Mathematical Methods for Engineers and Scientists 1: Complex Analysis and Linear Algebra

Preface to Second Edition
Preface to the First Edition
Contents
Part I Complex Analysis
1 Complex Numbers
1.1 Our Number System
1.1.1 Addition and Multiplication of Integers
1.1.2 Inverse Operations
1.1.3 Negative Numbers
1.1.4 Fractional Numbers
1.1.5 Irrational Numbers
1.1.6 Imaginary Numbers
1.2 Logarithm
1.2.1 Napier's Idea of Logarithm
1.2.2 Briggs' Common Logarithm
1.3 A Peculiar Number Called e
1.3.1 The Unique Property of e
1.3.2 The Natural Logarithm
1.3.3 Approximate Value of e
1.4 The Exponential Function as an Infinite Series
1.4.1 Compound Interest
1.4.2 The Limiting Process Representing e
1.4.3 The Exponential Function ex
1.5 Unification of Algebra and Geometry
1.5.1 The Remarkable Euler Formula
1.5.2 The Complex Plane
1.6 Polar Form of Complex Numbers
1.6.1 Powers and Roots of Complex Numbers
1.6.2 Trigonometry and Complex Numbers
1.6.3 Geometry and Complex Numbers
1.7 Elementary Functions of Complex Variable
1.7.1 Exponential and Trigonometric Functions of z
1.7.2 Hyperbolic Functions of z
1.7.3 Logarithm and General Power of z
1.7.4 Inverse Trigonomeric and Hyperbolic Functions
2 Complex Functions
2.1 Analytic Functions
2.1.1 Complex Function as Mapping Operation
2.1.2 Differentiation of a Complex Function
2.1.3 Cauchy-Riemann Conditions
2.1.4 Cauchy-Riemann Equations in Polar Coordinates
2.1.5 Analytic Function as a Function of z Alone
2.1.6 Analytic Function and Laplace's Equation
2.2 Complex Integration
2.2.1 Line Integral of a Complex Function
2.2.2 Parametric Form of Complex Line Integral
2.3 Cauchy's Integral Theorem
2.3.1 Green's Lemma
2.3.2 Cauchy-Goursat Theorem
2.3.3 Fundamental Theorem of Calculus
2.4 Consequences of Cauchy's Theorem
2.4.1 Principle of Deformation of Contours
2.4.2 The Cauchy Integral Formula
2.4.3 Derivatives of Analytic Function
3 Complex Series and Theory of Residues
3.1 A Basic Geometric Series
3.2 Taylor Series
3.2.1 The Complex Taylor Series
3.2.2 Convergence of Taylor Series
3.2.3 Analytic Continuation
3.2.4 Uniqueness of Taylor Series
3.3 Laurent Series
3.3.1 Uniqueness of Laurent Series
3.4 Theory of Residues
3.4.1 Zeros and Poles
3.4.2 Definition of the Residue
3.4.3 Methods of Finding Residues
3.4.4 Cauchy's Residue Theorem
3.4.5 Second Residue Theorem
3.5 Evaluation of Real Integrals with Residues
3.5.1 Integrals of Trigonometric Functions
3.5.2 Improper Integrals I: Closing the Contour with a Semicircle at Infinity
3.5.3 Fourier Integral and Jordan's Lemma
3.5.4 Improper Integrals II: Closing the Contour with Rectangular and Pie-Shaped Contour
3.5.5 Integration Along a Branch Cut
3.5.6 Principal Value and Indented Path Integrals
4 Conformal Mapping
4.1 Examples of Problems Solved by Conformal Mappings
4.2 Invariance of the Laplace Equation
4.3 Conformal Mapping
4.4 Complex Potential
4.5 Flow of Fluids
4.5.1 Irrotation Flow and Velocity Potential
4.5.2 Incompressibility of the Fluid
4.5.3 Stream Function and Stream Lines
4.5.4 Complex Velocity
4.6 The Joukowski Transformation
4.6.1 Properties of the Joukowski Transformation
