E-Book Overview
This book attempts to bridge the gap between the principles of pure mathematics and the applications in physical science. After the Mobius inversion formula had been considered as purely academic, or beyond what was useful in the physics community for more than 150 years, the apparently obscure result in classical mathematics suddenly appears to be connected to a variety of important inverse problems in physical science. This book only requires readers to have some background in elementary calculus and general physics, and the prerequisite knowledge of number theory is not needed. It will be attractive to our multidisciplinary readers interested in the Mobius technique, which is a tiny but important part of the number-theoretic methods. It will inspire many students and researchers in both physics and mathematics. In a practical problem, continuity and discreteness are often correlated, and few textbooks have given attention to this wide and important field as this one. Clearly, this book will be an essential supplement for many existing courses such as mathematical physics, elementary number theory and discrete mathematics.
E-Book Content
MÖBIUS INVERSION IN PHYSICS Tsinghua Report and Review in Physics Series Editor: Bangfen Zhu (Tsinghua University, China) Vol. 1 Möbius Inversion in Physics by Nanxian Chen ZhangFang - Mobius Inversion in Physics.pmd 2 4/6/2010, 9:00 AM MÖBIUS INVERSION IN PHYSICS Chen Nanxian Tsinghua University, China World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TA I P E I • CHENNAI Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. MÖBIUS INVERSION IN PHYSICS Tsinghua Report and Review in Physics — Vol. 1 Copyright © 2010 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-981-4291-62-0 ISBN-10 981-4291-62-5 Printed in Singapore. ZhangFang - Mobius Inversion in Physics.pmd 1 4/6/2010, 9:00 AM April 2, 2010 1:6 World Scientific Book - 9in x 6in To my parents Chen Xuan-shan and Lin De-yin. To my wife He Tong and my daughter Jasmine. v mainC April 2, 2010 1:6 World Scientific Book - 9in x 6in This page intentionally left blank mainC April 2, 2010 1:6 World Scientific Book - 9in x 6in Preface August Ferdinand M¨ obius gave three profound gifts to the scientific world: the M¨ obius strip in topology, the M¨ obius transformation in projective geometry, and the M¨ obius inverse formula in number theory. This book focuses on the last, and perhaps the least well known of these gifts, introducing the applications of the M¨ obius inverse formula to a variety of fields in physics, including mechanics, optics, statistical physics, solid state physics, astrophysics, parallel Fourier transform,