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A First Course in Analysis : Undergraduate Texts in Mathematics - George Pedrick

A First Course in Analysis

Undergraduate Texts in Mathematics


Published: 11th March 1994
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This is a new text covering advanced calculus. It discusses such topics as number systems, the extreme value problem, continuous functions, differentiation, integration and infinite series. The exposition is generally slower pace than that of most other texts, and it provides many examples which are important to students in a first analysis course. The reader will find that the focus of attention is shifted from the learning and applying of computational techniques to careful reasoning from hypothesis to conclusion. The book is intended both for a terminal course and as preparation for more advanced studies in mathematics, science, engineering and computation.

Notations and Conventions
Background: Number Systemsp. 1
Counting: The Natural Numbersp. 1
Measurement: The Rational Numbersp. 13
The Axioms of Ordered Fieldsp. 19
Decimal Representation. Irrationalsp. 25
Approximation: The Real Numbersp. 37
Least Upper Boundp. 38
Completeness. Nested Intervalsp. 40
Bounded Monotonic Sequencesp. 42
Cauchy Sequencesp. 47
The Real Number Systemp. 50
Countabilityp. 52
Appendix. The Fundamental Theorem of Algebra. Complex Numbersp. 55
The Extreme-Value Problemp. 64
Continuity, Compactness, and the Extreme-Value Theoremp. 65
Continuity of Rational Functions. Limits of Sequencesp. 72
Appendix. Completion of the Proof of the Fundamental Theorem of Algebrap. 76
Sequences and Series of Reals. The Number ep. 78
Sets of Reals. Limits of Functionsp. 90
Continuous Functionsp. 97
Implicit Functions. [actual symbol not reproducible] The Intermediate-Value Theoremp. 97
Inverse Functions. [actual symbol not reproducible]p. 101
Continuous Extension. Uniform Continuity. The Exponential and Logarithmp. 104
The Elementary Functionsp. 109
Uniformity. The Heine-Borel Theoremp. 112
Uniform Convergence. A Nowhere Differentiable Continuous Functionp. 118
The Weierstrass Approximation Theoremp. 124
Summary of the Main Properties of Continuous Functionsp. 128
Appendix. A Space-Filling Continuous Curvep. 128
Differentiationp. 133
Differential and Derivative. Tangent Linep. 135
The Foundations of Differentiationp. 140
Curve Sketching. The Mean-Value Theoremp. 146
Taylor's Theoremp. 154
Functions Defined Implicitlyp. 160
Integrationp. 169
Definitions. Darboux Theoremp. 171
Foundations of Integral Calculus. The Fundamental Theorem of Calculusp. 179
The Nature of Integrability. Lebesgue's Theoremp. 189
Improper Integralp. 197
Arclength. Bounded Variationp. 206
A Word About the Stieltjes Integral and Measure Theoryp. 215
Infinite Seriesp. 218
The Vibrating Stringp. 219
Convergence: General Considerationsp. 224
Convergence: Series of Positive Termsp. 230
Computation with Seriesp. 237
Power Seriesp. 244
Fourier Seriesp. 254
Bibliographyp. 266
Indexp. 269
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780387941080
ISBN-10: 0387941088
Series: Undergraduate Texts in Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 279
Published: 11th March 1994
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 1.91
Weight (kg): 1.35