Undergraduate Analysis, 2/e (Hardcover)

Serge Lang

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商品描述

Description:

This is a logically self-contained introduction to analysis, suitable for students who have had two years of calculus. The book centers around those properties that have to do with uniform convergence and uniform limits in the context of differentiation and integration. Topics discussed include the classical test for convergence of series, Fourier series, polynomial approximation, the Poisson kernel, the construction of harmonic functions on the disc, ordinary differential equation, curve integrals, derivatives in vector spaces, multiple integrals, and others. One of the author's main concerns is to achieve a balance between concrete examples and general theorems, augmented by a variety of interesting exercises. Some new material has been added in this second edition, for example: a new chapter on the global version of integration of locally integrable vector fields; a brief discussion of L1-Cauchy sequences, introducing students to the Lebesgue integral; more material on Dirac sequences and families, including a section on the heat kernel; a more systematic discussion of orders of magnitude; and a number of new exercises.

 

Table of Contents:

Chapter 0: Sets and Mappings Chapter 1: Real Numbers Chapter 2: Limits and Continuous Functions Chapter 3: Differentiation Chapter 4: Elementary Functions Chapter 5: The Elementary Real Integral Chapter 6: Normed Vector Spaces Chapter 7: Limits Chapter 8: Compactness Chapter 9: Series Chapter 10: The Integral in One Variable Appendix: The Lebesgue Integral Chapter 11: Approximation with Convolutions Chapter 12: Fourier Series Chapter 13, Improper Integrals Chapter 14: The Fourier Integral Chapter 15: Calculus in Vector Spaces Chapter 16: The Winding Number and Global Potential Functions Chapter 17: Derivatives in Vector Spaces Chapter 18: Inverse Mapping Theorem Chapter 19: Ordinary Differential Equations Chapter 20: Multiple Integration Chapter 22: Differential Forms Appendix

商品描述(中文翻譯)

描述:
這是一本邏輯上自成一體的分析入門書,適合已修習兩年微積分的學生。本書主要探討與一致收斂和一致極限相關的性質,並將其應用於微分和積分的背景下。討論的主題包括收斂級數的經典測試、傅立葉級數、多項式逼近、泊松核、在圓盤上構造調和函數、常微分方程、曲線積分、向量空間中的導數、多重積分等。作者的主要關注點之一是在具體例子和一般定理之間取得平衡,並配以各種有趣的練習題。第二版中新增了一些新內容,例如:關於局部可積向量場的全局積分的新章節;簡要介紹了L1-Cauchy序列,引導學生了解勒貝格積分;更多關於Dirac序列和家族的材料,包括一節關於熱核的內容;更系統地討論了量級的問題;以及一些新的練習題。

目錄:
第0章:集合和映射
第1章:實數
第2章:極限和連續函數
第3章:微分
第4章:初等函數
第5章:初等實積分
第6章:帶范數的向量空間
第7章:極限
第8章:緊致性
第9章:級數
第10章:一元積分
附錄:勒貝格積分
第11章:卷積逼近
第12章:傅立葉級數
第13章:不定積分
第14章:傅立葉積分
第15章:向量空間中的微積分
第16章:迴旋數和全局位能函數
第17章:向量空間中的導數
第18章:反函數定理
第19章:常微分方程
第20章:多重積分
第22章:微分形式
附錄