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Index
Cover
Title Page
Copyright Page
Dedication
Contents
Preface
Part I Real Variable Theory
Chapter 1 The Important Limit Processes
1.1. Functions and Functionals
1.2. Limits, Continuity, and Uniform Continuity
1.3. Differentiation
1.4. Integration
1.5. Asymptotic Notation and the “Big Oh”
1.6. Numerical Integration
1.7. Differentiation of Integrals Containing a Parameter; Leibnitz’s Rule
Exercises
Chapter 2 Infinite Series
2.1. Sequences and Series; Fundamentals and Tests for Convergence
2.2. Series with Terms of Mixed Sign
2.3. Series with Terms That Are Functions; Power Series
2.4. Taylor Series
2.5. That’s All Very Nice, But How Do We Sum the Series? Acceleration Techniques
*2.6. Asymptotic Expansions
Exercises
Chapter 3 Singular Integrals
3.1. Choice of Summability Criteria; Convergence and Cauchy Principal Value
3.2. Tests for Convergence
3.3. The Gamma Function
3.4. Evaluation of Singular Integrals
Exercises
Chapter 4 Interchange of Limit Processes and the Delta Function
4.1. Theorems on Limit Interchange
4.2. The Delta Function and Generalized Functions
Exercises
Chapter 5 Fourier Series and the Fourier Integral
5.1. The Fourier Series of f(x)
5.2. Pointwise Convergence of the Series
5.3. Termwise Integration and Differentiation of Fourier Series
5.4. Variations : Periods Other than 2π, and Finite Interval
5.5. Infinite Period ; the Fourier Integral
Exercises
Chapter 6 Fourier and Laplace Transforms
6.1. The Fourier Transform
6.2. The Laplace Transform
6.3. Other Transforms
Exercises
Chapter 7 Functions of Several Variables
7.1. Chain Differentiation
7.2. Taylor Series in Two or More Variables
Exercises
Chapter 8 Vectors, Surfaces, and Volumes
8.1. Vectors and Elementary Operations
8.2. Coordinate Systems, Base Vectors, and Components
8.3. Angular Velocity of a Rigid Body
8.4. Curvilinear Coordinate Representation of Surfaces
8.5. Curvilinear Coordinate Representation of Volumes
Exercises
Chapter 9 Vector Field Theory
9.1. Line Integrals
9.2. The Curves, Surfaces, and Regions Under Consideration
9.3. Divergence, Gradient, and Curl
9.4. The Divergence Theorem
9.5. Stokes’ Theorem
9.6. Irrotational and Solenoidal Fields
9.7. Noncartesian Systems
9.8. Fluid Mechanics; Irrotational Flow
9.9. The Gravitational Potential
Exercises
Chapter 10 The Calculus of Variations
10.1. Functions of One or More Variables
10.2. Constraints; Lagrange Multipliers
10.3. Functionals and the Calculus of Variations
10.4. Two or More Dependent Variables; Hamilton’s Principle
10.5. Two or More Independent Variables; Vibrating Strings and Membranes
10.6. The Ritz Method
10.7. Optimal Control
Exercises
Additional References for Part I
Part II Complex Variables
Chapter 11 Complex Numbers
11.1. The Algebra of Complex Numbers
11.2. The Complex Plane
Exercises
Chapter 12 Functions of a Complex Variable
12.1. Basic Notions
12.2. Differentiation and the Cauchy–Riemann Conditions for Analyticity
12.3. Harmonic Functions
12.4. Some Elementary Functions and Their Singularities
12.5. Multivaluedness and the Need for Branch Cuts
Exercises
Chapter 13 Integration, Cauchy’s Theorem, and the Cauchy Integral Formula
13.1. Integration in the Complex Plane
13.2. Bounds on Contour Integrals
13.3. Cauchy’s Theorem
13.4. An Important Little Integral
13.5. The Cauchy Integral Formula
Exercises
Chapter 14 Taylor and Laurent Series
14.1. Complex Series
14.2. Taylor Series
14.3. Laurent Series
14.4. Classification of Isolated Singularities
Exercises
Chapter 15 The Residue Theorem and Contour Integration
15.1. The Residue Theorem
15.2. The Calculation of Residues
