Bicomplex Holomorphic Functions The Algebra, Geometry and Analysis o
The purpose of this book is to develop the foundations of the theory of holomorphicity on the ring of bicomplex numbers. Accordingly, the main focus is on expressing the similarities with, and differences from, the classical theory of one complex variable
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M. Elena Luna-Elizarrarás Michael Shapiro Daniele C. Struppa Adrian Vajiac
Bicomplex Holomorphic Functions:
The Algebra, Geometry and Analysis
of Bicomplex Numbers
Frontiers in Mathematics
Advisory Editorial Board Leonid Bunimovich (Georgia Institute of Technology, Atlanta) William Y. C. Chen (Nankai University, Tianjin, China) Benoît Perthame (Université Pierre et Marie Curie, Paris) Laurent Saloff-Coste (Cornell University, Ithaca) Igor Shparlinski (Macquarie University, New South Wales) Wolfgang Sprössig (TU Bergakademie Freiberg) Cédric Villani (Institut Henri Poincaré, Paris)
M. Elena Luna-Elizarrarás • Michael Shapiro Daniele C. Struppa • Adrian Vajiac
Bicomplex Holomorphic Functions The Algebra, Geometry and Analysis of Bicomplex Numbers
M. Elena Luna-Elizarrarás Escuela Sup. de Física y Matemáticas Instituto Politécnico Nacional Mexico City, Mexico
Michael Shapiro Escuela Sup. de Física y Matemáticas Instituto Politécnico Nacional Mexico City, Mexico
Daniele C. Struppa Schmid College of Science and Technology Chapman University Orange, CA, USA
Adrian Vajiac Schmid College of Science and Technology Chapman University Orange, CA, USA
ISSN 1660-8046 ISSN 1660-8054 (electronic) Frontiers in Mathematics ISBN 978-3-319-24866-0 ISBN 978-3-319-24868-4 (eBook) DOI 10.1007/978-3-319-24868-4 Library of Congress Control Number: 2015954663 Mathematics Subject Classification (2010): 30G35, 32A30, 32A10 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 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.
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Contents Introduction 1
The 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
1
Bicomplex Numbers Definition of bicomplex numbers . . . . . . . . . . . . . . . . Versatility of different writings of bicomplex numbers . . . . . Conjugations of bicomplex numbers . . . . . . . . . . . . . .
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