Complex variables and applications

Cover......Page 1
Title page......Page 3
CONTENTS......Page 5
Preface......Page 9
Sums and Products......Page 13
Basic Algebraic Properties......Page 15
Further Properties......Page 17
Moduli......Page 20
Complex Conjugates......Page 23
Exponential Form......Page 27
Products and Quotients in Exponential Form......Page 29
Roots of Complex Numbers......Page 34
Examples......Page 37
Regions in the Complex Plane......Page 41
Functions of a Complex Variable......Page 45
Mappings......Page 48
Mappings by the Exponential Function......Page 52
Limits......Page 55
Theorems on Limits......Page 58
Limits Involving the Point at Infinity......Page 60
Continuity......Page 63
Derivatives......Page 66
Differentiation Formulas......Page 69
Cauchy-Riemann Equations......Page 72
Sufficient Conditions for Differentiability......Page 75
Polar Coordinates......Page 77
Analytic Functions......Page 82
Examples......Page 84
Harmonic Functions......Page 87
Uniquely Determined Analytic Functions......Page 92
Reflection Principle......Page 94
The Exponential Function......Page 99
The Logarithmic Function......Page 102
Branches and Derivatives of Logarithms......Page 104
Some Identities Involving Logarithms......Page 107
Complex Exponents......Page 109
Trigonometric Functions......Page 112
Hyperbolic Functions......Page 117
Inverse Trigonometric and Hyperbolic Functions......Page 120
Derivatives of Functions $w(t)$......Page 123
Definite Integrals of Functions $w(t)$......Page 125
Contours......Page 128
Contour Integrals......Page 134
Examples......Page 136
Upper Bounds for Moduli of Contour Integrals......Page 142
Antiderivatives......Page 147
Examples......Page 150
Cauchy-Goursat Theorem......Page 154
Proof of the Theorem......Page 156
Simply and Multiply Connected Domains......Page 161
Cauchy Integral Formula......Page 169
Derivatives of Analytic Functions......Page 170
Liouville's Theorem and the Fundamental Theorem of Algebra......Page 177
Maximum Modulus Principle......Page 179
Convergence of Sequences......Page 187
Convergence of Series......Page 190
Taylor Series......Page 194
Examples......Page 197
Laurent Series......Page 202
Examples......Page 207
Absolute and Uniform Convergence of Power Series......Page 212
Continuity of Sums of Power Series......Page 216
Integration and Differentiation of Power Series......Page 218
Uniqueness of Series Representations......Page 222
Multiplication and Division of Power Series......Page 227
Residues......Page 233
Cauchy's Residue Theorem......Page 237
Using a Single Residue......Page 239
The Three Types of Isolated Singular Points......Page 243
Residues at Poles......Page 246
Examples......Page 248
Zeros of Analytic Functions......Page 251
Zeros and Poles......Page 254
Behavior of $f$ Near Isolated Singular Points......Page 259
Evaluation of Improper Integrals......Page 263
Example......Page 266
Improper Integrals from Fourier Analysis......Page 271
Jordan's Lemma......Page 274
Indented Paths......Page 279
An Indentation Around a Branch Point......Page 282
Integration Along a Branch Cut......Page 285
Definite Integrals involving Sines and Cosines......Page 290
Argument Principle......Page 293
Rouche's Theorem......Page 296
Inverse Laplace Transforms......Page 300
Examples......Page 303
Linear Transformations......Page 311
The Transformation $w = 1/z$......Page 313
Mappings by $1/z$......Page 315
Linear Fractional Transformations......Page 319
An Implicit Form......Page 322
Mappings of the Upper Half Plane......Page 325
The Transformation $w = \sin z$......Page 330
Mappings by $z^2$ and Branches of $z^{1/2}$......Page 336
Square Roots of Polynomials......Page 341
Riemann Surfaces......Page 347
Surfaces for Related Functions......Page 350
Preservation of Angles......Page 355
Scale Factors......Page 358
Local Inverses......Page 360
Harmonic Conjugates......Page 363
Transformations of Harmonic Functions......Page 365
Transformations of Boundary Conditions......Page 367
Steady Temperatures......Page 373
Steady Temperatures in a Half Plane......Page 375
A Related Problem......Page 377
Temperatures in a Quadrant......Page 380
Electrostatic Potential......Page 385
Potential in a Cylindrical Space......Page 386
Two-Dimensional Fluid Flow......Page 391
The Stream Function......Page 393
Flows Around a Corner and Around a Cylinder......Page 395
Mapping the Real Axis onto a Polygon......Page 403
Schwarz-Christoffel Transformation......Page 405
Triangles and Rectangles......Page 409
Degenerate Polygons......Page 413
Fluid Flow in a Channel Through a Slit......Page 418
Flow in a Channel with an Offset......Page 420
Electrostatic Potential about an Edge of a Conducting Plate......Page 423
Poisson Integral Formula......Page 429
Dirichlet Problem for a Disk......Page 431
Related Boundary Value Problems......Page 435
Schwarz Integral Formula......Page 439
Dirichlet Problem for a Half Plane......Page 441
Neumann Problems......Page 445
Bibliography......Page 449
Table of Transformations of Regions......Page 453
Index......Page 463