15.3. Applications
15.4. Cauchy Principal Value
Exercises
Chapter 16 Conformai Mapping
16.1. The Fundamental Problem
16.2. Conformality
16.3. The Bilinear Transformation and Applications
Exercises
Additional References for Part II
Part III Linear Analysis
Chapter 17 Linear Spaces
17.1. Introduction; Extension to n tuples
17.2. The Definition of an Abstract Vector Space
Introducing a Norm and Inner Product
17.3. Linear Dependence, Dimension, and Bases
17.4. Function Space
*17.5. Continuity of the Inner Product
Exercises
Chapter 18 Linear Operators
18.1. Some Definitions
18.2. Operator Algebra
18.3. Further Discussion of Matrices
18.4. The Adjoint Operator
Exercises
Chapter 19 The Linear Equation Lx = c
19.1. Introduction
19.2. Existence and Uniqueness
19.3. The Inverse Operator L– 1
The Inverse of a Matrix
*Inverse of I + M where M is Small; the Neumann Series
Exercises
Chapter 20 The Eigenvalue Problem Lx = λx
20.1. Statement of the Eigenvalue Problem
20.2. Some Eigenhunts
20.3. The Sturm–Liouville Theory
20.4. Additional Discussion for the Matrix Case
The Inertia Tensor
20.5. The Inhomogeneous Problem Lx = c
20.6. Eigen Bounds and Estimates
Exercises
Additional References for Part III
Part IV Ordinary Differential Equations
Chapter 21 First-Order Equations
21.1. Standard Methods of Solution
Variables Separable
Homogeneous of Degree Zero
Exact Differentials and Integrating Factors
21.2. The Questions of Existence and Uniqueness
Exercises
Chapter 22 Higher-Order Systems
*22.1. Nonlinear Case; Existence and Uniqueness
22.2. The Linear Equation; Some Theory
22.3. The Linear Equation; Some Methods of Solution
22.4. Series Solution; Bessel and Legendre Functions
22.5. The Method of Green’s Functions
22.6. Some Nonlinear Problems and Techniques
Nonlinear Algebraic Equations
Application to Nonlinear Differential Equations
Exercises
Chapter 23 Qualitative Methods; The Phase Plane
23.1. The Phase Plane and Some Simple Examples
23.2. Singular Points
23.3. Additional Examples
Exercises
Chapter 24 Quantitative Methods
24.1. The Methods of Taylor and Euler
24.2. Improvements: Midpoint Rule and Runge–Kutta
24.3. Stability
24.4. Application to Higher-Order Systems
24.5. Boundary Value Problems
*Invariant Imbedding
24.6. The Method of Weighted Residuals
24.7. The Finite-Element Method
Exercises
Chapter 25 Perturbation Techniques
25.1. Introduction; the Regular Case
25.2. Singular Perturbations: Straining Methods
25.3. Singular Perturbations: Boundary Layer Methods
Exercises
Additional References for Part IV
Part V Partial Differential Equations
Chapter 26 Separation of Variables and Transform Methods
26.1. Introduction
26.2. Some Background; the Sturm-Liouville Theory
26.3. The Diffusion Equation
26.4. The Wave Equation
26.5. The Laplace Equation
Exercises
Chapter 27 Classification and the Method of Characteristics
27.1. Characteristics and Classification
27.2. The Hyperbolic Case
27.3. Reduction of Aϕxx + 2Bϕxy + Cϕyy = F to Normal Form
27.4. Comparison of Hyperbolic, Elliptic, and Parabolic Systems; Summary
Exercises
Chapter 28 Green’s Functions and Perturbation Techniques
28.1. The Green’s Function Approach
28.2. Perturbation Methods
Exercises
Chapter 29 Finite-Difference Methods
29.1. The Heat Equation; An Explicit Method and Its Convergence and Stability
29.2. The Heat Equation; Implicit Methods
29.3. Hyperbolic and Elliptic Problems
Exercises
Additional References for Part V
Survey-Type References
Answers to Selected Exercises
Index
Fourier and Laplace Transform Tables, Inside Front Cover
Some Frequently Needed Formulas, Inside Back Cover
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