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There are a great deal of books on introductory analysis in print today, many written by mathematicians of the first rank. The publication of another such book therefore warrants a defense. I have taught analysis for many years and have used a variety of texts during this time. These books were of excellent quality mathematically but did not satisfy the needs of the students I was teaching. They were written for mathematicians but not for those who were first aspiring to attain that status. The desire to fill this gap gave rise to the writing of this book. This book is intended to serve as a text for an introductory course in analysis. Its readers will most likely be mathematics, science, or engineering majors undertaking the last quarter of their undergraduate education. The aim of a first course in analysis is to provide the student with a sound foundation for analysis, to familiarize him with the kind of careful thinking used in advanced mathematics, and to provide him with tools for further work in it. The typical student we are dealing with has completed a three-semester calculus course and possibly an introductory course in differential equations. He may even have been exposed to a semester or two of modern algebra. All this time his training has most likely been intuitive with heuristics taking the place of proof. This may have been appropriate for that stage of his development.
List of contents
I Preliminaries.- II Functions.- III Real Sequences and Their Limits.- IV Infinite Series of Real Numbers.- V Limits of Functions.- VI Continuous Functions.- VII Derivatives.- VIII Convex Functions.- IX L'Hôpital's Rule-Taylor's Theorem.- X The Complex Numbers. Trigonometric Sums. Infinite Products.- XI More on Series: Sequences and Series of Functions.- XII Sequences and Series of Functions II.- XIII The Riemann Integral I.- XIV The Riemann Integral II.- XV Improper Integrals. Elliptic Integrals and Functions.
