[...]... can OK, so I do the odd bit of complex analysis, but only a few times that stu can really screw your head up for days but I can handle it it's OK really I can stop any time I want ' ( tim@bierman.demon.co.uk (Tim Bierman)) Contents 1 Fundamentals 9 1.1 A Little History 9 1.2 Why Bother With Complex Numbers and Functions? 11 1.3 What are Complex Numbers? ... any complex numbers in your life In order to explain what they are I am going to do a bit of very easy linear algebra The reasons for this will become clear fairly quickly 1.3 What are Complex Numbers? Complex numbers are points in the plane, together with a rule telling you how to multiply them They are two-dimensional, whereas the Real numbers are one dimensional, they form a line The fact that complex. .. Transforms 66 o 2.7 The Exponential Function 69 2.7.1 Digression: Innite Series 70 2.7.2 Back to Real exp 73 2.7.3 Back to Complex exp and Complex ln 76 2.8 Other powers 81 2.9 Trigonometric Functions 82 3 C - Dierentiable Functions 89 3.1 Two sorts of Dierentiability ... Principle; Rouch's Theorem 168 e 6.7 Concluding Remarks 174 8 CONTENTS Chapter 1 Fundamentals 1.1 A Little History If Complex Numbers had been invented thirty years ago instead of over three hundred, they wouldn't have been called `Complex Numbers' at all They'd have been called `Planar Numbers', or `Two-dimensional Numbers' or something similar, and there would have been none... people, such as Gauss, like nding out about Electricity and Magnetism, working out how to handle 1.2 WHY BOTHER WITH COMPLEX NUMBERS AND FUNCTIONS?11 calculations of orbits of asteroids and doing Pure Mathematics In these notes, I am going to rewrite history and give you the story about Complex Numbers and Functions as if they had been developed for the applications we now know they have This will short-circuit... right? But some people do it and feel good about themselves because they've done it Well, what works for your heart and lungs also applies to your brain Exercising it will make you feel better And Complex Analysis is one of the tougher and meatier bits of Mathematics Tough minded people usually like it But like physical exercise, it hurts the rst time you do it, and to get the benets you have to keep... Numbers? 12 1.4 Some Soothing Exercises 18 1.5 Some Classical Jargon 22 1.6 The Geometry of Complex Numbers 26 1.7 Conclusions 29 2 Examples of Complex Functions 33 2.1 A Linear Map 34 2.2 The function w = z2 36 1 2.3 The Square Root: w = z 2 ... mistake Your text books often have an introductory chapter explaining a bit of the historical development, and you should read this in order to be educated, but it isn't in the exam 1.2 Why Bother With Complex Numbers and Functions? In mastering the material in this book, you are going to have to do a lot of work This will consist mainly of chewing a pencil or pen as you struggle to do some sums Maths... Engineering Lecturers about what happens in their courses Generally, the more modern and intricate the material, the more Mathematics it uses Communication Engineering and Power Transmission both use Complex Functions; Filtering Theory in particular needs it Control Theory uses the subject extensively Whatever you think about Mathematicians, your lecturers in Engineering are practical people who wouldn't... number of moves.' The Engineer gave a cheerful grin `Maybe so,' he replied, `But in a nite number of moves, I can get as close as I need to be for all practical purposes.' And he made his moves *** The Complex Numbers were invented for purely mathematical reasons, just like the Reals, and were intended to make things neat and tidy in solving equations They were regarded with deep suspicion by the more