Complex Variables and The Laplace Transform For Engineers【2025|PDF下载-Epub版本|mobi电子书|kindle百度云盘下载】

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Chapter 1．Conceptual Structure of System Analysis1

1-1 Introduction1

1-2 Classical Steady-state Response of a Linear System1

1-3 Characterization of the System Function as a Function of a Complex Variable2

1-4 Fourier Series5

1-5 Fourier Integral6

1-6 The Laplace Integral8

1-7 Frequency,and the Generalized Frequency Variable10

1-8 Stability12

1-9 Convolution-type Integrals12

1-10 Idealized Systems13

1-11 Linear Systems with Time-varying Parameters14

1-12 Other Systems14

Problems14

Chapter 2．Introduction to Function Theory19

2-1 Introduction19

2-2 Definition of a Function24

2-3 Limit,Continuity26

2-4 Derivative of a Function29

2-5 Definition of Regularity,Singular Points,and Analyticity31

2-6 The Cauchy-Riemann Equations33

2-7 Transcendental Functions35

2-8 Harmonic Functions41

Problems42

Chapter 3．Conformal Mapping46

3-1 Introduction46

3-2 Some Simple Examples of Transformations46

3-3 Practical Applications52

3-4 The Function w=1/s56

3-5 The Function w=1/2(s+1/s)57

3-6 The Exponential Function61

3-7 Hyperbolic and Trigonometric Functions62

3-8 The Point at Infinity;The Riemann Sphere64

3-9 Further Properties of the Reciprocal Function66

3-10 The Bilinear Transformation70

3-11 Conformal Mapping73

3-12 Solution of Two-dimensional-field Problems77

Problems81

Chapter 4．Integration85

4-1 Introduction85

4-2 Some Definitions85

4-3 Integration88

4-4 Upper Bound of a Contour Integral94

4-5 Cauchy Integral Theorem94

4-6 Independence of Integration Path98

4-7 Significance of Connectivity99

4-8 Primitive Function(Antiderivative)100

4-9 The Logarithm102

4-10 Cauchy Integral Formulas105

4-11 Implications of the Cauchy Integral Formulas108

4-12 Morera＇s Theorem109

4-13 Use of Primitive Function to Evaluate a Contour Integral109

Problems110

Chapter 5．Infinite Series116

5-1 Introduction116

5-2 Series of Constants116

5-3 Series of Functions120

5-4 Integration of Series124

5-5 Convergence of Power Series125

5-6 Properties of Power Series128

5-7 Taylor Series129

5-8 Laurent Series134

5-9 Comparison of Taylor and Laurent Series136

5-10 Laurent Expansions about a Singular Point139

5-11 Poles and Essential Singularities;Residues142

5-12 Residue Theorem145

5-13 Analytic Continuation147

5-14 Classification of Single-valued Functions152

5-15 Partial-fraction Expansion153

5-16 Partial-fraction Expansion of Meromorphic Functions(Mittag-Leffler Theorem)157

