An engraving from Mario Bettini Aerarium phylosophiae matematicae, The prince provides money with which the Society of Jesus educates people in

Transcription

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

2 Mariana Giaquinta Giuseppe Madica Mathematical Analysis Approximation and Discrete Processes Springer Science+ Business Media, LLC

3 Mariano Giaquinta Scuola Normale Superiore Dipartimento di Matematica Pisa Italy Giuseppe Modica Universita degli Studi di Firenze Dipartimento di Matematica Applicata Firenze Italy Library of Congress Cataloging-In-Publication Data Giaquinta, Mariano, [Analisi matematica. 2, Approssimazione e processi discreti. English] Mathematical analysis: approximation and discrete processes / Mariano Giaquinta, Giuseppe Modica. p. cm. Includes bibliographical references and index. ISBN ISBN (ebook) DOI / Mathematical analysis. I. Modica, Giuseppe. II. Title. QA300.G dc CIP AMS Subject Classifications: OOA35, 0lA20, 01A35, OIA40, 01A45, 05-01, 11-01,26-01, 26A03, 26AI5, 26AI6, 26A18, 26A45, 26B05, 30BIO, 34-01, 37-01,40-01,41-01,60-01 ISBN Printed on acid-free paper. Printed on acid-free paper Springer Science+Business Media New York Originally published by Birkhiiuser Boston in 2004 \1)) Birkhiiuser f.qp) Softcover reprint of the hardcover 1st edition 2004 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhiiuser 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 I SPIN

4 Preface This volume! aims at introducing some basic ideas for studying approximation processes and, more generally, discrete processes. The study of discrete processes, which has grown together with the study of infinitesimal calculus, 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 introduce, 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 numbers 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 possible 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, We have discussed these notions in M. Giaquinta, G. Modica, Mathematical Analysis. Functions of One Variable, Birkhiiuser, Boston, In this volume we shall refer to this work as [GM1].

5 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 [GMt] 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 Alessandro 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 illustrations. Our special thanks go also to all members of the editorial technical staff of Birkhiiuser 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 addresses are giaquinta~sns.it modica~dma.unifi.it We shall try to keep up an errata corrige at the following webpage: Mariano Giaquinta Giuseppe Modica Pisa and Firenze October 2003

6 Preface... v 1. Real Numbers and Natural Numbers Introduction... 1 a. Numbers and measurement b. Never-ending processes c. Back to numbers d. An axiomatic or a constructive approach? The Axiomatic Approach to Real Numbers Algebraic and order properties a. Axioms for addition b. Axioms for multiplication c. The distributive law d. Order Continuity property a. Supremum b. The extended real line c. Dedekind cuts of lr Uniqueness of reals Natural Numbers a. Natural numbers and the principle of induction.. 17 b. Approximation of reals by rational numbers c. Recursive statements Summing Up Exercises Sequences of Real Numbers Sequences a. Limit of a sequence b. Properties of limits and calculus c. Limits of monotone sequences d. Sequences and supremum e. Subsequences Equivalent Formulations of the Continuity Axiom a. The principle of nested intervals or Cantor's principle

7 viii b. Cauchy criterion c. Upper and lower limits d. Bolzano-Weierstrass theorem e. The continuity property of the reals Limits of Sequences and Continuity a. Limits of sequences and limits of functions b. Continuity in terms of sequences Some Special Sequences a. Elementary limits b. Powers, exponentials and factorials c. Wallis and Stirling formulas d. Numerical integration An Alternative Definition of Exponentials and Logarithms. 59 a. A definition of ax using continuity..., 59 b. Euler's number e c. Derivative of the exponential Summing Up Exercises Integer Numbers: Congruences, Counting and Infinity Congruences Euclid's algorithm a. The greatest common divisor b. Integer solutions of first order equations Prime factorization Linear congruences Euler's function RSA Cryptography Combinatorics Samples, mappings and subsets a. Ordered samples and mappings b. Nonordered samples and subsets c. Ordered lists d. The formula of inclusion and exclusion e. Surjective maps Drawings Location problems The hypergeometric and multinomial distributions Infinity The mathematical analysis of infinity a. Cardinality b. Cantor-Bernstein theorem c. Denumerable sets d. The axiom of choice e. The power of the continuum f. The continuum hypothesis Some information on the theory of sets

