Table of Contents

Chapter 1 First-Order, Ordinary Differential Equations

• 1.1 Introduction
• 1.2 Systems Analysis - Mathematical Modeling of Physical Systems
• 1.3 Separable, First-Order Ordinary Differential Equations
• 1.4 Exact, First-Order Ordinary Differential Equations
• 1.5 Linear, First-Order Ordinary Differential Equations
• 1.6 Applications of First-Order, Linear Differential Equations
• 1.7 Orthogonal Trajectories
• 1.8 Direction Fields

Chapter 2 Second and Higher-Order, Ordinary Differential Equations

• 2.1 Introduction
• 2.2 Homogeneous, Second-Order Differential Equations with Constant Coefficients
• 2.3 Applications: Free Vibration of Mechanical Systems
• 2.4 Second-Order, Euler-Cauchy Differential Equation
• 2.5 Nonhomogeneous, Linear Second-Order Differential Equations
• 2.6 Method of Variation of Parameters
• 2.7 Applications: Forced Response
• 2.8 Linear, Higher-Order Differential Equations
• 2.9 Higher-Order, Homogeneous Equations with Constant Coefficients
• 2.10 Higher-Order, Nonhomogeneous Linear Differential Equations
• 2.11 Method of Variation of Parameters for Higher-Order ODE's

Chapter 3 Series Solutions of Ordinary Differential Equations

• 3.1 Introduction; Power Series
• 3.2 Power Series Solution Method
• 3.3 Legendre’s Equation and Polynomials
• 3.4 Extension of Series Solution Method; Frobenius Method
• 3.5 Bessel’s Equation and Functions
• 3.6 Properties of Bessel Functions
• 3.7 Equations Satisfied by Bessel Functions
• 3.8 Orthogonality of Functions; Sturm-Liouville Problem
• 3.9 Series Expansion of Functions in Terms of Orthogonal Functions

Chapter 4 Laplace Transformation

• 4.1 Introduction
• 4.2 Special Functions
• 4.3 Laplace Transform of Derivatives and Integrals
• 4.4 Inverse Laplace Transformation
• 4.5 Periodic Functions
• 4.6 System Response
• 4.7 Response to an Arbitrary Input; Convolution Integral
• Table 4.1 - Laplace Transform Pairs

Chapter 5 Fourier Analysis

• 5.1 Introduction
• 5.2 Even and Odd Periodic Extensions
• 5.3 Applications: Systems Forced Response
• 5.4 Fourier Integral
• 5.5 Fourier Transforms
• 5.6 Inverse Fourier Transformation
• 5.7 Fourier Transform of Generalized Functions
• 5.8 Connection Between Fourier and Laplace Transformations
• Table 5.3 Fourier Transform Pairs
• Tables 5.4, 5.5 Fourier Cosine and Sine Transform Pairs
• Table 5.6 Fourier Transform Pairs of Generalized Functions

Chapter 6 Partial Differential Equations

• 6.1 Introduction
• 6.2 One-Dimensional Wave Equation; Free Vibration of an Elastic String
• 6.3 Transverse Vibration of a Beam
• 6.4 One-Dimensional Heat Equation
• 6.5 Two-Dimensional Wave Equation; Vibration of Rectangular Membranes
• 6.6 Vibration of Circular Membranes
• 6.7 Laplacian in Spherical Coordinates
• 6.8 Solution of PDE’s via Laplace Transformation
• 6.9 Solution of PDE’s via Fourier Integrals and Transforms

Chapter 7 Complex Numbers, Variables, and Functions

• 7.1 Introduction
• 7.2 Polar Form Representation of Complex Numbers
• 7.3 Regions in the Complex Plane
• 7.4 Functions of a Complex Variable
• 7.5 Analytic Functions; Cauchy-Riemann Equations
• 7.6 Elementary Complex Functions; Exponential Function
• 7.7 Trigonometric and Hyperbolic Functions
• 7.8 Logarithmic Function
• 7.9 Geometrical Aspects: Mapping by Elementary Functions
• 7.10 Applications of Analytic Functions in Fluid Flow
• 7.11 Applications in Electrostatics and Heat

Chapter 8 Complex Integrals

• 8.1 Introduction
• 8.2 Integral Evaluation via Parametric Representation of the Path
• 8.3 Cauchy's Integral Theorem
• 8.4 Cauchy’s Integral Formula
• 8.5 Consequences of Cauchy’s Integral Formula

