Mathematical Analysis: Approximation and Discrete Processes

402 Pages·2004·31.91 MB·English

by  Mariano Giaquinta

An engraving from Mario Bettini Aerarium phylosophiae matematicae,
1648. The prince provides money with which the Society of Jesus edu
cates people in mathematics for aesthetic and practical purposes.

Mariano Giaquinta
Giuseppe Modica
Mathematical Analysis
Approximation and Discrete Processes
Birkhauser
Boston • Basel· Berlin
Mariano Giaquinta Giuseppe Modica
Scuola Normale Superiore Universita degli Studi di Firenze
Dipartimento di Matematica Dipartimento di Matematica Applicata
1-56100 Pisa 1-50139 Firenze
Italy Italy
Library of Congress Cataloging-in-Publication Data
Giaquinta, Mariano, 1947-
[Analisi matematica. 2, Approssimazione e processi discreti. English]
Mathematical analysis: approximation and discrete processes I Mariano Giaquinta,
Giuseppe Modica.
p. cm.
Includes bibliographical references and index.
ISBN 0-8176-4313-3 (alk. paper)
1. Mathematical analysis. I. Modica, Giuseppe. n. Title.
QA300.G49713 2004
515-<lc22
2004043696
CIP
AMS Subject Classifications: OOA35, 0lA20, 0lA35, 0lA40, 0lA45, 05-0 I, 11-01, 26-0 I, 26A03,
26AI5, 26AI6, 26AI8, 26A45, 26B05, 30BIO, 34-01, 37-01,40-01,41-01,60-01
ISBN 0-8176-4313-3 Printed on acid-free paper.
mil>
Printed on acid-free paper.
©2004 Birkhauser Boston Birkhiiuser l!(p)
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Birkhauser Boston, c/o Springer-Verlag New York, Inc., 175 Fifth
Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or
scholarly analysis. Use in connection with any form of information storage and retrieval, electronic
adaptation, computer software, or by similar or dissimilar methodology now known or hereafter
developed is forbidden.
The use in this publication of trade names, trademarks, service marks and similar terms, even if they
are not identified as such, is not to be taken as an expression of opinion as to whether or not they are
subject to property rights.
Printed in the United States of America. (TXQIHP)
987 6 5 432 1 SPIN 10890261
Birkhauser is a part of Springer Science+Business Media
www.birkhauser.com
Preface
This volumel aims at introducing some basic ideas for studying approxima
tion processes and, more generally, discrete processes. The study of discrete
processes, which has grown together with the study of infinitesimal calcu
lus, has become more and more relevant with the use of computers. The
volume is suitably divided in two parts.
In the first part we illustrate the numerical systems of reals, of integers
as a subset of the reals, and of complex numbers. In this context we intro
duce, in Chapter 2, the notion of sequence which invites also a rethinking
of the notions of limit and continuity2 in terms of discrete processes; then,
in Chapter 3, we discuss some elements of combinatorial calculus and the
mathematical notion of infinity. In Chapter 4 we introduce complex num
bers and illustrate some of their applications to elementary geometry; in
Chapter 5 we prove the fundamental theorem of algebra and present some
of the elementary properties of polynomials and rational functions, and of
finite sums of harmonic motions.
In the second part we deal with discrete processes, first with the process
of infinite summation, in the numerical case, i.e., in the case of numerical
series in Chapter 6, and in the case of power series in Chapter 7. The last
chapter provides an introduction to discrete dynamical systems; it should
be regarded as an invitation to further study.
We have tried to keep the treatment of topics as independent as pos
sible even at the cost of some repetition; usually, we assume as known
the content of [GMl], but, whenever possible, we provide an alternative
elementary treatment in order to allow the use of part of this volume on
sequences and series, independently from infinitesimal calculus.
The main body is formed by Chapter 1, Sections 2 and 3, Chapter 2,
Sections 1, 2, 3, and 4, Chapter 4, Sections 1 and 2, Chapter 6, Sections 1, 2,
3, and 4 and Chapter 7, Sections 1 and 2 for about a third of the whole. The
rest of the material may appear as heterogeneous; it develops in branches
that eventually meet, from which it is easy to select several paths. However,
1 This volume is a translation and revised edition of M. Giaquinta, G. Modica, Analisi
Matematica, II, Approssimazione e processi discreti, Pitagora Editrice, Bologna,
1999.
