Explores the interrelations between real and complex numbers by adopting both generalization and specialization methods to move between them, while simultaneously examining their analytic and geometric characteristics Engaging exposition with discussions, remarks, questions, and exercises to motivate understanding and critical thinking skills Encludes numerous examples and applications relevant to science and engineering students
Author(s): S. Ponnusamy, Herb Silverman
Edition: 1
Publisher: Birkhäuser Boston
Year: 2006
Language: English
Pages: 524
Tags: Математика;Комплексное исчисление;
Cover page......Page 1
Title page......Page 3
Preface......Page 6
Contents......Page 9
1.1 The Complex Field......Page 12
1.2 Rectangular Representation......Page 16
1.3 Polar Representation......Page 26
2.1 Point Sets in the Plane......Page 36
2.2 Sequences......Page 43
2.3 Compactness......Page 50
2.4 Stereographic Projection......Page 55
2.5 Continuity......Page 59
3.1 Basic Mappings......Page 72
3.2 Linear Fractional Transformations......Page 77
3.3 Other Mappings......Page 96
4.1 The Exponential Function......Page 102
4.2 Mapping Properties......Page 111
4.3 The Logarithmic Function......Page 119
4.4 Complex Exponents......Page 125
5.1 Cauchy–Riemann Equation......Page 132
5.2 Analyticity......Page 141
5.3 Harmonic Functions......Page 152
6.1 Sequences Revisited......Page 164
6.2 Uniform Convergence......Page 175
6.3 Maclaurin and Taylor Series......Page 184
6.4 Operations on Power Series......Page 197
7.1 Curves......Page 206
7.2 Parameterizations......Page 218
7.3 Line Integrals......Page 228
7.4 Cauchy’s Theorem......Page 237
8.1 Cauchy’s Integral Formula......Page 254
8.2 Cauchy’s Inequality and Applications......Page 274
8.3 Maximum Modulus Theorem......Page 286
9.1 Laurent Series......Page 296
9.2 Classification of Singularities......Page 304
9.3 Evaluation of Real Integrals......Page 319
9.4 Argument Principle......Page 342
10.1 Comparison with Analytic Functions......Page 360
10.2 Poisson Integral Formula......Page 369
10.3 Positive Harmonic Functions......Page 382
11.1 Conformal Mappings......Page 390
11.2 Normal Families......Page 401
11.3 Riemann Mapping Theorem......Page 406
11.4 The Class S......Page 416
12.1 Infinite Products......Page 422
12.2 Weierstrass’ Product Theorem......Page 433
12.3 Mittag-Leffler Theorem......Page 448
13.1 Basic Concepts......Page 456
13.2 Special Functions......Page 469
References and Further Reading......Page 484
Index of Special Notations......Page 486
Index......Page 490
Hints for Selected Questions and Exercises......Page 496
