademie Freiberg Freiberg, Germany
ISBN 978-3-0348-0964-1 (eBook) ISBN 978-3-0348-0962-7 DOI 10.1007/978-3-0348-0964-1 Library of Congress Control Number: 2016942573 Mathematics Subject Classification 2010: 30AXX, 30CXX, 30GXX, 33CXX, 35CXX, 35JXX, 35FXX, 35KXX, 43AXX, 62PXX, 74BXX, 76-XX, 78-XX © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This book is published under the trade name Birkhäuser. The registered company is Springer International Publishing AG Switzerland (www.birkhauser-science.com)
Contents Preface
xi
1
Basic properties of holomorphic functions 1.1 Number systems . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Real Clifford numbers . . . . . . . . . . . . . . . . 1.1.2 Quaternion algebra . . . . . . . . . . . . . . . . . . 1.1.3 On rotations . . . . . . . . . . . . . . . . . . . . . 1.1.4 Complex quaternions . . . . . . . . . . . . . . . . . 1.1.5 Clifford’s geometric algebra . . . . . . . . . . . . . 1.1.6 The ± split with respect to two square roots of −1 1.1.7 Bicomplex numbers . . . . . . . . . . . . . . . . . 1.2 Classical function spaces in quaternions . . . . . . . . . . 1.3 New types of holomorphic functions . . . . . . . . . . . . 1.3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Construction of holomorphic functions . . . . . . . 1.4 Integral theorems . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 General integral theorems . . . . . . . . . . . . . . 1.4.2 Integral theorems for holomorphic functions . . . . 1.5 Polynomial systems . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Fueter polynomials . . . . . . . . . . . . . . . . . .
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