Fundamental Ideas of Analysis

by Reed, Michael C.

• ISBN13: 9780471159964
• ISBN10: 0471159964
• Edition: 1st
• Format: Paperback
• Copyright: 1997-09-17
• Publisher: Wiley

Summary

The ideas and methods of mathematics, long central to the physical sciences, now play an increasingly important role in a wide variety of disciplines. Analysis provides theorems that prove that results are true and provides techniques to estimate the errors in approximate calculations. The ideas and methods of analysis play a fundamental role in ordinary differential equations, probability theory, differential geometry, numerical analysis, complex analysis, partial differential equations, as well as in most areas of applied mathematics.

Author Biography

Michael C. Reed is a practicing anthropologist who has worked since 1992 for CSR Incorporated, a Washington, DC, research and evaluation firm. He received his PhD in sociocultural anthropology in 1988, following 15 months of Fulbright fellowship-funded fieldwork in francophone Africa. His dissertation was titled An Ethnohistorical Study of the Political Economy of Ndjole', Gabon. Since 1990, Reed has evaluated several U.S. programs, such as youth substance abuse prevention, school-age child care, secondary education, and community development corporations. He reviews articles for the American Evaluation Association journal Evaluation Practice.

Table of Contents

Preface
CHAPTER 1 Preliminaries	1	(26)
1.1 The Real Numbers	1	(5)
1.2 Sets and Functions	6	(9)
1.3 Cardinality	15	(5)
1.4 Methods of Proof	20	(7)
CHAPTER 2 Sequences	27	(46)
2.1 Convergence	27	(8)
2.2 Limit Theorems	35	(5)
2.3 Two-state Markov Chains*	40	(4)
2.4 Cauchy Sequences	44	(8)
2.5 Supremum and Infimum	52	(3)
2.6 The Bolzano-Weierstrass Theorem	55	(5)
2.7 The Quadratic Map*	60	(8)
Projects	68	(5)
CHAPTER 3 The Riemann Integral	73	(48)
3.1 Continuity	73	(7)
3.2 Continuous Functions on Closed Intervals	80	(7)
3.3 The Riemann Integral	87	(8)
3.4 Numerical Methods*	95	(8)
3.5 Discontinuities	103	(10)
3.6 Improper Integrals	113	(6)
Projects	119	(2)
CHAPTER 4 Differentiation	121	(42)
4.1 Differentiable Functions	121	(8)
4.2 The Fundamental Theorem of Calculus	129	(5)
4.3 Taylor's Theorem	134	(6)
4.4 Newton's Method*	140	(7)
4.5 Inverse Functions	147	(4)
4.6 Functions of Two Variables*	151	(8)
Projects	159	(4)
CHAPTER 5 Sequences of Functions	163	(60)
5.1 Pointwise and Uniform Convergence	163	(6)
5.2 Limit Theorems	169	(6)
5.3 The Supremum Norm	175	(8)
5.4 Integral Equations*	183	(5)
5.5 The Calculus of Variations*	188	(8)
5.6 Metric Spaces	196	(7)
5.7 The Contraction Mapping Principle	203	(7)
5.8 Normed Linear Spaces*	210	(9)
Projects	219	(4)
CHAPTER 6 Series of Functions	223	(50)
6.1 Lim sup and Lim inf	223	(5)
6.2 Series of Real Constants	228	(10)
6.3 The Weierstrass M-test	238	(7)
6.4 Power Series	245	(7)
6.5 Complex Numbers*	252	(8)
6.6 Infinite Products and Prime Numbers*	260	(10)
Projects	270	(3)
CHAPTER 7 Differential Equations*	273	(26)
7.1 Local Existence	273	(10)
7.2 Global Existence	283	(6)
7.3 The Error Estimate for Euler's Method	289	(7)
Projects	296	(3)
CHAPTER 8 Complex Analysis*	299	(24)
8.1 Analytic Functions	299	(6)
8.2 Integration on Paths	305	(7)
8.3 Cauchy's Theorem	312	(8)
Projects	320	(3)
CHAPTER 9 Fourier Series*	323	(36)
9.1 The Heat Equation	323	(8)
9.2 Definitions and Examples	331	(6)
9.3 Pointwise Convergence	337	(8)
9.4 Mean-square Convergence	345	(10)
Projects	355	(4)
CHAPTER 10 Probability Theory*	359	(44)
10.1 Discrete Random Variables	359	(9)
10.2 Coding Theory	368	(8)
10.3 Continuous Random Variables	376	(10)
10.4 The Variation Metric	386	(12)
Projects	398	(5)
Bibliography	403	(3)
Symbol Index	406	(3)
Index	409