Complex variables

Preface
......Page 4
Hints to the reader
......Page 6
Contents
......Page 10
1.1 Real numbers......Page 16
1.3 Complex numbers as marks in a plane......Page 18
1.5 Addition and subtraction......Page 21
1.6 Multiplication and division......Page 23
1.7 Summary and notation......Page 27
1.8 Conjugate numbers......Page 30
1.9 Vectorial operations......Page 32
1.10 Limits......Page 35
1.11 Additional examples and comments on Chapter one......Page 38
2.1 Extension to the complex domain......Page 50
2.2 Exponential function......Page 51
2.3 Trigonometric functions......Page 53
2.4 Consequences of Euler's theorem......Page 56
2.5 Further applications of Euler's theorem......Page 58
2.6 Logarithms......Page 61
2.7 Powers......Page 64
2.8 Inverse trigonometric functions......Page 67
2.9 General remarks......Page 68
2.10 Complex function of a real variable: kinematic representation......Page 70
2.11 Real functions of a complex variable: graphical representation......Page 72
2.12 Complex functions of a complex variable: graphical representation on two planes......Page 74
2.13. Complex functions of a complex variable: physical representation in one plane......Page 76
2.14 Additional examples and comments on Chapter two......Page 78
3.1 Derivatives......Page 90
3.2 Rules for differentiation......Page 92
3.3 Analytic condition for differentiability: the Cauchy-Riemann equations......Page 95
3.4 Graphical interpretation of differentiability: conformal mapping......Page 100
3.5 Physical interpretation of differentiability: sourceless and irrotational vector-fields......Page 103
3.6 Divergence and curl......Page 106
3.7 Laplace's equation......Page 110
3.8 Analytic functions......Page 112
3.9 Summary and outlook......Page 113
3.10 Additional examples and comments on Chapter three......Page 114
4.1 The stereographic or Ptolemy projection......Page 128
4.2 Properties of the stereographic projection......Page 132
4.3 The bilinear transformation......Page 135
4.4 Properties of the bilinear transformation......Page 137
4.5 The transformation w = z^2......Page 143
4.6 The transformation w = e^z......Page 144
4.7 The Mercator map......Page 146
4.8 Additional examples and comments on Chapter four......Page 147
5.1 Work and flux......Page 158
5.2 The main theorem......Page 161
5.3 Complex line integrals......Page 162
5.4 Rules for integration......Page 167
5.5 The divergence theorem......Page 170
5.6 A more formal proof of Cauchy's theorem......Page 172
5.7 Other forms of Cauchy's theorem......Page 173
5.8 The indefinite integral in the complex domain......Page 177
5.9 Geometric language......Page 182
5.10 Additional examples and comments on Chapter five......Page 184
6.1 Cauchy's integral formula......Page 192
6.2 A first application to the evaluation of definite integrals......Page 195
6.3 Some consequences of the Cauchy formula: higher derivatives......Page 199
6.4 More consequences of the Cauchy formula: the principle of maximum modulus......Page 202
6.5 Taylor's theorem, MacLaurin's theorem......Page 203
6.6 Laurent's theorem......Page 210
6.7 Singularities of analytic functions......Page 217
6.8 The residue theorem......Page 221
6.9 Computation of residues......Page 223
6.10 Evaluation of definite integrals......Page 225
6.11 Additional examples and comments on Chapter six......Page 234
7.1 Analytic continuation......Page 246
7.2 The gamma function......Page 250
7.3 Schwarz reflection principle......Page 255
7.4 The general mapping problem: Riemann's mapping theorem......Page 258
7.5 The Schwarz-Christoffel mapping......Page 260
7.6 A discussion of the Schwarz-Christoffel formula......Page 266
7.7 Degenerate polygons......Page 270
7.8 Additional examples and comments on Chapter seven......Page 275
8.1 The equations of hydrodynamics......Page 280
8.2 The complex potential......Page 282
8.3 Flow in channels: sources, sinks, and dipoles......Page 285
8.4 Flow in channels: conformal mapping......Page 287
8.5 Flows past fixed bodies......Page 293
8.6 Flows with free boundaries......Page 297
9.1 Asymptotic series......Page 306
9.2 Notation and definitions......Page 309
9.3 Manipulating asymptotic series......Page 311
9.4 Laplace's asymptotic formula......Page 317
9.5 Perron's extension of Laplace's formula......Page 322
9.6 The saddle-point method......Page 329
9.10 Additional examples and comments on Chapter nine......Page 337
Index......Page 340