Chapter 1．Conceptual Structure of System Analysis 1
1-1 Introduction 1
1-2 Classical Steady-state Response of a Linear System 1
1-3 Characterization of the System Function as a Function of a Complex Variable 2
1-4 Fourier Series 5
1-5 Fourier Integral 6
1-6 The Laplace Integral 8
1-7 Frequency,and the Generalized Frequency Variable 10
1-8 Stability 12
1-9 Convolution-type Integrals 12
1-10 Idealized Systems 13
1-11 Linear Systems with Time-varying Parameters 14
1-12 Other Systems 14
Problems 14
Chapter 2．Introduction to Function Theory 19
2-1 Introduction 19
2-2 Definition of a Function 24
2-3 Limit,Continuity 26
2-4 Derivative of a Function 29
2-5 Definition of Regularity,Singular Points,and Analyticity 31
2-6 The Cauchy-Riemann Equations 33
2-7 Transcendental Functions 35
2-8 Harmonic Functions 41
Problems 42
Chapter 3．Conformal Mapping 46
3-1 Introduction 46
3-2 Some Simple Examples of Transformations 46
3-3 Practical Applications 52
3-4 The Function w=1/s 56
3-5 The Function w=1/2(s+1/s) 57
3-6 The Exponential Function 61
3-7 Hyperbolic and Trigonometric Functions 62
3-8 The Point at Infinity;The Riemann Sphere 64
3-9 Further Properties of the Reciprocal Function 66
3-10 The Bilinear Transformation 70
3-11 Conformal Mapping 73
3-12 Solution of Two-dimensional-field Problems 77
Problems 81
Chapter 4．Integration 85
4-1 Introduction 85
4-2 Some Definitions 85
4-3 Integration 88
4-4 Upper Bound of a Contour Integral 94
4-5 Cauchy Integral Theorem 94
4-6 Independence of Integration Path 98
4-7 Significance of Connectivity 99
4-8 Primitive Function(Antiderivative) 100
4-9 The Logarithm 102
4-10 Cauchy Integral Formulas 105
4-11 Implications of the Cauchy Integral Formulas 108
4-12 Morera＇s Theorem 109
4-13 Use of Primitive Function to Evaluate a Contour Integral 109
Problems 110
Chapter 5．Infinite Series 116
5-1 Introduction 116
5-2 Series of Constants 116
5-3 Series of Functions 120
5-4 Integration of Series 124
5-5 Convergence of Power Series 125
5-6 Properties of Power Series 128
5-7 Taylor Series 129
5-8 Laurent Series 134
5-9 Comparison of Taylor and Laurent Series 136
5-10 Laurent Expansions about a Singular Point 139
5-11 Poles and Essential Singularities;Residues 142
5-12 Residue Theorem 145
5-13 Analytic Continuation 147
5-14 Classification of Single-valued Functions 152
5-15 Partial-fraction Expansion 153
5-16 Partial-fraction Expansion of Meromorphic Functions(Mittag-Leffler Theorem) 157
Problems 162
Chapter 6．Multivalued Functions 169
6-1 Introduction 169
6-2 Examples of Inverse Functions Which Are Multivalued 170
6-3 The Logarithmic Function 176
6-4 Differentiability of Multivalued Functions 177
6-5 Integration around a Branch Point 180
6-6 Position of Branch Cut 185
6-7 The Function w=s+(s2-1)1/2 185
6-8 Locating Branch Points 186
6-9 Expansion of Multivalued Functions in Series 188
6-10 Application to Root Locus 190
Problems 197
Chapter 7．Some Useful Theorems 201
7-1 Introduction 201
7-2 Properties of Real Functions 201
7-3 Gauss Mean-value Theorem(and Related Theorems) 205
7-4 Principle of the Maximum and Minimum 207
7-5 An Application to Network Theory 208
7-6 The Index Principle 211
7-7 Applications of the Index Principle,Nyquist Criterion 213
7-8 Poisson＇s Integrals 215
7-9 Poisson＇s Integrals Transformed to the Imaginary Axis 220
7-10 Relationships between Real and Imaginary Parts,for Real Frequencies 223
7-11 Gain and Angle Functions 229
Problems 231
Chapter 8．Theorems on Real Integrals 234
8-1 Introduction 234
8-2 Piecewise Continuous Functions of a Real Variable 234
8-3 Theorems and Definitions for Real Integrals 236
8-4 Improper Integrals 237
8-5 Almost Piecewise Continuous Functions 240
8-6 Iterated Integrals of Functions of Two Variables(Finite Limits) 242
8-7 Iterated Integrals of Functions of Two Variables(Infinite Limits) 247
8-8 Limit under the Integral for Improper Integrals 250
8-9 M Test for Uniform Convergence of an Improper Integral of the First Kind 251
8-10 A Theorem for Trigonometric Integrals 252
8-11 Two Theorems on Integration over Large Semicircles 254
8-12 Evaluation of Improper Real Integrals by Contour Integration 259
Problems 263
Chapter 9．The Fourier Integral 268
9-1 Introduction 268
9-2 Derivation of the Fourier Integral Theorem 268
9-3 Some Properties of the Fourier Transform 273
9-4 Remarks about Uniqueness and Symmetry 273
9-5 Parseval＇s Theorem 279