Problems162

Chapter 6．Multivalued Functions169

6-1 Introduction169

6-2 Examples of Inverse Functions Which Are Multivalued170

6-3 The Logarithmic Function176

6-4 Differentiability of Multivalued Functions177

6-5 Integration around a Branch Point180

6-6 Position of Branch Cut185

6-7 The Function w=s+(s2-1)1/2185

6-8 Locating Branch Points186

6-9 Expansion of Multivalued Functions in Series188

6-10 Application to Root Locus190

Problems197

Chapter 7．Some Useful Theorems201

7-1 Introduction201

7-2 Properties of Real Functions201

7-3 Gauss Mean-value Theorem(and Related Theorems)205

7-4 Principle of the Maximum and Minimum207

7-5 An Application to Network Theory208

7-6 The Index Principle211

7-7 Applications of the Index Principle,Nyquist Criterion213

7-8 Poisson＇s Integrals215

7-9 Poisson＇s Integrals Transformed to the Imaginary Axis220

7-10 Relationships between Real and Imaginary Parts,for Real Frequencies223

7-11 Gain and Angle Functions229

Problems231

Chapter 8．Theorems on Real Integrals234

8-1 Introduction234

8-2 Piecewise Continuous Functions of a Real Variable234

8-3 Theorems and Definitions for Real Integrals236

8-4 Improper Integrals237

8-5 Almost Piecewise Continuous Functions240

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 Integrals250

8-9 M Test for Uniform Convergence of an Improper Integral of the First Kind251

8-10 A Theorem for Trigonometric Integrals252

8-11 Two Theorems on Integration over Large Semicircles254

8-12 Evaluation of Improper Real Integrals by Contour Integration259

Problems263

Chapter 9．The Fourier Integral268

9-1 Introduction268

9-2 Derivation of the Fourier Integral Theorem268

9-3 Some Properties of the Fourier Transform273

9-4 Remarks about Uniqueness and Symmetry273

9-5 Parseval＇s Theorem279

Problems282

Chapter 10．The Laplace Transform285

10-1 Introduction285

10-2 The Two-sided Laplace Transform285

10-3 Functions of Exponential Order287

10-4 The Laplace Integral for Functions of Exponential Order288

10-5 Convergence of the Laplace Integral for the General Case289

10-6 Further Ideas about Uniform Convergence293

10-7 Convergence of the Two-sided Laplace Integral295

10-8 The One-and Two-sided Laplace Transforms297

10-9 Significance of Analytic Continuation in Evaluating the Laplace Integral298

10-10 Linear Combinations of Laplace Transforms299

10-11 Laplace Transforms of Some Typical Functions300

10-12 Elementary Properties of F(s)306

10-13 The Shifting Theorems309

10-14 Laplace Transform of the Derivative of f(t)311

10-15 Laplace Transform of the Integral of a Function312

10-16 Initial-and Final-value Theorems314

10-17 Nonuniqueness of Function Pairs for the Two-sided Laplace Transform315

10-18 The Inversion Formula318

10-19 Evaluation of the Inversion Formula322

10-20 Evaluating the Residues(The Heaviside Expansion Theorem)324

10-21 Evaluating the Inversion Integral When F(s)Is Multivalued326

Problems328

Chapter 11．Convolution Theorems336

11-1 Introduction336

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 Modulation347

11-7 Convolution in the 3 Plane(Two-sided Transform)349

Problems350

Chapter 12．Further Properties of the Laplace Transform353

12-1 Introduction353

12-2 Behavior of F(s)at Infinity353

12-3 Functions of Exponential Type357

12-4 A Special Class of Piecewise Continuous Functions362

12-5 Laplace Transform of the Derivative of a Piecewise Continuous Function of Exponential Order367

12-6 Approximation of f(t)by Polynomials370

12-7 Initial-and Final-value Theorems372

12-8 Conditions Sufficient to Make F(s)a Laplace Transform374

12-9 Relationships between Properties of f(t) and F(s)376

Problems378

Chapter 13．Solution of Ordinary Linear Equations with Constant Coefficients381

13-1 Introduction381

13-2 Existence of a Laplace Transform Solution for a Second-order Equation381

13-3 Solution of Simultaneous Equations384

13-4 The Natural Response388

13-5 Stability390

13-6 The Forced Response390

13-7 Illustrative Examples391

13-8 Solution for the Integral Function395

13-9 Sinusoidal Steady-state Response397

13-10 Immittance Functions398

13-11 Which Is the Driving Function?400

13-12 Combination of Immittance Functions400

13-13 Helmholtz Theorem403

13-14 Appraisal of the Immittance Concept and the Helmholtz Theorem405

13-15 The System Function406

Problems407

Chapter 14．Impulse Functions410

14-1 Introduction410

14-2 Examples of an Impulse Response410

14-3 Impulse Response for the General Case412

14-4 Impulsive Response415

14-5 Impulse Excitation Occurring at t=T1418

14-6 Generalization of the＂Laplace Transform＂of the Derivative419

14-7 Response to the Derivative and Integral of an Excitation422

14-8 The Singularity Functions424

14-9 Interchangeability of Order of Differentiation and Integration425

14-10 Integrands with Impulsive Factors426

14-11 Convolution Extended to Impulse Functions428

14-12 Superposition430

14-13 Summary431

Problems433

Chapter 15．Periodic Functions435

15-1 Introduction435

15-2 Laplace Transform of a Periodic Function436

15-3 Application to the Response of a Physical Lumped-parameter System438

15-4 Proof That ？-1[P(s)] Is Periodic440

15-5 The Case Where H(s)Has a Pole at Infinity441

15-6 Illustrative Example442

Problems444

Chapter 16．The Z Transform445

16-1 Introduction445

16-2 The Laplace Transform of f＊(t)446

16-3 Z Transform of Powers of t448

16-4 Z Transform of a Function Multiplied by e-at449

16-5 The Shifting Theorem450

16-6 Initial-and Final-value Theorems450

16-7 The Inversion Formula451

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 Output456

16-10 Convolution457

16-11 The Two-sided Z Transform458

16-12 Systems with Sampled Input and Continuous Output459

16-13 Discontinuous Functions462

Problems462

Appendix A465

Appendix B468

Bibliography469

Index471