Calculus with Complex Numbers

Transcription

1 Calculus with Complex Numbers

2 Calculus with Complex Numbers John B. Reade

3 First published 2003 by Taylor & Francis 11 New Fetter Lane, London EC4P 4EE Simultaneously published in the USA and Canada by Taylor & Francis Inc. 29 West 35th Street, New York, NY Taylor & Francis is an imprint of the Taylor & Francis Group Typeset in Times New Roman by Newgen Imaging Systems (P) Ltd, Chennai, India Printed and bound in Great Britain by MPG Books Ltd, Bodmin All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data A catalog record for this book has been requested ISBN (hbk) ISBN X (pbk)

4 Contents Preface 1 Complex numbers 1 2 Complex functions 14 3 Derivatives 24 4 Integrals 36 5 Evaluation of finite real integrals 49 6 Evaluation of infinite real integrals 53 7 Summation of series 65 8 Fundamental theorem of algebra 75 Solutions to examples 82 Appendix 1: Cauchy s theorem 93 Appendix 2: Half residue theorem 95 Bibliography 97 vii

5 Preface This book is based on the premise that the learning curve is isomorphic to the historical curve. In other words, the learning order of events is the same as the historical order of events. For example, we learn arithmetic before we learn algebra. We learn how before we learn why. Historically, calculus with real numbers came first, initiated by Newton and Leibnitz in the seventeenth century. Even though complex numbers had been known about from the time of Fibonacci in the thirteenth century, nobody thought of doing calculus with complex numbers until the nineteenth century. Here the pioneers were Cauchy and Riemann. Rigorous mathematics as we know it today did not come into existence until the twentieth century. It is important to observe that the nineteenth century mathematicians had the right theorems, even if they didn t always have the right proofs. The learning process proceeds similarly. Real calculus comes first, followed by complex calculus. In both cases we learn by using calculus to solve problems. It is when we have seen what a piece of mathematics can do that we begin to ask whether it is rigorous. Practice always comes before theory. The emphasis of this book therefore is on the applications of complex calculus, rather than on the foundations of the subject. A working knowledge of real calculus is assumed, also an acquaintance with complex numbers. A background not unlike that of an average mathematician in Equivalently, a British student just starting at university. The approach is to ask what happens if we try to do calculus with complex numbers instead of with real numbers. We find that parts are the same, whilst other parts are strikingly different. The most powerful result is the residue theorem for evaluating complex integrals. Students wishing to study the subject at a deeper level should not find that they have to unlearn anything presented here. I would like to thank the mathematics students at Manchester University for sitting patiently through lectures on this material over the years. Also for their feedback (positive and negative) which has been invaluable. The book is respectfully dedicated to them. John B. Reade June 2002