Table of contents for Mathematical methods for engineers and scientists / K.T. Tang.


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Part I Complex Analysis
1   Complex Numbers ............. ..........................    3
1.1  Our Number System  ....................................  3
1.1.1 Addition and Multiplication of Integers .............. 4
1.1.2 Inverse Operations ................................ 5
1.1.3 Negative Numbers ...............................  6
1.1.4 Fractional Numbers ................................  7
1.1.5 Irrational Numbers ................................ 8
1.1.6  Imaginary  Numbers  ..................... .........  9
1.2  Logarithm  .................................... ........  13
1.2.1  Napier's Idea of Logarithm  .........................  13
1.2.2 Briggs' Common Logarithm ........................ 15
1.3 A Peculiar Number Called e ........................... 18
1.3.1  The Unique Property of e  ..........................  18
1.3.2  The Natural Logarithm  ............................  19
1.3.3  Approximate Value of e  ...........................  21
1.4 The Exponential Function as an Infinite Series .............. 21
1.4.1 Compound Interest ................................ 21
1.4.2 The Limiting Process Representing e................. 23
1.4.3 The Exponential Function ex . ...................... 24
1.5 Unification of Algebra and Geometry ...................... 24
1.5.1 The Remarkable Euler Formula ..................... 24
1.5.2  The Complex  Plane  ...............................  25
1.6 Polar Form of Complex Numbers .......................... 28
1.6.1 Powers and Roots of Complex Numbers .............. 30
1.6.2 Trigonometry and Complex Numbers ................ 33
1.6.3 Geometry and Complex Numbers ................... 40
1.7 Elementary Functions of Complex Variable ................. 46
1.7.1 Exponential and Trigonometric Functions of z ........ 46
1.7.2  Hyperbolic Functions of z  ..........................  48
1.7.3 Logarithm and General Power of z .................. 50
1.7.4 Inverse Trigonometric and Hyperbolic Functions....... 55
Exercises .............................................    58
2   Com plex  Functions  ............ ............................  61
2.1 Analytic Functions .................................. 61
2.1.1 Complex Function as Mapping Operation ............ 62
2.1.2  Differentiation of a Complex Function ................  62
2.1.3 Cauchy-Riemann Conditions ....................... 65
2.1.4 Cauchy-Riemann Equations in Polar Coordinates ..... 67
2.1.5 Analytic Function as a Function of z Alone ........... 69
2.1.6 Analytic Function and Laplace's Equation ............ 74
2.2 Complex Integration.................................. 81
2.2.1 Line Integral of a Complex Function ................. 81
2.2.2 Parametric Form of Complex Line Integral ........... 84
2.3  Cauchy's Integral Theorem  ...............................  87
2.3.1  Green's Lemma  ...................................  87
2.3.2 Cauchy-Goursat Theorem .......................... 89
2.3.3 Fundamental Theorem of Calculus ................... 90
2.4 Consequences of Cauchy's Theorem ........................ 93
2.4.1 Principle of Deformation of Contours ................ 93
2.4.2 The Cauchy Integral Formula ....................... 94
2.4.3 Derivatives of Analytic Function .................... 96
Exercises .....................  . ..........................103
3   Complex Series and Theory of Residues .................. 107
3.1  A  Basic  Geometric Series  ................................ 107
3.2 Taylor Series .................. ......................108
3.2.1  The Complex Taylor Series ......................... 108
3.2.2  Convergence of Taylor Series  ....................... 109
3.2.3  Analytic  Continuation  ............................. 111
3.2.4  Uniqueness of Taylor Series ......................... 112
3.3  Laurent Series ............... .. ....................117
3.3.1  Uniqueness of Laurent Series........................ 120
3.4  Theory  of Residues  ............ .......................... 126
3.4.1  Zeros  and  Poles  ................................... 126
3.4.2  Definition  of the Residue  ........................... 128
3.4.3  Methods of Finding Residues  ....................... 129
3.4.4  Cauchy's Residue Theorem  ......................... 133
3.4.5  Second  Residue Theorem  ........................... 134
3.5 Evaluation of Real Integrals with Residues .................. 141
3.5.1  Integrals of Trigonometric Functions ................. 141
3.5.2 Improper Integrals I: Closing the Contour
with  a Semicircle at Infinity  ........................ 144
3.5.3 Fourier Integral and Jordan's Lemma ............... 147
3.5.4 Improper Integrals II: Closing the Contour
with Rectangular and Pie-shaped Contour ............ 153
3.5.5 Integration Along a Branch Cut ..................... 158
3.5.6 Principal Value and Indented Path Integrals .......... 160
