Complex Variables and Applications, James Ward Brown, Ruel V. Churchill

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complex variables and applications

Complex Variables and Applications کتاب در زمینه متغیرهای اعداد مختلط و کاربردهای اعداد مختلط است.

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زبان: انگلیسی

حجم فایل: 5 مگابایت

فهرست:

Complex Numbers 1

Sums and Products 1

Basic Algebraic Properties 3

Further Properties 5

Vectors and Moduli 9

Complex Conjugates 13

Exponential Form 16

Products and Powers in Exponential Form 18

Arguments of Products and Quotients 20

Roots of Complex Numbers 24

Examples 27

Regions in the Complex Plane 31

Analytic Functions 35

Functions of a Complex Variable 35

Mappings 38

Mappings by the Exponential Function 42

Limits 45

Theorems on Limits 48

Limits Involving the Point at Infinity 50

Continuity 53

Derivatives 56

Differentiation Formulas 60

Cauchy–Riemann Equations 63

Sufficient Conditions for Differentiability 66

Polar Coordinates 68

Analytic Functions 73

Examples 75

Harmonic Functions 78

Uniquely Determined Analytic Functions 83

Reflection Principle 85

Elementary Functions 89

The Exponential Function 89

The Logarithmic Function 93

Branches and Derivatives of Logarithms 95

Some Identities Involving Logarithms 98

Complex Exponents 101

Trigonometric Functions 104

Hyperbolic Functions 109

Inverse Trigonometric and Hyperbolic Functions 112

Integrals 117

Derivatives of Functions w(t) 117

Definite Integrals of Functions w(t) 119

Contours 122

Contour Integrals 127

Some Examples 129

Examples with Branch Cuts 133

Upper Bounds for Moduli of Contour Integrals 137

Antiderivatives 142

Proof of the Theorem 146

Cauchy–Goursat Theorem 150

Proof of the Theorem 152

Simply Connected Domains 156

Multiply Connected Domains 158

Cauchy Integral Formula 164

An Extension of the Cauchy Integral Formula 165

Some Consequences of the Extension 168

Liouville’s Theorem and the Fundamental Theorem of Algebra 172

Maximum Modulus Principle 175

Series 181

Convergence of Sequences 181

Convergence of Series 184

Taylor Series 189

Proof of Taylor’s Theorem 190

Examples 192

Laurent Series 197

Proof of Laurent’s Theorem 199

Examples 202

Absolute and Uniform Convergence of Power Series 208

Continuity of Sums of Power Series 211

Integration and Differentiation of Power Series 213

Uniqueness of Series Representations 217

Multiplication and Division of Power Series 222

Residues and Poles 229

Isolated Singular Points 229

Residues 231

Cauchy’s Residue Theorem 234

Residue at Infinity 237

The Three Types of Isolated Singular Points 240

Residues at Poles 244

Examples 245

Zeros of Analytic Functions 249

Zeros and Poles 252

Behavior of Functions Near Isolated Singular Points 257

Applications of Residues 261

Evaluation of Improper Integrals 261

Example 264

Improper Integrals from Fourier Analysis 269

Jordan’s Lemma 272

Indented Paths 277

An Indentation Around a Branch Point 280

Integration Along a Branch Cut 283

Definite Integrals Involving Sines and Cosines 288

Argument Principle 291

Roache’s Theorem 294

Inverse Laplace Transforms 298

Examples 301

Mapping by Elementary Functions 311

Linear Transformations 311

The Transformation w = 1/z 313

Mappings by 1/z 315

Linear Fractional Transformations 319

An Implicit Form 322

Mappings of the Upper Half Plane 325

The Transformation w = sin z 330

Mappings by z2 and Branches of z1/2 336

Square Roots of Polynomials 341

Riemann Surfaces 347

Surfaces for Related Functions 351

Conformal Mapping 355

Preservation of Angles 355

Scale Factors 358

Local Inverses 360

Harmonic Conjugates 363

Transformations of Harmonic Functions 365

Transformations of Boundary Conditions 367

Applications of Conformal Mapping 373

Steady Temperatures 373

Steady Temperatures in a Half Plane 375

A Related Problem 377

Temperatures in a Quadrant 379

Electrostatic Potential 385

Potential in a Cylindrical Space 386

Two-Dimensional Fluid Flow 391

The Stream Function 393

Flows Around a Corner and Around a Cylinder 395

The Schwarz–Christoffel Transformation 403

Mapping the Real Axis onto a Polygon 403

Schwarz–Christoffel Transformation 405

Triangles and Rectangles 408

Degenerate Polygons 413

Fluid Flow in a Channel Through a Slit 417

Flow in a Channel with an Offset 420

Electrostatic Potential About an Edge of a Conducting Plate 422

Integral Formulas of the Poisson Type 429

Poisson Integral Formula 429

Dirichlet Problem for a Disk 432

Related Boundary Value Problems 437

Schwarz Integral Formula 440

Dirichlet Problem for a Half Plane 441

Neumann Problems 445

Appendixes 449

Bibliography 449

Table of Transformations of Regions 452

Index 461

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Complex Variables and Applications, James Ward Brown, Ruel V. Churchill

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