Mathematical Methods for Engineers and Scientists 1: Complex Analysis, Determinants and Matrices

Copyright\r......Page 4
Preface......Page 6
Table of Contents\r......Page 8
Part I: Complex Analysis......Page 12
1.1 Our Number System......Page 14
1.1.1 Addition and Multiplication of Integers......Page 15
1.1.2 Inverse Operations......Page 16
1.1.3 Negative Numbers......Page 17
1.1.4 Fractional Numbers......Page 18
1.1.5 Irrational Numbers......Page 19
1.1.6 Imaginary Numbers......Page 20
1.2.1 Napier’s Idea of Logarithm......Page 24
1.2.2 Briggs’ Common Logarithm......Page 26
1.3.1 The Unique Property of e......Page 29
1.3.2 The Natural Logarithm......Page 30
1.4.1 Compound Interest......Page 32
1.4.2 The Limiting Process Representing e......Page 34
1.5.1 The Remarkable Euler Formula......Page 35
1.5.2 The Complex Plane......Page 36
1.6 Polar Form of Complex Numbers......Page 39
1.6.1 Powers and Roots of Complex Numbers......Page 41
1.6.2 Trigonometry and Complex Numbers......Page 44
1.6.3 Geometry and Complex Numbers......Page 51
1.7.1 Exponential and Trigonometric Functions of z......Page 57
1.7.2 Hyperbolic Functions of z......Page 59
1.7.3 Logarithm and General Power of z......Page 61
1.7.4 Inverse Trigonometric and Hyperbolic Functions......Page 66
2.1 Analytic Functions......Page 72
2.1.2 Differentiation of a Complex Function......Page 73
2.1.3 Cauchy–Riemann Conditions......Page 76
2.1.4 Cauchy–Riemann Equations in Polar Coordinates......Page 78
2.1.5 Analytic Function as a Function of z Alone......Page 80
2.1.6 Analytic Function and Laplace’s Equation......Page 85
2.2.1 Line Integral of a Complex Function......Page 92
2.2.2 Parametric Form of Complex Line Integral......Page 95
2.3.1 Green’s Lemma......Page 98
2.3.2 Cauchy–Goursat Theorem......Page 100
2.3.3 Fundamental Theorem of Calculus......Page 101
2.4.1 Principle of Deformation of Contours......Page 104
2.4.2 The Cauchy Integral Formula......Page 105
2.4.3 Derivatives of Analytic Function......Page 107
3.1 A Basic Geometric Series......Page 118
3.2.1 The Complex Taylor Series......Page 119
3.2.2 Convergence of Taylor Series......Page 120
3.2.3 Analytic Continuation......Page 122
3.2.4 Uniqueness of Taylor Series......Page 123
3.3 Laurent Series......Page 128
3.3.1 Uniqueness of Laurent Series......Page 131
3.4.1 Zeros and Poles......Page 137
3.4.2 Definition of the Residue......Page 139
3.4.3 Methods of Finding Residues......Page 140
3.4.4 Cauchy’s Residue Theorem......Page 144
3.4.5 Second Residue Theorem......Page 145
3.5.1 Integrals of Trigonometric Functions......Page 152
3.5.2 Improper Integrals I: Closing the Contour with a Semicircle at Infinity\r......Page 155
3.5.3 Fourier Integral and Jordan’s Lemma......Page 158
3.5.4 Improper Integrals II: Closing the Contour with Rectangular and Pie-shaped Contour\r......Page 164
3.5.5 Integration Along a Branch Cut......Page 169
3.5.6 Principal Value and Indented Path Integrals......Page 171
Part II: Determinants and Matrices\r......Page 182
4.1.1 Solution of Two Linear Equations......Page 184
4.1.3 Solution of Three Linear Equations......Page 186
4.2.1 Notations......Page 190
4.2.2 Definition of a nth Order Determinant......Page 192
4.2.3 Minors, Cofactors......Page 194
4.2.4 Laplacian Development of Determinants by a Row (or a Column)\r......Page 195
4.3 Properties of Determinants......Page 199
4.4.1 Nonhomogeneous Systems......Page 204
4.4.2 Homogeneous Systems......Page 206
4.5 Block Diagonal Determinants......Page 207
4.6 Laplacian Developments by Complementary Minors......Page 209
4.7 Multiplication of Determinants of the Same Order......Page 213
4.8 Differentiation of Determinants......Page 214
4.9 Determinants in Geometry......Page 215
5.1.1 Definition......Page 224
5.1.2 Some Special Matrices......Page 225
5.1.3 Matrix Equation......Page 227
5.1.4 Transpose of a Matrix......Page 229
5.2.1 Product of Two Matrices......Page 231
5.2.2 Motivation of Matrix Multiplication......Page 234
5.2.3 Properties of Product Matrices......Page 236
5.2.4 Determinant of Matrix Product......Page 241
5.2.5 The Commutator......Page 243
5.3 Systems of Linear Equations......Page 244
5.3.1 Gauss Elimination Method......Page 245
5.3.2 Existence and Uniqueness of Solutions of Linear Systems......Page 248
5.4.1 Nonsingular Matrix......Page 252
5.4.2 Inverse Matrix by Cramer’s Rule......Page 254
5.4.3 Inverse of Elementary Matrices......Page 257
5.4.4 Inverse Matrix by Gauss–Jordan Elimination......Page 259
6.1.1 Secular Equation......Page 266
6.1.2 Properties of Characteristic Polynomial......Page 273
6.1.3 Properties of Eigenvalues......Page 276
6.2 Some Terminology......Page 277
6.2.1 Hermitian Conjugation......Page 278
6.2.2 Orthogonality......Page 279
6.2.3 Gram–Schmidt Process......Page 280
6.3.1 Unitary Matrix......Page 282
6.3.2 Properties of Unitary Matrix......Page 283
6.3.3 Orthogonal Matrix......Page 284
6.3.4 Independent Elements of an Orthogonal Matrix......Page 285
6.3.5 Orthogonal Transformation and Rotation Matrix......Page 286
6.4.1 Similarity Transformation......Page 289
6.4.2 Diagonalizing a Square Matrix......Page 292
6.4.3 Quadratic Forms......Page 295
6.5.1 Definitions......Page 297
6.5.2 Eigenvalues of Hermitian Matrix......Page 298
6.5.3 Diagonalizing a Hermitian Matrix......Page 299
6.5.4 Simultaneous Diagonalization......Page 307
6.6 Normal Matrix......Page 309
6.7.1 Polynomial Functions of a Matrix......Page 311
6.7.2 Evaluating Matrix Functions by Diagonalization......Page 312
6.7.3 The Cayley–Hamilton Theorem......Page 316
References......Page 324
Index......Page 326