Real Analysis A Historical Approach

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.

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.