Chapter 9 Complex Series

• 9.1 Introduction
• 9.2 Complex Series
• 9.3 Uniformly Convergent Complex Series
• 9.4 Power Series
• 9.5 Taylor’s Series
• 9.6 More Creative Methods to Write Taylor's Series

Chapter 10 Laurent Series and Residue Theory

• 10.1 Introduction
• 10.2 Singularities and Zeros of Functions
• 10.3 Residue Theory
• 10.4 Real Integral Evaluation
• 10.5 Evaluation of Two More Classes of Real Integrals

Chapter 11 Conformal Mapping

• 11.1 Introduction
• 11.2 The Bilinear Transformation
• 11.3 Regional Mappings by the Bilinear Transformation
• 11.4 The Schwarz-Christoffel Transformation
• 11.5 Applications of Conformal Mapping

Chapter 12 Linear Algebra; Linear Systems of Equations

• 12.1 Introduction
• 12.2 Linear Systems of Algebraic Equations
• 12.3 Rank of a Matrix
• 12.4 Role of Rank in the Solution of Linear Systems
• 12.5 Determinant of a Matrix
• 12.6 Inverse of a Matrix

Chapter 13 Matrix Eigenvalue Problem

• 13.1 Introduction
• 13.2 Eigenvalue Properties of Special Matrices
• 13.3 Similarity Transformation and Matrix Diagonalization
• 13.4 Modal Decomposition
• 13.5 Simultaneous Diagonalization
• 13.6 Systems of Linear, First-Order Differential Equations
• 13.7 State-Variable Equations
• 13.8 Phase Plane Method for Linear Systems
• 13.9 Phase Plane Analysis of Nonlinear Systems

Chapter 14 Fundamentals of Numerical Methods

• 14.1 Introduction
• 14.2 Iterative Methods; Rate of Convergence and Stability
• 14.3 Bracketing and Fixed-Point Methods
• 14.4 Polynomial Approximation and Interpolation
• 14.5 Cubic Spline Interpolation
• 14.6 Fourier Approximation and Interpolation
• 14.7 Numerical Differentiation and Integration

Chapter 15 Numerical Methods for Ordinary Differential Equations

• 15.1 Introduction; Numerical Methods for First-Order Ordinary Differential Equations
• 15.2 Multi-step Methods
• 15.3 Numerical Solutions of Systems of First-Order ODE's

Chapter 16 Numerical Methods for Partial Differential Equations

• 16.1 Introduction
• 16.2 More on the Numerical Solution of Elliptic Equations
• 16.3 Numerical Solution of Parabolic and Hyperbolic PDE’s

Chapter 17 Numerical Methods in Linear Algebra

• 17.1 Introduction; Direct Methods for Solving Linear Systems
• 17.2 Factorization Methods for Solving Linear Systems
• 17.3 Error Analysis
• 17.4 Iterative Methods for Solving Linear Algebraic Systems
• 17.5 Solution of Overdetermined Linear Systems; Least-Squares Method
• 17.6 Localization of Eigenvalues
• 17.7 Approximation of the Dominant Eigenvalue; Power Method
• 17.8 Deflation Methods; Inverse Power Method
• 17.9 Householder and Lanczos Tridiagonalization Methods; QR-Factorization

Chapter 18 Probability and Statistics

• 18.1 Introduction
• 18.2 Probability, Conditional Probability, Independence
• 18.3 Counting Techniques: Permutations, Combinations
• 18.4 Random Variables, Probability Distributions
• 18.5 Characteristics of Distributions
• 18.6 Binomial, Hypergeometric, and Poisson Probability Distributions
• 18.7 Normal (Gaussian) Distribution
• 18.8 Joint Probability Distributions
• 18.9 Statistical Analysis
• 18.10 Parameter Estimation: Point Estimates
• 18.11 Parameter Estimation: Interval Estimates
• 18.12 Hypothesis Testing Based on a Single Sample
• 18.13 Inferences Based on Two Samples
• 18.14 Goodness of Fit Test
• 18.15 Linear Regression Analysis

Appendix 1 References

Appendix 2 Useful Formulas

Appendix 3 Answers to Odd-Numbered Problems

Appendix 4 Tables

• Table 1 Gamma Function
• Table 2A Bessel Functions of the First Kind
• Table 2B Bessel Functions of the Second Kind
• Table 3 Binomial Distribution
• Table 4 Poisson Distribution
• Table 5 (Standard) Normal Distribution
• Table 6 t-Distribution
• Table 7 x²-Distribution
• Table 8 F Distribution

Index