This book introduces the main ideas and fundamental methods of analysis. A key feature of this lively yet rigorous and systematic exposition is the historical accounts of ideas and methods pertaining to the relevant topics. Most interesting and useful are the connections developed between analysis and other mathematical disciplines, in this case, numerical analysis and probability theory. "Mathematical Analysis: Approximation and Discrete Processes" features beautiful illustrations, examples, and exercises at the end of each chapter.

This fairly self-contained work embraces a broad range of topics in analysis at the graduate level, requiring only a sound knowledge of calculus and the functions of one variable Replete with beautiful illustrations, examples, exercises at the end of each chapter, and a comprehensive index May be used in graduate seminars and courses, or as a reference text by mathematicians, physicists, and engineers

Klappentext

This volume! 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.

Inhalt

1. Real Numbers and Natural Numbers.- 1.1 Introduction.- 1.2 The Axiomatic Approach to Real Numbers.- 1.3 Natural Numbers.- 1.4 Summing Up.- 1.5 Exercises.- 2. Sequences of Real Numbers.- 2.1 Sequences.- 2.2 Equivalent Formulations of the Continuity Axiom.- 2.3 Limits of Sequences and Continuity.- 2.4 Some Special Sequences.- 2.5 An Alternative Definition of Exponentials and Logarithms.- 2.6 Summing Up.- 2.7 Exercises.- 3. Integer Numbers: Congruences, Counting and Infinity.- 3.1 Congruences.- 3.2 Combinatorics.- 3.3 Infinity.- 3.4 Summing Up.- 3.5 Exercises.- 4. Complex Numbers.- 4.1 Complex Numbers.- 4.2 Sequences of Complex Numbers.- 4.3 Some Elementary Applications.- 4.4 Summing Up.- 4.5 Exercises.- 5. Polynomials, Rational Functions and Trigonometric Polynomials.- 5.1 Polynomials.- 5.2 Solutions of Polynomial Equations.- 5.3 Rational Functions.- 5.4 Sinusoidal Functions and Their Sums.- 5.5 Summing Up.- 5.6 Exercises.- 6. Series.- 6.1 Basic Facts.- 6.2 Taylor Series, e and ?.- 6.3 Series of Nonnegative Terms.- 6.4 Series of Terms of Arbitrary Sign.- 6.5 Series of Products.- 6.6 Products of Series.- 6.7 Rearrangements.- 6.8 Summing Up.- 6.9 Exercises.- 7. Power Series.- 7.1 Basic Theory.- 7.2 Further Results.- 7.3 Some Applications.- 7.4 Further Applications.- 7.5 Summing Up.- 7.6 Exercises.- 8. Discrete Processes.- 8.1 Recurrences.- 8.2 One-Dimensional Dynamical Systems.- 8.3 Two-Dimensional Dynamical Systems.- 8.4 Exercises.- A. Mathematicians and Other Scientists.- B. Bibliographical Notes.- C. Index.