2 We have discussed these notions in M. Giaquinta, G. Modica, Mathematical Analysis.
Functions of One Variable, Birkhauser, Boston, 2003. In this volume we shall refer
to this work as [GM1].
vi Preface
we believe that the whole of the material is, besides its intrinsic interest,
fundamentally basic for any further study of mathematical analysis.
As in [GMl] an appropriate number of exercises are distributed in the
text and at the end of each chapter. They are marked by the symbol';
the double " indicates exercises that are more difficult.
We are greatly indebted to Cecilia Conti for her help in polishing our
first draft and we warmly thank her. We would like to thank also Alessan
dro Berarducci, Roberto Conti, Pietro Majer and Stefano Marmi for their
comments when preparing the Italian edition, and Stefan Hildebrandt for
his comments and suggestions concerning especially the choice of illustra
tions. Our special thanks go also to all members of the editorial technical
staff of Birkhauser for the excellent quality of their work and especially to
the executive editor Ann Kostant.
Note: We have tried to avoid misprints and errors. But, as most authors,
we are imperfect authors. We will be very grateful to anybody who wants
to inform us about errors or just misprints or wants to express criticism
or other comments. Our e-mail addresses are
giaquinta~sns.it modica~dma.unifi.it
We shall try to keep up an errata corrige at the following webpage:
http://www.sns.it/-giaquinta
Mariano Giaquinta
Giuseppe Modica
Pisa and Firenze
October 2003
Contents
Preface..................................................... v
1. Real Numbers and Natural Numbers................... 1
1.1 Introduction......................................... 1
a. Numbers and measurement. . . . .... . . . . .. . . . . . . 2
b. Never-ending processes. . . . . . . . . . . . . . . . . . . . . . . . 4
c. Back to numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
d. An axiomatic or a constructive approach? ....... 8
1.2 The Axiomatic Approach to Real Numbers. . . . . . . . . . . . . . 9
1.2.1 Algebraic and order properties. . . . . . . . . . . . . . . . . . . . 9
a. Axioms for addition .......................... 10
b. Axioms for multiplication. . . . . . . . . . . . . . . . . . . . .. 10
c. The distributive law .......................... 11
d. Order ....................................... 12
1.2.2 Continuity property. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13
a. Supremum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13
b. The extended real line ........................ 15
c. Dedekind cuts of R . . . . . . . . . . . . . . . . . . . . . . . . . .. 15
1.2.3 Uniqueness of reals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 16
1.3 Natural Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17
a. Natural numbers and the principle of induction .. 17
b. Approximation of reals by rational numbers. . . . .. 20
c. Recursive statements. . . . . . . . . . . . . . . . . . . . . . . . .. 21
1.4 Summing Up ........................................ 25
1.5 Exercises............................................ 26
2. Sequences of Real Numbers ............................ 31
2.1 Sequences ........................................... 31
a. Limit of a sequence . . . . . . . . . . . . . . . . . . . . . . . . . .. 35
b. Properties of limits and calculus. . . . . . . . . . . . . . .. 36
c, Limits of monotone sequences . . . . . . . . . . . . . . . . .. 39
d. Sequences and supremum. . . . . . . . . . . . . . . . . . . . .. 40
e. Subsequences ............................. . .. 40
2.2 Equivalent Formulations of the Continuity Axiom ........ 41
a. The principle of nested intervals
or Cantor's principle. . . . . . . . . . . . . . . . . . . . . . . . .. 41
viii Contents
b. Cauchy criterion ............................. 42
c. Upper and lower limits ........................ 44
d. Bolzano-Weierstrass theorem .................. 46
e. The continuity property of the reals. . . . . . . . . . . .. 46
2.3 Limits of Sequences and Continuity ..................... 47
a. Limits of sequences and limits of functions. . . . . .. 47
b. Continuity in terms of sequences ............... 48
2.4 Some Special Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 49
a. Elementary limits ............................ 50
b. Powers, exponentials and factorials ............. 53
c. Wallis and Stirling formulas. . . . . . . . . . . . . . . . . . .. 55
d. Numerical integration. . . . . . . . . . . . . . . . . . . . . . . .. 57
2.5 An Alternative Definition of Exponentials and Logarithms. 59
a. A definition of aX using continuity. . . . . . . . . . . . .. 59
