Complex Variables and Applications - student soluions manual

Cover......Page 1
Complex Variables and Applications, Seventh Edition......Page 2
Preface......Page 7
Sums and Products......Page 11
Basic Algebraic Properties......Page 13
Further Properties......Page 15
Moduli......Page 18
Complex Conjugates......Page 21
Exponential Form......Page 25
Products and Quotients in Exponential Form......Page 27
Roots of Complex Numbers......Page 32
Examples......Page 35
Regions in the Complex Plane......Page 39
Functions of a Complex Variable......Page 43
Mappings......Page 46
Mappings by the Exponential Function......Page 50
Limits......Page 53
Theorems on Limits......Page 56
Limits Involving the Point at Infinity......Page 58
Continuity......Page 61
Derivatives......Page 64
Differentiation Formu]as......Page 67
Cauchy-Riemann Equations......Page 70
Sufficient Conditions for Differentiability......Page 73
Polar Coordinates......Page 75
Analytic Functions......Page 80
Examples......Page 82
Harmonic Functions......Page 85
Uniquely Determined Analytic Functions......Page 90
Reflection Principle......Page 92
The Exponential Function......Page 97
The Logarithmic Function......Page 100
Branches and Derivatives of Logarithms......Page 102
Some Identities Involving Logarithms......Page 105
Complex Exponents......Page 107
Trigonometric Functions......Page 110
Hyperbolic Functions......Page 115
Inverse Trigonometric and Hyperbolic Functions......Page 118
Derivatives of Functions w(t)......Page 121
Definite Integrals of Functions w(t)......Page 123
Contours......Page 126
Contour Integrals......Page 132
Examples......Page 134
Upper Bounds for Moduli of Contour Integrals......Page 140
Antiderivatives......Page 145
Examples......Page 148
Cauchy-Goursat Theorem......Page 152
Proof of the Theorem......Page 154
Simply and Multiply Connected Domains......Page 159
Cauchy Integral Formula......Page 167
Derivatives of Analytic Functions......Page 168
Liouville\'s Theorem and the Fundamental Theorem of Algebra......Page 175
Maximum Modulus Principle......Page 177
Convergence of Sequences......Page 185
Convergence of Series......Page 188
Taylor Series......Page 192
Examples......Page 195
Laurent Series......Page 200
Examples......Page 205
Abso1ute and Uniform Convergence of Power Series......Page 210
Continuity of Sums of Power Series......Page 214
Integration and Differentiation of Power Series......Page 216
Uniqueness of Series Representations......Page 220
Multiplication and Division of Power Series......Page 225
Residues......Page 231
Cauchy\'s Residue Theorem......Page 235
Using a Single Residue......Page 237
The Three TYpes of Isolated Singular Points......Page 241
Residues at Poles......Page 244
Examples......Page 246
Zeros of Analytic Functions......Page 249
Zeros and Poles......Page 252
Behavior off Near Isolated Singular Points......Page 257
Evaluation of Improper Integrals......Page 261
Example......Page 264
Improper Integrals from Fourier Analysis......Page 269
Jordan\'s Lemma......Page 272
Indented Paths......Page 277
An Indentation Around a Branch Point......Page 280
Integration Along a Branch Cut......Page 283
Definite Integrals involving Sines and Cosines......Page 288
Argument Principle......Page 291
Rouche\'s Theorem......Page 294
Inverse Laplace Transforms......Page 298
Examples......Page 301
Linear Transformations......Page 309
The Transformation w = l/z......Page 311
Mappings by 1/z......Page 313
Linear Fractional Transformations......Page 317
An Implicit Form......Page 320
Mappings of the Upper Half Plane......Page 323
The Transformation w = sin z......Page 328
Mappings by z^2 and Branches of z^1/2......Page 334
Square Roots of Polynomials......Page 339
Riemann Surfaces......Page 345
Surfaces for Related Functions......Page 348
Preservation of Angles......Page 353
Scale Factors......Page 356
Local Inverses......Page 358
Harmonic Conjugates......Page 361
Transformations of Harmonic Functions......Page 363
Transformations of Boundary Conditions......Page 365
Steady Temperatures......Page 371
Steady Temperatures in a Half Plane......Page 373
A Related Problem......Page 375
Temperatures in a Quadrant......Page 378
Electrostatic Potential......Page 383
Potential in a Cylindrical Space......Page 384
Two-Dimensional Fluid Flow......Page 389
The Stream Function......Page 391
Flows Around a Corner and Around a Cylinder......Page 393
Mapping the Real Axis onto a Polygon......Page 401
Schwarz-Christoffel Transformation......Page 403
Triangles and Rectangles......Page 407
Degenerate Polygons......Page 411
Fluid Flow in a Channel Through a Slit......Page 416
Flow in a Channel with an Offset......Page 418
Electrostatic Potential about an Edge of a Conducting Plate......Page 421
Poisson Integral Formula......Page 427
Dirichlet Problem for a Disk......Page 429
Related Boundary Value Problems......Page 433
Schwarz Integral Formula......Page 437
Dirichlet Problem for a Half Plane......Page 439
Neumann Problems......Page 443
Bibliography......Page 447
Table of Transformations of Regions......Page 451
Index......Page 461