8 ix 3.4 Summing Up Exercises Complex Numbers Complex Numbers a. The system of complex numbers b. The n-th roots c. Complex exponential and logarithm Sequences of Complex Numbers a. Definitions b. Weierstrass's theorem Some Elementary Applications A few applications of the complex notation A few applicatons to elementary Euclidean geometry 135 a. Special points of a triangle b. Equilateral triangles Summing Up Exercises Polynomials, Rational Functions and Trigonometric Polynomials... '" Polynomials The Division Algorithm a. Euclid's algorithm and Bezout identity b. Factorization c. The factor theorem The fundamental theorem of algebra a. Factorization in C b. Simple and multiple roots of a polynomial c. Factorization in lr Solutions of Polynomial Equations Solutions by radicals Distribution of the roots of a polynomial a. Descartes's law of signs b. Sturm's theorem Rational Functions a. Decomposition in C b. Decomposition in lr c. Integration of rational functions Sinusoidal Functions and Their Sums Trigonometric polynomials a. Periodic functions b. Trigonometric polynomials c. Spectrum and energy identity d. Sampling Sums of sinusoidal functions Summing Up

9 x 5.6 Exercises Series Basic Facts a. Definitions and examples b. A necessary condition for convergence c. Series and improper integrals d. Decimals Taylor Series, e and 7r a. The number 7r. 198 b. More on the number e Series of Nonnegative Terms a. Series of positive decreasing terms b. The root and ratio tests c. Viete's formula for 7r d. Euler and Wallis formulas Series of Terms of Arbitrary Sign a. Absolute convergence b. Series of complex terms Series of Products a. Alternating series b. Summation by parts c. Sequences of bounded total variation d. Dirichlet and Abel theorems Products of Series Rearrangements Summing Up Exercises Power Series Basic Theory Circle of convergence a. The disc and the domain of convergence Continuity of the sum a. Uniform Convergence b. Continuity of uniform limits c. Uniform convergence of power series Differentiation and integration a. Series of derivatives and of integrals b. Real power series c. Power series and Taylor series d. Complex series Further Results Boundary values Product and composition of power series a. Weierstrass's double series theorem Taylor series: examples

10 xi 7.3 Some Applications Complex functions An alternate definition of Jr, e and of elementary functions Series solutions of differential equations Generating functions and combinatorics a. Generating functions b. Enumerators c. Exponential enumerators d. A few location problems e. Partitions of a set Further Applications Euler-MacLaurin summation formula a. Bernoulli numbers b. Bernoulli polynomials c. Euler-MacLaurin formula and Stirling's approximation Euler r function a. Definition and characterizations b. Functional relations c. Asymptotics of rand 'lj; Summing Up Exercises Discrete Processes Recurrences Linear difference equations a. First order linear difference equations b. Second order homogeneous difference equations c. Second order nonhomogeneous difference equations d. Z-transform and Laplace transform e. Fibonacci's numbers Some nonlinear examples a. Simple examples b. Evaluating algorithm performance c. Rate of convergence Continued fractions a. Definitions and elementary properties b. Developments as continuous fractions c. Infinite continued fractions d. Irrationals and approximations by rationals e. Order of approximation and transcendental numbers One-Dimensional Dynamical Systems Discretization and models a. Euler's method

11 xii b. Runge-Kutta method c. Models Examples of one-dimensional dynamics a. Expansive dynamics b. Contractive dynamics: fixed points c. Sinks and sources d. Periodic orbits e. Periodic-doubling cascade transition to chaos f. The intermittency phenomenon g. Ergodic dynamics Chaotic dynamics a. Sensitive dependence on initial conditions and the Lyapunov exponent b. Chaotic orbits c. Bernoulli's shift d. The triangular map e. Conjugate maps Chaotic attractors, basins of attraction Cantor sets and other self-similar sets a. Measure and dimension b. Cantor sets c. Iterated function systems d. Dimension of the invariant set Two-Dimensional Dynamical Systems Game of life Fractal boundaries a. Julia sets b. Mandelbrot set Fractals on the computer Exercises A. Mathematicians and Other Scientists B. Bibliographical Notes C. Index