Real Analysis A Historical Approach

by Stahl, Saul

• ISBN13: 9780470878903
• ISBN10: 0470878908
• Edition: 2nd
• Format: Hardcover
• Copyright: 2011-08-30
• Publisher: Wiley
• More Book Details
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Summary

Based on reviewer and user feedback, this Second Edition features a new chapter on the Riemann integral including the subject of uniform continuity as well as a discussion of epsilon-delta convergence and a section that details the modern preference for convergence of sequences over convergence of series. The first third of this book describes the utility of infinite, power, and trigonometric series in both pure and applied mathematics through several snapshots from the works of Archimedes, Fermat, Newton, and Euler and offers glimpses of the Greeks' method of exhaustion, preNewtonian calculus, Newton's concerns, and Euler's miraculously effective, though often logically unsound, mathematical wizardry. The infinite geometric progression is the scarlet thread that unifies the early chapters wherein the nondifferentiability of Euler's Trigonometric Series provides the crucial counterexample that clarifies the need for this careful examination of the foundations of calculus. The second third of the book consists of a fairly conventional discussion of various aspects of the completeness of the real number system. These culminate in Cauchy's criterion for the convergence of infinite series, which is in turn applied to both power and trigonometric series. The last third of the book addresses sequential continuity and differentiability, the maximum principle for continuous functions, and the mean value theorem for differentiable ones. A considerable amount of discussion and quoted material is included to shed some light on the concerns of the mathematicians who developed the key concepts and on the difficulties they faced.

Author Biography

SAUL STAHL, PhD, is Professor in the Department of Mathematics at The University of Kansas. He has published numerous journal articles in his areas of research interest, which include combinatorics, discrete mathematics, and topological graph theory. Dr. Stahl is the author of Introductory Modern Algebra: A Historical Approach and Introduction to Topology and Geometry, both published by Wiley. He was awarded the Carl B. Allendoerfer Award from the Mathematical Association of America for expository articles in both 1986 and 2006.

Table of Contents

Preface to the Second Edition	p. xi
Acknowledgments	p. xv
Archimedes and the Parabola	p. 1
The Area of the Parabolic Segment	p. 1
The Geometry of the Parabola	p. 8
Fermat, Differentiation, and Integration	p. 13
Fermat's Calculus	p. 13
Newton's Calculus (Part 1)	p. 19
The Fractional Binomial Theorem	p. 19
Areas and Infinite Series	p. 23
Newton's Proofs	p. 30
Newton's Calculus (Part 2)	p. 35
The Solution of Differential Equations	p. 35
The Solution of Algebraic Equations	p. 40
Chapter appendix: Mathematica Implementations of Newton's Algorithm	p. 48
Euler	p. 51
Trigonometric Series	p. 51
The Real Numbers	p. 61
An Informal Introduction	p. 61
Ordered Fields	p. 64
Completeness and Irrational Numbers	p. 71
The Euclideait Process	p. 76
Functions	p. 80
Sequences and Their Limits	p. 85
The Definitions	p. 85
Limit Theorems	p. 92
The Cauchy Property	p. 103
Limits of Monotone Sequences	p. 103
The Cauchy Property	p. 111
The Convergence of Infinite Series	p. 115
Stock Series	p. 115
Series of Positive Terms	p. 120
Series of Arbitrary Terms	p. 124
The Most Celebrated Problem	p. 132
Series of Functions	p. 139
Power Series	p. 139
Trigonometric Series	p. 145
Continn	p. 149
An Informal Introduction	p. 149
The Limit of a Function	p. 150
Continuity	p. 155
Properties of Continuous Functions	p. 163
Differentiability	p. 169
An Informal Introduction to Differentiation	p. 169
The Derivative	p. 171
The Consequences of Differentiability	p. 177
Integrability	p. 182
Uniform Convergence	p. 187
Uniform and Nonuniform Convergence	p. 187
Consequences of Uniform Convergence	p. 197
The Vindication	p. 207
Trigonometric Series	p. 207
Power Series	p. 212
The Riemann Integral	p. 217
Continuity Revisited	p. 217
Lower and Upper Sums	p. 221
Integrability	p. 227
Excerpts from "Quadrature of the Parabola"by Archimedes	p. 237
On a Method for the Evaluation of Maxima and Minima by Pierre de Fermat	p. 245
From a Letter to Henry Oldenburg on the Binomial Series (June 13, 1676) by Isaac Newton	p. 247
From a Letter to Henry Oldenburg on the Binomial Series (October 24, 1676) by Isaac Newton	p. 249
Excerpts from "Of Analysis by Equations of an Infinite Number of Terms" by Isaac Newton	p. 253
Excerpts from "Subsiduum Calculi Sinuum" by Leonhard Euler	p. 265
Solutions to Selected Exercises	p. 267
Bibliography	p. 287
Index	p. 291
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