Real and Complex Analysis

569 Pages·2009·4.72 MB·English

by  Apelian

Real and Complex
Analysis
C8067_FM.indd 1 11/3/09 1:42:10 PM
PURE AND APPLIED MATHEMATICS
A Program of Monographs, Textbooks, and Lecture Notes
EXECUTIVE EDITORS
Earl J. Taft Zuhair Nashed
Rutgers University University of Central Florida
Piscataway, New Jersey Orlando, Florida
EDITORIAL BOARD
M. S. Baouendi Anil Nerode
University of California, Cornell University
San Diego
Freddy van Oystaeyen
Jane Cronin University of Antwerp,
Rutgers University Belgium
Jack K. Hale Donald Passman
Georgia Institute of Technology University of Wisconsin,
Madison
S. Kobayashi
University of California, Fred S. Roberts
Berkeley Rutgers University
Marvin Marcus David L. Russell
University of California, Virginia Polytechnic Institute
Santa Barbara and State University
W. S. Massey Walter Schempp
Yale University Universität Siegen
C8067_FM.indd 2 11/3/09 1:42:10 PM
MONOGRAPHS AND TEXTBOOKS IN
PURE AND APPLIED MATHEMATICS
Recent Titles
Christof Eck, Jiri Jarusek, and Miroslav Krbec, Unilateral Contact Problems: Variational
Methods and Existence Theorems (2005)
M. M. Rao, Conditional Measures and Applications, Second Edition (2005)
A. B. Kharazishvili, Strange Functions in Real Analysis, Second Edition (2006)
Vincenzo Ancona and Bernard Gaveau, Differential Forms on Singular Varieties:
De Rham and Hodge Theory Simplified (2005)
Santiago Alves Tavares, Generation of Multivariate Hermite Interpolating Polynomials
(2005)
Sergio Macías, Topics on Continua (2005)
Mircea Sofonea, Weimin Han, and Meir Shillor, Analysis and Approximation of Contact
Problems with Adhesion or Damage (2006)
Marwan Moubachir and Jean-Paul Zolésio, Moving Shape Analysis and Control:
Applications to Fluid Structure Interactions (2006)
Alfred Geroldinger and Franz Halter-Koch, Non-Unique Factorizations: Algebraic,
Combinatorial and Analytic Theory (2006)
Kevin J. Hastings, Introduction to the Mathematics of Operations Research
with Mathematica®, Second Edition (2006)
Robert Carlson, A Concrete Introduction to Real Analysis (2006)
John Dauns and Yiqiang Zhou, Classes of Modules (2006)
N. K. Govil, H. N. Mhaskar, Ram N. Mohapatra, Zuhair Nashed, and J. Szabados,
Frontiers in Interpolation and Approximation (2006)
Luca Lorenzi and Marcello Bertoldi, Analytical Methods for Markov Semigroups (2006)
M. A. Al-Gwaiz and S. A. Elsanousi, Elements of Real Analysis (2006)
Theodore G. Faticoni, Direct Sum Decompositions of Torsion-Free Finite
Rank Groups (2007)
R. Sivaramakrishnan, Certain Number-Theoretic Episodes in Algebra (2006)
Aderemi Kuku, Representation Theory and Higher Algebraic K-Theory (2006)
Robert Piziak and P. L. Odell, Matrix Theory: From Generalized Inverses to
Jordan Form (2007)
Norman L. Johnson, Vikram Jha, and Mauro Biliotti, Handbook of Finite
Translation Planes (2007)
Lieven Le Bruyn, Noncommutative Geometry and Cayley-smooth Orders (2008)
Fritz Schwarz, Algorithmic Lie Theory for Solving Ordinary Differential Equations (2008)
Jane Cronin, Ordinary Differential Equations: Introduction and Qualitative Theory,
Third Edition (2008)
Su Gao, Invariant Descriptive Set Theory (2009)
Christopher Apelian and Steve Surace, Real and Complex Analysis (2010)
C8067_FM.indd 3 11/3/09 1:42:10 PM
C8067_FM.indd 4 11/3/09 1:42:10 PM
Real and Complex
Analysis
Christopher Apelian
Steve Surace
with Akhil Mathew
C8067_FM.indd 5 11/3/09 1:42:10 PM
Chapman & Hall/CRC
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2010 by Taylor and Francis Group, LLC
Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Printed in the United States of America on acid-free paper
10 9 8 7 6 5 4 3 2 1
International Standard Book Number-13: 978-1-58488-807-9 (Ebook-PDF)
This book contains information obtained from authentic and highly regarded sources. Reasonable efforts
have been made to publish reliable data and information, but the author and publisher cannot assume
responsibility for the validity of all materials or the consequences of their use. The authors and publishers
have attempted to trace the copyright holders of all material reproduced in this publication and apologize to
copyright holders if permission to publish in this form has not been obtained. If any copyright material has
not been acknowledged please write and let us know so we may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmit-
ted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented,
including photocopying, microfilming, and recording, or in any information storage or retrieval system,
without written permission from the publishers.
For permission to photocopy or use material electronically from this work, please access www.copyright.
com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood
Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and
registration for a variety of users. For organizations that have been granted a photocopy license by the CCC,
a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used
only for identification and explanation without intent to infringe.
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
TomywifePaula.
Forthesacrificesshemadewhilethisbookwasbeingwritten.
AndtoEllie.
Forbeingexactlywhosheis.
-CA
Tofamilyandfriends.
-SS

CONTENTS
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Acknowledgments. . . . . . . . . . . . . . . . . . . . . xvii
TheAuthors . . . . . . . . . . . . . . . . . . . . . . . . xix
1. TheSpacesR, Rk,andC . . . . . . . . . . . . . . . . . 1
1 THEREALNUMBERSR . . . . . . . . . . . . . . . . . . . . . . . 1
PropertiesoftheRealNumbersR,2. TheAbsoluteValue,7. Intervals
inR,10
2 THEREALSPACESRk. . . . . . . . . . . . . . . . . . . . . . . . 10
PropertiesoftheRealSpacesRk,11. InnerProductsandNormson
Rk,14. IntervalsinRk,18
3 THECOMPLEXNUMBERSC. . . . . . . . . . . . . . . . . . . . . 19
An Extension of R2, 19. Properties of Complex Numbers, 21. A
Norm on C and the Complex Conjugate of z, 24. Polar Notation
andtheArgumentsofz,26. Circles,Disks,Powers,andRoots,30.
MatrixRepresentationofComplexNumbers,34
4 SUPPLEMENTARYEXERCISES . . . . . . . . . . . . . . . . . . . . 35
2. Point-SetTopology . . . . . . . . . . . . . . . . . . . . 41
1 BOUNDEDSETS . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
BoundedSetsinX,42. BoundedSetsinR,44. Spheres,Balls,and
Neighborhoods,47
2 CLASSIFICATIONOFPOINTS . . . . . . . . . . . . . . . . . . . . 50
Interior, Exterior, and Boundary Points, 50. Limit Points and Iso-
latedPoints,53
3 OPENANDCLOSEDSETS . . . . . . . . . . . . . . . . . . . . . . 55
OpenSets,55. ClosedSets,58. RelativelyOpenandClosedSets,61.
Density,62
ix