4.6.2 Joukowski Profiles
4.7 The Möbius Transformation
4.7.1 Linear Transformation
4.7.2 Inversion
4.7.3 General Bilinear Transformation
4.7.4 Some Properties of Bilinear Transformation
4.7.5 Mapping Two Distinct Circles into Concentric Circles
4.8 The Schwarz-Christoffel Transformation
4.8.1 Formulation of the Schwarz-Christoffel Transformation
4.8.2 Convention Regarding the Polygon
4.8.3 Examples
Part II Linear Algebra
5 Linear Algebra and Vector Space
5.1 Linear System of Equations
5.1.1 Cramer's Rule
5.1.2 Gaussian Elimination
5.1.3 LU Decomposition
5.1.4 Geometry and Linear Equations
5.2 Vector Space
5.2.1 Definition of Vector Space
5.2.2 Dot Product and Length of a Vector
5.2.3 Essence of Finite Dimensional Vector Space
5.2.4 Infinite Dimensional Function Space
6 Determinants
6.1 Systems of Linear Equations
6.1.1 Solution of Two Linear Equations
6.1.2 Properties of Second-Order Determinants
6.1.3 Solution of Three Linear Equations
6.2 General Definition of Determinants
6.2.1 Notations
6.2.2 Definition of a nnth-Order Determinant
6.2.3 Minors, Cofactors
6.2.4 Laplacian Development of Determinants by a Row (or a Column)
6.3 Properties of Determinants
6.4 Cramer's Rule
6.4.1 Nonhomogeneous Systems
6.4.2 Homogeneous Systems
6.5 Block Diagonal Determinants
6.6 Laplacian Developments by Complementary Minors
6.7 Multiplication of Determinants of the Same Order
6.8 Differentiation of Determinants
6.9 Determinants in Geometry
7 Matrix Algebra
7.1 Matrix Notation
7.1.1 Definition
7.1.2 Some Special Matrices
7.1.3 Matrix Equation
7.1.4 Transpose of a Matrix
7.2 Matrix Multiplication
7.2.1 Product of Two Matrices
7.2.2 Motivation of Matrix Multiplication
7.2.3 Properties of Product Matrices
7.2.4 Determinant of Matrix Product
7.2.5 The Commutator
7.3 Systems of Linear Equations
7.3.1 Gauss Elimination Method
7.3.2 Existence and Uniqueness of Solutions of Linear Systems
7.4 Inverse Matrix
7.4.1 Non-singular Matrix
7.4.2 Inverse Matrix by Cramer's Rule
7.4.3 Inverse of Elementary Matrices
7.4.4 Inverse Matrix by Gauss-Jordan Elimination
8 Eigenvalue Problems of Matrices
8.1 Eigenvalues and Eigenvectors
8.1.1 Secular Equation
8.1.2 Properties of Characteristic Polynomial
8.1.3 Properties of Eigenvalues
8.2 Some Terminology
8.2.1 Hermitian Conjugation
8.2.2 Orthogonality
8.2.3 Gram-Schmidt Process
8.3 Unitary Matrix and Orthogonal Matrix
8.3.1 Unitary Matrix
8.3.2 Properties of Unitary Matrix
8.3.3 Orthogonal Matrix
8.3.4 Independent Elements of an Orthogonal Matrix
8.3.5 Orthogonal Transformation and Rotation Matrix
8.4 Diagonalization
8.4.1 Similarity Transformation
8.4.2 Diagonalizing a Square Matrix
8.4.3 Quadratic Forms
8.5 Hermitian Matrix and Symmetric Matrix
8.5.1 Definitions
8.5.2 Eigenvalues of Hermitian Matrix
8.5.3 Diagonalizing a Hermitian Matrix
8.5.4 Simultaneous Diagonalization
8.6 Normal Matrix
8.7 Functions of a Matrix
8.7.1 Polynomial Functions of a Matrix
8.7.2 Evaluating Matrix Functions by Diagonalization
8.7.3 The Cayley-Hamilton Theorem
Appendix References
This bibliograph includes the references cited in the text and a few other books and tables that might be useful.
Index