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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