This unusual and lively textbook offers a clear and intuitive approach to the classical and beautiful theory of complex variables. With very little dependence on advanced concepts from several-variable calculus and topology, the text focuses on the authentic complex-variable ideas and techniques. Accessible to students at their early stages of mathematical study, this full first year course in complex analysis offers new and interesting motivations for classical results and introduces related topics stressing motivation and technique. Numerous illustrations, examples, and now 300 exercises, enrich the text. Students who master this textbook will emerge with an excellent grounding in complex analysis, and a solid understanding of its wide applicability.
Preface to the Third Edition
Preface to the Second Edition
1 The Complex Numbers 1
Introduction 1
1.1 The Field of Complex Numbers 1
1.2 The Complex Plane 4
1.3 The Solution of the Cubic Equation 9
1.4 Topological Aspects of the Complex Plane 12
1.5 Stereographic Projection; The Point at Infinity 16
Exercises 18
2 Functions of the Complex Variable z 21
Introduction 21
2.1 Analytic Polynomials 21
2.2 Power Series 25
2.3 Differentiability and Uniqueness of Power Series 28
Exercises 32
3 Analytic Functions 35
3.1 Analyticity and the Cauchy-Riemann Equations 35
3.2 The Functions ez, sin z, cos z 40
Exercises 41
4 Line Integrals and Entire Functions 45
Introduction 45
4.1 Properties of the Line Integral 45
4.2 The Closed Curve Theorem for Entire Functions 52
Exercises 56
5 Properties of Entire Functions 59
5.1 The Cauchy Integral Formula and Taylor Expansion for Entire Functions
