复变函数及应用 英文版2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

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1 Complex Numbers1

Sums and Products1

Basic Algebraic Properties3

Further Properties5

Vectors and Moduli9

Complex Conjugates13

Exponential Form16

Products and Powers in Exponential Form18

Arguments of Products and Quotients20

Roots of Complex Numbers24

Examples27

Regions in the Complex Plane31

2 Analytic Functions35

Functions of a Complex Variable35

Mappings38

Mappings by the Exponential Function42

Limits45

Theorems on Limits48

Limits Involving the Point at Infinity50

Continuity53

Derivatives56

Differentiation Formulas60

Cauchy-Riemann Equations63

Sufficient Conditions for Differentiability66

Polar Coordinates68

Analytic Functions73

Examples75

Harmonic Functions78

Uniquely Determined Analytic Functions83

Reflection Principle85

3 Elementary Functions89

The Exponential Function89

The Logarithmic Function93

Branches and Derivatives of Logarithms95

Some Identities Involving Logarithms98

Complex Exponents101

Trigonometric Functions104

Hyperbolic Functions109

Inverse Trigonometric and Hyperbolic Functions112

4 Integrals117

Derivatives of Functions w（t）117

Definite Integrals of Functions w（t）119

Contours122

Contour Integrals127

Some Examples129

Examples with Branch Cuts133

Upper Bounds for Moduli of Contour Integrals137

Antiderivatives142

Proof of the Theorem146

Cauchy-Goursat Theorem150

Proof of the Theorem152

Simply Connected Domains156

Multiply Connected Domains158

Cauchy Integral Formula164

An Extension of the Cauchy Integral Formula165

Some Consequences of the Extension168

Liouville’s Theorem and the Fundamental Theorem of Algebra172

Maximum Modulus Principle175

5 Series181

Convergence of Sequences181

Convergence of Series184

Taylor Series189

Proof of Taylor’s Theorem190

Examples192

Laurent Series197

Proof of Laurent’s Theorem199

Examples202

Absolute and Uniform Convergence of Power Series208

Continuity of Sums of Power Series211

Integration and Differentiation of Power Series213

Uniqueness of Series Representations217

Multiplication and Division of Power Series222

6 Residues and Poles229

Isolated Singular Points229

Residues231

Cauchy’s Residue Theorem234

Residue at Infinity237

The Three Types of Isolated Singular Points240

Residues at Poles244

Examples245

Zeros of Analytic Functions249

Zeros and Poles252

Behavior of Functions Near Isolated Singular Points257

7 Applications of Residues261

Evaluation of Improper Integrals261

Example264

Improper Integrals from Fourier Analysis269

Jordan’s Lemma272

Indented Paths277

An Indentation Around a Branch Point280

Integration Along a Branch Cut283

Definite Integrals Involving Sines and Cosines288

Argument Principle291

Rouché’s Theorem294

Inverse Laplace Transforms298

Examples301

8 Mapping by Elementary Functions311

Linear Transformations311

The Transformation w＝1/z313

Mappings by 1/z315

Linear Fractional Transformations319

An Implicit Form322

Mappings of the Upper Half Plane325

The Transformation w＝sin z330

Mappings by z2 and Branches of z1/2336

Square Roots of Polynomials341

Riemann Surfaces347

Surfaces for Related Functions351

9 Conformal Mapping355

Preservation of Angles355

Scale Factors358

Local Inverses360

Harmonic Conjugates363

Transformations of Harmonic Functions365

Transformations of Boundary Conditions367

10 Applications of Conformal Mapping373

Steady Temperatures373

Steady Temperatures in a Half Plane375

A Related Problem377

Temperatures in a Quadrant379

Electrostatic Potential385

Potential in a Cylindrical Space386

Two-Dimensional Fluid Flow391

The Stream Function393

Flows Around a Corner and Around a Cylinder395

11 The Schwarz-Christoffel Transformation403

Mapping the Real Axis Onto a Polygon403

Schwarz-Christoffel Transformation405

Triangles and Rectangles408

Degenerate Polygons413

Fluid Flow in a Channel Through a Slit417

Flow in a Channel With an Offset420

Electrostatic Potential About an Edge of a Conducting Plate422

12 Integral Formulas of the Poisson Type429

Poisson Integral Formula429

Dirichlet Problem for a Disk432

Related Boundary Value Problems437

Schwarz Integral Formula440

Dirichlet Problem for a Half Plane441

Neumann Problems445

Appendixes449

Bibliography449

Table of Transformations of Regions452

Index461