Complex variables and applications

Cover......Page 1
Title......Page 4
Copyright......Page 5
Contents......Page 10
Preface......Page 16
Sums and Products......Page 18
Basic Algebraic Properties......Page 20
Further Algebraic Properties......Page 22
Vectors and Moduli......Page 25
Triangle Inequality......Page 28
Complex Conjugates......Page 31
Exponential Form......Page 34
Products and Powers in Exponential Form......Page 37
Arguments of Products and Quotients......Page 38
Roots of Complex Numbers......Page 42
Examples......Page 45
Regions in the Complex Plane......Page 49
Functions and Mappings......Page 54
The Mappingw = z[sup(2)]......Page 57
Limits......Page 61
Theorems on Limits......Page 64
Limits Involving the Point at Infinity......Page 67
Continuity......Page 69
Derivatives......Page 72
Rules for Differentiation......Page 76
Cauchy–Riemann Equations......Page 79
Examples......Page 81
Sufficient Conditions for Differentiability......Page 82
Polar Coordinates......Page 85
Analytic Functions......Page 89
Further Examples......Page 91
Harmonic Functions......Page 94
Uniquely Determined Analytic Functions......Page 97
Reflection Principle......Page 99
The Exponential Function......Page 104
The Logarithmic Function......Page 107
Examples......Page 109
Branches and Derivatives of Logarithms......Page 110
Some Identities Involving Logarithms......Page 114
The Power Function......Page 117
Examples......Page 118
The Trigonometric Functions sin z and cos z......Page 120
Zeros and Singularities of Trigonometric Functions......Page 122
Hyperbolic Functions......Page 126
Inverse Trigonometric and Hyperbolic Functions......Page 129
Derivatives of Functions w(t)......Page 132
Definite Integrals of Functions w(t)......Page 134
Contours......Page 137
Contour Integrals......Page 142
Some Examples......Page 144
Examples Involving Branch Cuts......Page 148
Upper Bounds for Moduli of Contour Integrals......Page 152
Antiderivatives......Page 157
Proof of the Theorem......Page 161
Cauchy–Goursat Theorem......Page 165
Proof of the Theorem......Page 167
Simply Connected Domains......Page 171
Multiply Connected Domains......Page 173
Cauchy Integral Formula......Page 179
An Extension of the Cauchy Integral Formula......Page 181
Verification of the Extension......Page 183
Some Consequences of the Extension......Page 185
Liouville's Theorem and the Fundamental Theorem of Algebra......Page 189
Maximum Modulus Principle......Page 190
Convergence of Sequences......Page 196
Convergence of Series......Page 199
Taylor Series......Page 203
Proof of Taylor's Theorem......Page 204
Examples......Page 206
Negative Powers of (z — z0)......Page 210
Laurent Series......Page 214
Proof of Laurent's Theorem......Page 216
Examples......Page 219
Absolute and Uniform Convergence of Power Series......Page 225
Continuity of Sums of Power Series......Page 228
Integration and Differentiation of Power Series......Page 230
Uniqueness of Series Representations......Page 233
Multiplication and Division of Power Series......Page 238
Isolated Singular Points......Page 244
Residues......Page 246
Cauchy's Residue Theorem......Page 250
Residue at Infinity......Page 252
The Three Types of Isolated Singular Points......Page 255
Examples......Page 257
Residues at Poles......Page 259
Examples......Page 261
Zeros of Analytic Functions......Page 265
Zeros and Poles......Page 268
Behavior of Functions Near Isolated Singular Points......Page 272
Evaluation of Improper Integrals......Page 276
Example......Page 279
Improper Integrals from Fourier Analysis......Page 284
Jordan's Lemma......Page 286
An Indented Path......Page 291
An Indentation Around a Branch Point......Page 294
Integration Along a Branch Cut......Page 297
Definite Integrals Involving Sines and Cosines......Page 301
Argument Principle......Page 304
Rouche's Theorem......Page 307
Inverse Laplace Transforms......Page 311
Linear Transformations......Page 316
The Transformationw w = 1/z......Page 318
Mappings by 1/z......Page 320
Linear Fractional Transformations......Page 324
An Implicit Form......Page 327
Mappings of the Upper Half Plane......Page 330
Examples......Page 332
Mappings by the Exponential Function......Page 335
Mapping Vertical Line Segments by w = sin z......Page 337
Mapping Horizontal Line Segments by w = sin z......Page 339
Some Related Mappings......Page 341
Mappings by z[sup(2)]......Page 343
Mappings by Branches of z[sup(1/2)]......Page 345
Square Roots of Polynomials......Page 349
Riemann Surfaces......Page 355
Surfaces for Related Functions......Page 358
Preservation of Angles and Scale Factors......Page 362
Further Examples......Page 365
Local Inverses......Page 367
Harmonic Conjugates......Page 371
Transformations of Harmonic Functions......Page 374
Transformations of Boundary Conditions......Page 377
Steady Temperatures......Page 382
Steady Temperatures in a Half Plane......Page 384
A Related Problem......Page 386
Temperatures in a Quadrant......Page 388
Electrostatic Potential......Page 393
Examples......Page 394
Two-Dimensional Fluid Flow......Page 399
The Stream Function......Page 401
Flows Around a Corner and Around a Cylinder......Page 403
Mapping the Real Axis onto a Polygon......Page 410
Schwarz–Christoffel Transformation......Page 412
Triangles and Rectangles......Page 416
Degenerate Polygons......Page 419
Fluid Flow in a Channel through a Slit......Page 424
Flow in a Channel with an Offset......Page 426
Electrostatic Potential about an Edge of a Conducting Plate......Page 429
Poisson Integral Formula......Page 434
Dirichlet Problem for a Disk......Page 437
Examples......Page 439
Related Boundary Value Problems......Page 443
Schwarz Integral Formula......Page 445
Dirichlet Problem for a Half Plane......Page 447
Neumann Problems......Page 450
Bibliography......Page 454
Table of Transformations of Regions......Page 458
B......Page 468
C......Page 469
D......Page 470
F......Page 471
I......Page 472
L......Page 473
N......Page 474
Q......Page 475
S......Page 476
T......Page 477
Z......Page 478