Understanding Analysis 2/e (Hardcover)

This lively introductory text exposes the student to the rewards of a rigorous study of functions of a real variable. In each chapter, informal discussions of questions that give analysis its inherent fascination are followed by precise, but not overly formal, developments of the techniques needed to make sense of  them. By focusing on the unifying themes of approximation and the resolution of paradoxes that arise in the transition from the finite to the infinite, the text turns what could be a daunting cascade of definitions and theorems into a coherent and engaging progression of ideas. Acutely aware of the need for rigor, the student is much better prepared to understand what constitutes a proper mathematical proof and how to write one.

Fifteen years of classroom experience with the first edition of Understanding Analysis have solidified and refined the central narrative of the second edition. Roughly 150 new exercises join a selection of the best exercises from the first edition, and three more project-style sections have been added. Investigations of Euler’s computation of ζ(2), the Weierstrass Approximation ­ Theorem, and the gamma function are now among the book’s cohort of seminal results serving as motivation and payoff for the beginning student to master the methods of analysis.

Review of the first edition:

“This is a dangerous book. Understanding Analysis is so well-written and the development of the theory so w

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ated that exposing students to it could well lead them to expect such excellence in all their textbooks. … Understanding Analysis is perfectly titled; if your students read it, that’s what’s going to happen. … This terrific book will become the text of choice for the single-variable introductory analysis course … ”   

― Steve Kennedy, MAA Reviews                 

這本生動的入門教材讓學生體驗到嚴謹研究實變函數的回報。每一章節都以非正式的討論引出分析學的迷人之處，接著以精確但不過於正式的方式發展出理解這些問題所需的技巧。著重於近似和從有限到無限的轉變中產生的悖論的解決，本書將原本可能令人生畏的定義和定理串聯成一個有條理且引人入勝的思想進展。學生對嚴謹性有敏銳的認識，因此更能理解何謂正確的數學證明以及如何撰寫證明。

第二版的核心敘述在第一版的十五年教學經驗中得到鞏固和完善。大約有150個新練習題加入了第一版中最好的練習題，並增加了三個項目式的章節。對歐拉計算ζ(2)、Weierstrass逼近定理和伽瑪函數的探討現在成為本書的重要結果之一，為初學者掌握分析方法提供了動機和回報。

第一版的評論如下：
「這是一本危險的書。《理解分析》寫得非常好，理論的發展也非常有動機，讓學生對所有教科書都期望有這樣的卓越表現...《理解分析》的標題非常貼切；如果你的學生讀了這本書，就會發生這種情況...這本了不起的書將成為單變量入門分析課程的首選教材...」- Steve Kennedy, MAA Reviews