| Complex Analysis | |
| Complex Numbers | p. 3 |
| Our Number System | p. 3 |
| Addition and Multiplication of Integers | p. 4 |
| Inverse Operations | p. 5 |
| Negative Numbers | p. 6 |
| Fractional Numbers | p. 7 |
| Irrational Numbers | p. 8 |
| Imaginary Numbers | p. 9 |
| Logarithm | p. 13 |
| Napier's Idea of Logarithm | p. 13 |
| Briggs' Common Logarithm | p. 15 |
| A Peculiar Number Called e | p. 18 |
| The Unique Property of e | p. 18 |
| The Natural Logarithm | p. 19 |
| Approximate Value of e | p. 21 |
| The Exponential Function as an Infinite Series | p. 21 |
| Compound Interest | p. 21 |
| The Limiting Process Representing e | p. 23 |
| The Exponential Function e[superscript x] | p. 24 |
| Unification of Algebra and Geometry | p. 24 |
| The Remarkable Euler Formula | p. 24 |
| The Complex Plane | p. 25 |
| Polar Form of Complex Numbers | p. 28 |
| Powers and Roots of Complex Numbers | p. 30 |
| Trigonometry and Complex Numbers | p. 33 |
| Geometry and Complex Numbers | p. 40 |
| Elementary Functions of Complex Variable | p. 46 |
| Exponential and Trigonometric Functions of z | p. 46 |
| Hyperbolic Functions of z | p. 48 |
| Logarithm and General Power of z | p. 50 |
| Inverse Trigonometric and Hyperbolic Functions | p. 55 |
| Exercises | p. 58 |
| Complex Functions | p. 61 |
| Analytic Functions | p. 61 |
| Complex Function as Mapping Operation | p. 62 |
| Differentiation of a Complex Function | p. 62 |
| Cauchy-Riemann Conditions | p. 65 |
| Cauchy-Riemann Equations in Polar Coordinates | p. 67 |
| Analytic Function as a Function of z Alone | p. 69 |
| Analytic Function and Laplace's Equation | p. 74 |
| Complex Integration | p. 81 |
| Line Integral of a Complex Function | p. 81 |
| Parametric Form of Complex Line Integral | p. 84 |
| Cauchy's Integral Theorem | p. 87 |
| Green's Lemma | p. 87 |
| Cauchy-Goursat Theorem | p. 89 |
| Fundamental Theorem of Calculus | p. 90 |
| Consequences of Cauchy's Theorem | p. 93 |
| Principle of Deformation of Contours | p. 93 |
| The Cauchy Integral Formula | p. 94 |
| Derivatives of Analytic Function | p. 96 |
| Exercises | p. 103 |
| Complex Series and Theory of Residues | p. 107 |
| A Basic Geometric Series | p. 107 |
| Taylor Series | p. 108 |
| The Complex Taylor Series | p. 108 |
| Convergence of Taylor Series | p. 109 |
| Analytic Continuation | p. 111 |
| Uniqueness of Taylor Series | p. 112 |
| Laurent Series | p. 117 |
| Uniqueness of Laurent Series | p. 120 |
| Theory of Residues | p. 126 |
| Zeros and Poles | p. 126 |
| Definition of the Residue | p. 128 |
| Methods of Finding Residues | p. 129 |
| Cauchy's Residue Theorem | p. 133 |
| Second Residue Theorem | p. 134 |
| Evaluation of Real Integrals with Residues | p. 141 |
| Integrals of Trigonometric Functions | p. 141 |
| Improper Integrals I: Closing the Contour with a Semicircle at Infinity | p. 144 |
| Fourier Integral and Jordan's Lemma | p. 147 |
| Improper Integrals II: Closing the Contour with Rectangular and Pie-shaped Contour | p. 153 |
| Integration Along a Branch Cut | p. 158 |
| Principal Value and Indented Path Integrals | p. 160 |
| Exercises | p. 165 |
| Determinants and Matrices | |
| Determinants | p. 173 |
| Systems of Linear Equations | p. 173 |
| Solution of Two Linear Equations | p. 173 |
| Properties of Second-Order Determinants | p. 175 |
| Solution of Three Linear Equations | p. 175 |
| General Definition of Determinants | p. 179 |
| Notations | p. 179 |
| Definition of a nth Order Determinant | p. 181 |
| Minors, Cofactors | p. 183 |
| Laplacian Development of Determinants by a Row (or a Column) | p. 184 |
| Properties of Determinants | p. 188 |
| Cramer's Rule | p. 193 |
| Nonhomogeneous Systems | p. 193 |
| Homogeneous Systems | p. 195 |
| Block Diagonal Determinants | p. 196 |
| Laplacian Developments by Complementary Minors | p. 198 |
| Multiplication of Determinants of the Same Order | p. 202 |
| Differentiation of Determinants | p. 203 |
| Determinants in Geometry | p. 204 |
| Exercises | p. 208 |
| Matrix Algebra | p. 213 |
| Matrix Notation | p. 213 |
| Definition | p. 213 |
| Some Special Matrices | p. 214 |
| Matrix Equation | p. 216 |
| Transpose of a Matrix | p. 218 |
| Matrix Multiplication | p. 220 |
| Product of Two Matrices | p. 220 |
| Motivation of Matrix Multiplication | p. 223 |
| Properties of Product Matrices | p. 225 |
| Determinant of Matrix Product | p. 230 |
| The Commutator | p. 232 |
| Systems of Linear Equations | p. 233 |
| Gauss Elimination Method | p. 234 |
| Existence and Uniqueness of Solutions of Linear Systems | p. 237 |
| Inverse Matrix | p. 241 |
| Nonsingular Matrix | p. 241 |
| Inverse Matrix by Cramer's Rule | p. 243 |
| Inverse of Elementary Matrices | p. 246 |
| Inverse Matrix by Gauss-Jordan Elimination | p. 248 |
| Exercises | p. 250 |
| Eigenvalue Problems of Matrices | p. 255 |
| Eigenvalues and Eigenvectors | p. 255 |
| Secular Equation | p. 255 |
| Properties of Characteristic Polynomial | p. 262 |
| Properties of Eigenvalues | p. 265 |
| Some Terminology | p. 266 |
| Hermitian Conjugation | p. 267 |
| Orthogonality | p. 268 |
| Gram-Schmidt Process | p. 269 |
| Unitary Matrix and Orthogonal Matrix | p. 271 |
| Unitary Matrix | p. 271 |
| Properties of Unitary Matrix | p. 272 |
| Orthogonal Matrix | p. 273 |
| Independent Elements of an Orthogonal Matrix | p. 274 |
| Orthogonal Transformation and Rotation Matrix | p. 275 |
| Diagonalization | p. 278 |
| Similarity Transformation | p. 278 |
| Diagonalizing a Square Matrix | p. 281 |
| Quadratic Forms | p. 284 |
| Hermitian Matrix and Symmetric Matrix | p. 286 |
| Definitions | p. 286 |
| Eigenvalues of Hermitian Matrix | p. 287 |
| Diagonalizing a Hermitian Matrix | p. 288 |
| Simultaneous Diagonalization | p. 296 |
| Normal Matrix | p. 298 |
| Functions of a Matrix | p. 300 |
| Polynomial Functions of a Matrix | p. 300 |
| Evaluating Matrix Functions by Diagonalization | p. 301 |
| The Cayley-Hamilton Theorem | p. 305 |
| Exercises | p. 309 |
| References | p. 313 |
| Index | p. 315 |
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