Contents

Preface to the Second Edition ix

Preface to the First Edition xi

Chapter I. Complex Numbers 1

§I.1. Deﬁnition of C 2

§I.2. Field Axioms 2

§I.3. Embedding of R in C. The Imaginary Unit 3

§I.4. Geometric Representation 3

§I.5. Triangle Inequality 4

§I.6. Parallelogram Equality 5

§I.7. Plane Geometry via Complex Numbers 5

§I.8. C as a Metric Space 6

§I.9. Polar Form 6

§I.10. De Moivre’s Formula 7

§I.11. Roots 8

§I.12. Stereographic Projection 9

§I.13. Spherical Metric 10

§I.14. Extended Complex Plane 11

Chapter II. Complex Diﬀerentiation 13

§II.1. Deﬁnition of the Derivative 13

§II.2. Restatement in Terms of Linear Approximation 14

§II.3. Immediate Consequences 14

iii