Problems 282
Chapter 10．The Laplace Transform 285
10-1 Introduction 285
10-2 The Two-sided Laplace Transform 285
10-3 Functions of Exponential Order 287
10-4 The Laplace Integral for Functions of Exponential Order 288
10-5 Convergence of the Laplace Integral for the General Case 289
10-6 Further Ideas about Uniform Convergence 293
10-7 Convergence of the Two-sided Laplace Integral 295
10-8 The One-and Two-sided Laplace Transforms 297
10-9 Significance of Analytic Continuation in Evaluating the Laplace Integral 298
10-10 Linear Combinations of Laplace Transforms 299
10-11 Laplace Transforms of Some Typical Functions 300
10-12 Elementary Properties of F(s) 306
10-13 The Shifting Theorems 309
10-14 Laplace Transform of the Derivative of f(t) 311
10-15 Laplace Transform of the Integral of a Function 312
10-16 Initial-and Final-value Theorems 314
10-17 Nonuniqueness of Function Pairs for the Two-sided Laplace Transform 315
10-18 The Inversion Formula 318
10-19 Evaluation of the Inversion Formula 322
10-20 Evaluating the Residues(The Heaviside Expansion Theorem) 324
10-21 Evaluating the Inversion Integral When F(s)Is Multivalued 326
Problems 328
Chapter 11．Convolution Theorems 336
11-1 Introduction 336
11-2 Convolution in the t Plane(Fourier Transform) 337
11-3 Convolution in the t Plane(Two-sided Laplace Transform) 338
11-4 Convolution in the t Plane(One-sided Transform) 342
11-5 Convolution in the s Plane(One-sided Transform) 343
11-6 Application of Convolution in the s Plane to Amplitude Modulation 347
11-7 Convolution in the 3 Plane(Two-sided Transform) 349
Problems 350
Chapter 12．Further Properties of the Laplace Transform 353
12-1 Introduction 353
12-2 Behavior of F(s)at Infinity 353
12-3 Functions of Exponential Type 357
12-4 A Special Class of Piecewise Continuous Functions 362
12-5 Laplace Transform of the Derivative of a Piecewise Continuous Function of Exponential Order 367
12-6 Approximation of f(t)by Polynomials 370
12-7 Initial-and Final-value Theorems 372
12-8 Conditions Sufficient to Make F(s)a Laplace Transform 374
12-9 Relationships between Properties of f(t) and F(s) 376
Problems 378
Chapter 13．Solution of Ordinary Linear Equations with Constant Coefficients 381
13-1 Introduction 381
13-2 Existence of a Laplace Transform Solution for a Second-order Equation 381
13-3 Solution of Simultaneous Equations 384
13-4 The Natural Response 388
13-5 Stability 390
13-6 The Forced Response 390
13-7 Illustrative Examples 391
13-8 Solution for the Integral Function 395
13-9 Sinusoidal Steady-state Response 397
13-10 Immittance Functions 398
13-11 Which Is the Driving Function? 400
13-12 Combination of Immittance Functions 400
13-13 Helmholtz Theorem 403
13-14 Appraisal of the Immittance Concept and the Helmholtz Theorem 405
13-15 The System Function 406
Problems 407
Chapter 14．Impulse Functions 410
14-1 Introduction 410
14-2 Examples of an Impulse Response 410
14-3 Impulse Response for the General Case 412
14-4 Impulsive Response 415
14-5 Impulse Excitation Occurring at t=T1 418
14-6 Generalization of the＂Laplace Transform＂of the Derivative 419
14-7 Response to the Derivative and Integral of an Excitation 422
14-8 The Singularity Functions 424
14-9 Interchangeability of Order of Differentiation and Integration 425
14-10 Integrands with Impulsive Factors 426
14-11 Convolution Extended to Impulse Functions 428
14-12 Superposition 430
14-13 Summary 431
Problems 433
Chapter 15．Periodic Functions 435
15-1 Introduction 435
15-2 Laplace Transform of a Periodic Function 436
15-3 Application to the Response of a Physical Lumped-parameter System 438
15-4 Proof That ？-1[P(s)] Is Periodic 440
15-5 The Case Where H(s)Has a Pole at Infinity 441
15-6 Illustrative Example 442
Problems 444
Chapter 16．The Z Transform 445
16-1 Introduction 445
16-2 The Laplace Transform of f＊(t) 446
16-3 Z Transform of Powers of t 448
16-4 Z Transform of a Function Multiplied by e-at 449
16-5 The Shifting Theorem 450
16-6 Initial-and Final-value Theorems 450
16-7 The Inversion Formula 451
16-8 Periodic Properties of F＊(s),and Relationship to F(s) 453
16-9 Transmission of a System with Synchronized Sampling of Input and Output 456
16-10 Convolution 457
16-11 The Two-sided Z Transform 458
16-12 Systems with Sampled Input and Continuous Output 459
16-13 Discontinuous Functions 462
Problems 462
Appendix A 465
Appendix B 468
Bibliography 469
Index 471