E x ercises  .. ......... ........................................  165
Part II Determinants and Matrices
4   Determinants...........................................173
4.1  Systems of Linear Equations  ............................. 173
4.1.1  Solution of Two Linear Equations  ................... 173
4.1.2 Properties of Second-Order Determinants ............. 175
4.1.3  Solution of Three Linear Equations .................. 175
4.2 General Definition of Determinants ........................ 179
4.2.1 Notations ...................  ..................179
4.2.2 Definition of a nth Order Determinant ............... 181
4.2.3 Minors, Cofactors ..............................183
4.2.4 Laplacian Development of Determinants by a Row
(or a  Column)  ........................... ........184
4.3  Properties of Determinants  ............................... 188
4.4  Cramer's Rule .................  .....................193
4.4.1  Nonhomogeneous Systems .......................... 193
4.4.2  Homogeneous Systems  ............................. 195
4.5 Block Diagonal Determinants ............................. 196
4.6 Laplacian Developments by Complementary Minors .......... 198
4.7 Multiplication of Determinants of the Same Order ........... 202
4.8  Differentiation of Determinants ............................ 203
4.9  Determinants in  Geometry ................................ 204
Exercises ...............................................208
5   Matrix Algebra........................................213
5.1 Matrix Notation...................   ..................213
5.1.1  D efinition  ............. ........................... 213
5.1.2  Some  Special Matrices  ............................. 214
5.1.3  M atrix  Equation  .................................. 216
5.1.4  Transpose of a  Matrix  ............................. 218
5.2 Matrix Multiplication ............  .................... 220
5.2.1  Product of Two  Matrices ........................... 220
5.2.2  Motivation of Matrix Multiplication  ................. 223
5.2.3  Properties of Product Matrices .................... 225
5.2.4  Determinant of Matrix Product ................... . 230
5.2.5  The  Commutator . . ........ ...................... 232
5.3 Systems of Linear Equations ...........................  233
5.3.1  Gauss Elimination  Method  ......................... 234
5.3.2 Existence and Uniqueness of Solutions
of Linear Systems ..............................  237
5.4  Inverse Matrix  .......................................... 241
5.4.1  Nonsingular M atrix  ........ ........................ 241
5.4.2 Inverse Matrix by Cramer's Rule .................... 243
5.4.3 Inverse of Elementary Matrices ...................... 246
5.4.4 Inverse Matrix by Gauss-Jordan Elimination ......... 248
Exercises .........  ....................................  250
6   Eigenvalue Problems of Matrices .......................... 255
6.1 Eigenvalues and Eigenvectors .......................... 255
6.1.1 Secular Equation ................... .............255
6.1.2 Properties of Characteristic Polynomial .............. 262
6.1.3  Properties of Eigenvalues ............... .......... . 265
6.2 Some Terminology .................................. 266
6.2.1 Hermitian Conjugation ............................ 267
6.2.2  Orthogonality ..................................... 268
6.2.3  Gram-Schmidt Process  ........................ ... 269
6.3 Unitary Matrix and Orthogonal Matrix .................... 271
6.3.1 Unitary Matrix ................................  271
6.3.2 Properties of Unitary Matrix. ....................... 272
6.3.3  Orthogonal Matrix  ................................273
6.3.4 Independent Elements of an Orthogonal Matrix ....... 274
6.3.5 Orthogonal Transformation and Rotation Matrix ...... 275
6.4 Diagonalization .........................................78
6.4.1  Similarity  Transformation  .......................... 278
6.4.2  Diagonalizing a Square Matrix  ................... .. 281
6.4.3 Quadratic Forms ............... ...............284
6.5 Hermitian Matrix and Symmetric Matrix ................... 286
6.5.1 Definitions ...................................... 286
6.5.2 Eigenvalues of Hermitian Matrix .................... 287
6.5.3 Diagonalizing a Hermitian Matrix ................... 288
6.5.4  Simultaneous Diagonalization  ....................... 296
6.6 Normal Matrix .......................................... 298
6.7  Functions of a Matrix  .................................... 300
6.7.1  Polynomial Functions of a Matrix  ................... 300
6.7.2 Evaluating Matrix Functions by Diagonalization ....... 301
6.7.3 The Cayley-Hamilton Theorem ................... . 305
Exercises .............................................. 309
References ............................................ 313
Index....................................................315



Library of Congress subject headings for this publication: Mathematical physics Textbooks, Engineering mathematics Textbooks, Mathematical models Textbooks