Mariano Giaquinta, Giuseppe Modica

Mathematical Analysis (eBook, PDF)

Approximation and Discrete Processes

• Format: PDF

Produktbeschreibung

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.

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Produktdetails

• Verlag: Birkhäuser Boston
• Seitenzahl: 388
• Erscheinungstermin: 31. August 2012
• Englisch
• ISBN-13: 9780817644147
• Artikelnr.: 44193605

Autorenporträt

Mariano Giaquinta, Scuola Normale Superiore, Pisa, Italy / Giuseppe Modica, Università degli Studi di Firenze, Italy

Inhaltsangabe

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.

Rezensionen

"This self-contained book aims to introduce the main ideas for studying approximation processes, more generally discrete processes at graduate level. The use of computers induces a growing need for studying discrete processes.... A key feature this lively yet rigorous and systematic treatment is the historical accounts of ideas and methods of the subject. Ideas in mathematics develop in cultural, historical and economical contexts, thus the authors made brief accounts of those aspects and used a large number of beautiful illustrations.... Each chapter has a short summary where the most important facts discussed are collected and described. There is also a large number of exercises inserted at various points into the text....The book is meant principally for graduate students in mathematics, physics, engineering, and computer science, but it can be used at technological and scientific faculties by anyone who wants to approach these topics. It may also be used in graduate seminars and courses or as a reference text by mathematicians, physicists, and engineers." -Zentralblatt MATH

"Mathematical Analysis does contain a substantial amount of material that is unusual in terms of an introductory text in real analysis...These are all interesting topics that have gained increasing importance in modern applications of mathematics, albeit outside the traditional area of analysis. It is very nice to have these topics developed outside a specialized textbook, in, e.g., combinatorics, dynamical systems, or number theory. The authors do a very good job presenting this material...Mathematical Analysis includes substantial amounts of historical background...The book also contains a lot of examples...I can happily recommend Mathematical Analysis as a good resource for instructors of introductory analysis courses, especially in terms of providing some unusual applications of analysis and developments of some basic classic topics that are oftenshortchanged in standard texts." -SIAM Review

"This is the second volume of a series on analysis. ... a real hodgepodge that could only be the primary textbook for a course specifically based on it. ... In this series Giaquinta and Modica have set themselves the formidable task of constructing from scratch an analysis sequence of several years length. ... they have more regard for classical topics and arguments than most authors writing analysis books today. ... I enjoyed reading this volume ... ." (Warren Johnson, The Mathematical Association of America, January, 2010)