Foundations of Analysis Second Edition

, by Belding, David F; Mitchell, Kevin J

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ISBN: 9780486462967 | 048646296X
Cover: Paperback
Copyright: 2/29/2008

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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.

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Preface	p. vii
The Real Number System	p. 1
Introduction	p. 1
Irrational Numbers	p. 2
Constructing the Real Numbers	p. 12
An Axiom System for the Real Numbers	p. 25
The Heine-Borel Theorem	p. 37
Functions, Limits, and Continuity	p. 45
Introduction	p. 45
Functions	p. 46
Limits	p. 52
Limit Theory	p. 65
Other Types of Limits	p. 71
Continuity	p. 75
Continuity on Closed Intervals	p. 81
Differentiation and Integration	p. 91
Introduction	p. 91
The Derivative	p. 92
Elementary Laws of Differentiation	p. 100
The Mean Value Theorem	p. 106
Integration	p. 114
Properties of the Integral	p. 130
The Fundamental Theorems of Calculus	p. 141
Taylor Polynomials	p. 149
Sequences and Series	p. 161
Introduction	p. 161
Infinite Sequences	p. 162
Monotone and Cauchy Sequences	p. 168
Infinite Series and Convergence Tests	p. 179
Absolute and Conditional Convergence	p. 189
Sequences of Functions	p. 198
Series of Functions	p. 207
Calculus in Two Dimensions	p. 219
Introduction	p. 219
The Dot Product	p. 220
Vector-Valued Functions	p. 230
Functions of a Vector Variable	p. 250
The Derivative for Vector Functions	p. 263
Integration	p. 280
Line Integrals and Green's Theorem	p. 303
Introduction	p. 303
The Fundamental Theorem of Calculus: Part I	p. 304
Line Integrals	p. 306
The Fundamental Theorem of Calculus: Part II	p. 326
Green's Theorem	p. 333
Path Independence and Potential Functions	p. 350
Complex Analysis	p. 359
Introduction	p. 359
The Complex Numbers	p. 361
Complex Functions and Limits	p. 372
Differentiability of Complex Functions	p. 380
The Cauchy-Riemann Equations	p. 385
Integration	p. 391
The Cauchy Integral Theorem	p. 400
The Fundamental Theorem of Algebra	p. 410
Consequences of the Cauchy Integral Formula	p. 414
Bibliography	p. 419
Symbol Index	p. 421
Index	p. 422
Table of Contents provided by Ingram. All Rights Reserved.

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Foundations of Analysis Second Edition

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Table of Contents

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