[...]... two, and I thought \Why not? I can handle it" Then one day someone said \Hey, man, that's kidstu - try some calculus" so I tried some dierentials then I went on to integrals even the occasional volume of revolution but I can stop any time I want to I know I 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... 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 at it for a while... 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 some of the mystery, but will be... years ago wanted to ban them, is now trying to keep the confusion It's a funny old world, and no 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... periodic waveforms Also, the functions which arise between them are very useful for talking about solutions of some Partial Dierential Equations So don't look down on Pure Mathematicians for wanting to have things clean and cool It pays o in very unexpected ways The Universe also seems to like things clean and cool And most supersmart people, such as Gauss, like nding out about Electricity and Magnetism,... 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 numbers form a plane is probably the most important thing to know... matrices transform points in the plane To be denite, take   x y for a point, or if you prefer vector in R 2 and let  a c b d  be a 2  2 matrix Placing the matrix to the left of the vector:  a c b d  x y  1.3 WHAT ARE COMPLEX NUMBERS? 13 and doing matrix multiplication gives a new vector:   ax + cy bx + dy This is all old stu which you ought to be good at by now1 Now I am going to look at... Mathematician, who was thinking much the same but wasn't as forthright `And why is it impossible?' asked the Engineer belligerently `Because,' said the Mathematician, thinking quickly, `In order to get to her, you will rst have to get halfway And then you will have to get half of the rest of the distance, and then half of that And so on; in short, you can never get there in a nite number of moves.' The Engineer... 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 conservative folk for a century or so It turns out that they are very cool things to have for `measuring'... in the form of a story: A (male) Mathematician and a (male) Engineer who knew each other, had both been invited to the same party They were standing at one corner of the room and eyeing a particularly attractive girl in the opposite corner `Wow, she looks pretty good,' said the Engineer `I think I'll go over there and try my luck.' `Impossible, and out of the question!' said the Mathematician, who