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9780486462967

Foundations of Analysis Second Edition

by ;
  • ISBN13:

    9780486462967

  • ISBN10:

    048646296X

  • Edition: 2nd
  • Format: Paperback
  • Copyright: 2008-02-29
  • Publisher: Dover Publications

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Summary

This treatment develops the real number system and the theory of calculus on the real line, extending the theory to real and complex planes. Designed for students with one year of calculus, it features extended discussions of key ideas and detailed proofs of difficult theorems. 1991 edition.

Author Biography

The authors are Professors of Mathematics at Hobart and William Smith College in Geneva, New York.

Table of Contents

Prefacep. vii
The Real Number Systemp. 1
Introductionp. 1
Irrational Numbersp. 2
Constructing the Real Numbersp. 12
An Axiom System for the Real Numbersp. 25
The Heine-Borel Theoremp. 37
Functions, Limits, and Continuityp. 45
Introductionp. 45
Functionsp. 46
Limitsp. 52
Limit Theoryp. 65
Other Types of Limitsp. 71
Continuityp. 75
Continuity on Closed Intervalsp. 81
Differentiation and Integrationp. 91
Introductionp. 91
The Derivativep. 92
Elementary Laws of Differentiationp. 100
The Mean Value Theoremp. 106
Integrationp. 114
Properties of the Integralp. 130
The Fundamental Theorems of Calculusp. 141
Taylor Polynomialsp. 149
Sequences and Seriesp. 161
Introductionp. 161
Infinite Sequencesp. 162
Monotone and Cauchy Sequencesp. 168
Infinite Series and Convergence Testsp. 179
Absolute and Conditional Convergencep. 189
Sequences of Functionsp. 198
Series of Functionsp. 207
Calculus in Two Dimensionsp. 219
Introductionp. 219
The Dot Productp. 220
Vector-Valued Functionsp. 230
Functions of a Vector Variablep. 250
The Derivative for Vector Functionsp. 263
Integrationp. 280
Line Integrals and Green's Theoremp. 303
Introductionp. 303
The Fundamental Theorem of Calculus: Part Ip. 304
Line Integralsp. 306
The Fundamental Theorem of Calculus: Part IIp. 326
Green's Theoremp. 333
Path Independence and Potential Functionsp. 350
Complex Analysisp. 359
Introductionp. 359
The Complex Numbersp. 361
Complex Functions and Limitsp. 372
Differentiability of Complex Functionsp. 380
The Cauchy-Riemann Equationsp. 385
Integrationp. 391
The Cauchy Integral Theoremp. 400
The Fundamental Theorem of Algebrap. 410
Consequences of the Cauchy Integral Formulap. 414
Bibliographyp. 419
Symbol Indexp. 421
Indexp. 422
Table of Contents provided by Ingram. All Rights Reserved.

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