# Mathematical Analysis

## Approximation and Discrete Processes

• Mariano Giaquinta
• Giuseppe Modica
Textbook

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

### 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

• DOI https://doi.org/10.1007/978-0-8176-4414-7
• Copyright Information Birkhäuser Boston 2004
• Publisher Name Birkhäuser, Boston, MA
• eBook Packages
• Print ISBN 978-0-8176-4337-9
• Online ISBN 978-0-8176-4414-7
• Buy this book on publisher's site
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