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Analysis of Functions of a Single Variable by Lawrence Baggett - HTML preview
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Mathematics (Academic)
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Analysis of Functions of a Single Variable
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Analysis of Functions of a Single Variable
Table of Contents
Preface to Analysis of Functions of a Single Variable: A Detailed Development
1
.
Chapter
1
.
The Real and Complex Numbers
1.1
.
Definition of the Numbers 1, i, and the square root of 2
1.2
.
The Natural Numbers and the Integers
1.3
.
The Rational Numbers
1.4
.
The Real Numbers
1.5
.
Properties of the Real Numbers
1.6
.
Intervals and Approximation
1.7
.
The Geometric Progression and the Binomial Theorem
1.8
.
The Complex Numbers
Chapter
2
.
The Limit of a Sequence of Numbers
2.1
.
Definition of the Number e
2.2
.
Sequences and Limits
2.3
.
Existence of Certain Fundamental Limits
2.4
.
Definition of e
2.5
.
Properties of Convergent Sequences
2.6
.
Subsequences and Cluster Points
2.7
.
A Little Topology
2.8
.
Infinite Series
Chapter
3
.
Functions and Continuity
3.1
.
Functions and Continuity Definition of the Number π
3.2
.
Functions
3.3
.
Polynomial Functions
3.4
.
Continuity
3.5
.
Continuity and Topology
3.6
.
Deeper Analytic Properties of Continuous Functions
3.7
.
Power Series Functions
3.8
.
The Elementary Transcendental Functions
3.9
.
Analytic Functions and Taylor Series
3.10
.
Uniform Convergence
Chapter
4
.
Differentiation, Local Behavior
4.1
.
Differentiation, Local Behavior E^iπ = -1.
4.2
.
The Limit of a Function
4.3
.
The Derivative of a Function
4.4
.
Consequences of Differentiability, the Mean Value Theorem
4.5
.
The Exponential and Logarithm Functions
4.6
.
The Trigonometric and Hyperbolic Functions
4.7
.
L'Hopital's Rule
4.8
.
Higher Order Derivatives
4.9
.
Taylor Polynomials and Taylor's Remainder Theorem
4.10
.
The General Binomial Theorem
4.11
.
More on Partial Derivatives
Chapter
5
.
Integration, Average Behavior
5.1
.
Integration, Average Behavior A=π r^2
5.2
.
Integrals of Step Functions
5.3
.
Integrable Functions
5.4
.
The Fundamental Theorem of Calculus
5.5
.
Consequences of the Fundamental Theorem
5.6
.
Area of Regions in the Plane
5.7
.
Extending the Definition of Integrability
5.8
.
Integration in the Plane
Chapter
6
.
Integration over Smooth Curves in the Plane
6.1
.
Integration Over Smooth Curves in the Plane C=2π r
6.2
.
Smooth Curves in the Plane
6.3
.
Arc Length
6.4
.
Integration with Respect to Arc Length
6.5
.
Contour Integrals
6.6
.
Vector Fields, Differential Forms, and Line Integrals
6.7
.
Integration Around Closed Curves, and Green's Theorem
Chapter
7
.
The Fundamental Theorem of Algebra, and The Fundamental Theorem of Analysis
7.1
.
The Fundamental Theorem of Algebra, and the Fundamental Theorem of Analysis
7.2
.
Cauchy's Theorem
7.3
.
Basic Applications of the Cauchy Integral Formula
7.4
.
The Fundamental Theorem of Algebra
7.5
.
The Maximum Modulus Principle
7.6
.
The Open Mapping Theorem and the Inverse Function Theorem
7.7
.
Uniform Convergence of Analytic Functions
7.8
.
Isolated Singularities, and the Residue Theorem
Appendix
A
.
Appendix: Existence and Uniqueness of a Complete Ordered Field
A.1
.
Index
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