Mathematical Tripos: IA Algebra & Geometry (Part I) Contents 0 Introduction i 0.1 Schedule . i 0.2 Lectures . ii 0.3 Printed Notes . ii 0.4 Example Sheets . iii 0.5 Acknowledgements. iii 0.6 Revision. iv 1 Complex Numbers 1 1.0 Why Study This? . 1 1.1 Introduction . 1 1.1.1 Real numbers . 1 1.1.2 The general solution of a quadratic equation . 1 1.1.3 Complex numbers . 1 1.2 Algebraic Manipulation of Complex Numbers . 2 1.3 Functions of Complex Numbers . 2 1.3.1 Complex conjugate . 2 1.3.2 Modulus . 3 1.4 The Argand Diagram . 3 1.5 Polar (Modulus/Argument) Representation . 5 1.5.1 Geometric interpretation of multiplication . 5 1.6 The Exponential Function . 6 1.6.1 The real exponential function . 6 1.6.2 The complex exponential function . 7 1.6.3 The complex trigonometric functions . 7 1.6.4 Relation to modulus/argument form . 8 This is a supervisor’s copy of the notes. It is not to be distributed to students. 1.6.5 Modulus/argument expression for 1 . 8 1.7 RootsofUnity .............................................. 8 1.8 De Moivre’s Theorem . 9 1.9 Logarithms and Complex Powers . 10 1.9.1 Complex powers . 10 1.10 Lines and Circles in the Complex Plane . 11 1.10.1 Lines . 11 1.10.2 Circles . 11 1.11 M¨obius Transformations . 11 1.11.1 Composition . 12 Mathematical Tripos: IA Algebra & Geometry (Part I) a c
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