b. Euler's number e . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 62
c. Derivative of the exponential. . . . . . . . . . . . . . . . . .. 62
2.6 Summing Up ........................................ 63
2.7 Exercises............................................ 65
3. Integer Numbers: Congruences, Counting and Infinity .. 71
3.1 Congruences......................................... 71
3.1.1 Euclid's algorithm .............................. 71
a. The greatest common divisor .................. 72
b. Integer solutions of first order equations . . . . . . . .. 74
3.1.2 Prime factorization. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 77
3.1.3 Linear congruences. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 79
3.1.4 Euler's function ¢ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 82
3.1.5 RSA Cryptography .............................. 84
3.2 Combinatorics....................................... 88
3.2.1 Samples, mappings and subsets ................... 89
a. Ordered samples and mappings. . . . . . . . . . . . . . . .. 89
b. Nonordered samples and subsets. . . . . . . . . . . . . . .. 91
c. Ordered lists. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 92
d. The formula of inclusion and exclusion . . . . . . . . .. 93
e. Surjective maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 94
3.2.2 Drawings ...................................... 95
3.2.3 Location problems .............................. 95
3.2.4 The hypergeometric and multinomial distributions .. 97
3.3 Infinity.............................................. 99
3.3.1 The mathematical analysis of infinity. . . . . . . . . . . . .. 99
a. Cardinality .................................. 100
b. Cantor-Bernstein theorem ..................... 102
c. Denumerable sets ............................. 103
d. The axiom of choice .......................... 104
e. The power of the continuum ................... 105
f. The continuum hypothesis ..................... 106
3.3.2 Some information on the theory of sets ............ 107
Contents ix
3.4 Summing Up ........................................ 111
3.5 Exercises ............................................ 113
4. Complex Numbers ...................................... 121
4.1 Complex Numbers ............ '" ..................... 122
a. The system of complex numbers ................ 122
b. The n-th roots ............................... 128
c. Complex exponential and logarithm ............. 129
4.2 Sequences of Complex Numbers ........................ 131
a. Definitions ................................... 131
b. Weierstrass's theorem ......................... 132
4.3 Some Elementary Applications ......................... 133
4.3.1 A few applications of the complex notation ......... 133
4.3.2 A few applicatons to elementary Euclidean geometry 135
a. Special points of a triangl~ ..................... 136
b. Equilateral triangles .......................... 138
4.4 Summing Up ........................................ 140
4.5 Exercises ............................................ 142
5. Polynomials, Rational Functions
and Trigonometric Polynomials ......................... 145
5.1 Polynomials ......................................... 145
5.1.1 The Division Algorithm .......................... 147
a. Euclid's algorithm and Bezout identity .......... 148
b. Factorization ................................. 150
c. The factor theorem ........................... 150
5.1.2 The fundamental theorem of algebra .............. 153
a. Factorization in C ............................ 153
b. Simple and multiple roots of a polynomial ....... 156
c. Factorization in ~ ............................ 157
5.2 Solutions of Polynomial Equations ...................... 158
5.2.1 Solutions by radicals ............................ 158
5.2.2 Distribution of the roots of a polynomial ........... 163
a. Descartes's law of signs ........................ 164
b. Sturm's theorem ............................. 164
5.3 Rational Functions ................................... 166
a. Decomposition in C ........................... 166
b. Decomposition in ~ ........................... 170
c. Integration of rational functions ................ 171
5.4 Sinusoidal Functions and Their Sums ................... 173
5.4.1 Trigonometric polynomials ....................... 174
a. Periodic functions ............................ 174
b. Trigonometric polynomials ..................... 175
c. Spectrum and energy identity .................. 176
d. Sampling .................................... 178
5.4.2 Sums of sinusoidal functions ...................... 181
5.5 Summing Up ........................................ 183