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Mathematical Analysis

Approximation and Discrete Processes

  • Mariano Giaquinta
  • Giuseppe Modica
Textbook

Table of contents

  1. Front Matter
    Pages i-xii
  2. Mariano Giaquinta, Giuseppe Modica
    Pages 1-30
  3. Mariano Giaquinta, Giuseppe Modica
    Pages 31-70
  4. Mariano Giaquinta, Giuseppe Modica
    Pages 71-120
  5. Mariano Giaquinta, Giuseppe Modica
    Pages 121-144
  6. Mariano Giaquinta, Giuseppe Modica
    Pages 145-186
  7. Mariano Giaquinta, Giuseppe Modica
    Pages 187-234
  8. Mariano Giaquinta, Giuseppe Modica
    Pages 235-295
  9. Mariano Giaquinta, Giuseppe Modica
    Pages 297-376
  10. Back Matter
    Pages 377-388

About this book

Introduction

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.

Keywords

Complex analysis: one variable Probability theory Taylor series approximation/probability calculus discrete dynamical systems logarithm numerical analysis ordinary differential equations

Authors and affiliations

  • Mariano Giaquinta
    • 1
  • Giuseppe Modica
    • 2
  1. 1.Dipartimento di MatematicaScuola Normale SuperiorePisaItaly
  2. 2.Dipartimento di Matematica ApplicataUniversità degli Studi di FirenzeFirenzeItaly

Bibliographic information

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