Real Analysis: A Historical Approach

by Saul Stahl

• ISBN13: 9780471318521
• ISBN10: 0471318523
• Format: Hardcover
• Copyright: 1999-08-01
• Publisher: Wiley-Interscience
• More Book Details
• Purchase Benefits

Summary

A provocative look at the tools and history of real analysis This new work from award-winning author Saul Stahl offers a real treat for students of analysis. Combining historical coverage with a superb introductory treatment, Real Analysis: A Historical Approach helps readers easily make the transition from concrete to abstract ideas. The book begins with an exciting sampling of classic and famous problems first posed by some of the greatest mathematicians of all time. Archimedes, Fermat, Newton, and Euler are each summoned in turn-illuminating the utility of infinite, power, and trigonometric series in both pure and applied mathematics. Next, Dr. Stahl develops the basic tools of advanced calculus, introducing the various aspects of the completeness of the real number system, sequential continuity and differentiability, as well as uniform convergence. Finally, he presents applications and examples to reinforce concepts and demonstrate the validity of many of the historical methods and results. Ample exercises, illustrations, and appended excerpts from the original historical works complete this focused, unconventional, highly interesting book. It is an invaluable resource for mathematicians and educators seeking to gain insight into the true language of mathematics.

Author Biography

<b>SAUL STAHL, PhD</b>, is Professor of Mathematics at the University of Kansas. The recipient of the Carl S. Allendoerfer Award from the Mathematical Association of America in 1986 for excellence in expository writing, Dr. Stahl has published over 30 articles as well as three books, including Introductory Modern Algebra: A Historical Approach, also from Wiley.

Table of Contents

Preface	v
Acknowledgments	ix
Archimedes and the Parabola	1	(12)
The Area of the Parabolic Segment	1	(7)
The Geometry of the Parabola	8	(5)
Fermat, Differntiation, and Integration	13	(6)
Fermat's Calculus	13	(6)
Newton's Calculus (Part 1)	19	(16)
The Fractional Binomial Theorem	19	(4)
Areas and Infinite Series	23	(7)
Newton's Proofs	30	(5)
Newton's Calculus (Part 2)	35	(16)
The solution of Differential Equations	35	(5)
The solution of Algebraic Equations	40	(8)
Chapter Appendix: Mathematica implementations of Newton's algorithm	48	(3)
Euler	51	(10)
Trigonometric Series	51	(10)
The Real Numbers	61	(24)
An Informal Introduction	61	(3)
Ordered Fields	64	(7)
Completeness and Irrational Numbers	71	(5)
The Euclidean Process	76	(4)
Functions	80	(5)
Sequences and Their Limits	85	(18)
The Definitions	85	(7)
Limit Theorems	92	(11)
The Cauchy Property	103	(12)
Limits of Monotone Sequences	103	(8)
The Cauchy Property	111	(4)
The Convergence of Infinite Series	115	(24)
Stock Series	115	(5)
Series of Positive Terms	120	(4)
Series of Arbitrary Terms	124	(8)
The Most Celebrated Problem	132	(7)
Series of Functions	139	(10)
Power Series	139	(6)
Trigonometric Series	145	(4)
Continuity	149	(20)
An Informal Introduction	149	(1)
The Limit of a Function	150	(5)
Continuity	155	(8)
Properties of Continuous Functions	163	(6)
Differentiability	169	(18)
An Informal Introductio to Differentiation	169	(2)
The Derivative	171	(6)
The Consequences of Differentiability	177	(5)
Integrability	182	(5)
Uniform Convergence	187	(20)
Uniform and Non-Uniform Convergence	187	(10)
Consequences of Uniform Convergence	197	(10)
The Vindication	207	(10)
Trigonometric Series	207	(5)
Power Series	212	(5)
Appendix A: Excerpts from ``Quadrature of the Parabola'' by Archimedes	217	(8)
Appendix B: On a Method for the Evaluation of Maxima and Minima	225	(2)
Pierre de Fermat
Appendix C: From a Letter to Henry Oldernburg on the Binomial Series (June 13, 1676)	227	(2)
Isaac Newton
Appendix D: From a Letter to Henry Oldenbury on the Binomial Series (October 24, 1676)	229	(4)
Isaac Newton
Appendix E: Excerpts from ``Of Analysis by Equations of an Infinite Number of Terms''	233	(12)
Isaac Newton
Appendix F: Excerpts from ``Subsiduum Colculi Sinuum''	245	(2)
Leonhard Euler
Solutions to Selected Exercises	247	(17)
Bibliography	264	(3)
Index	267