How do you prepare? Well, a very important part of my preparation was that I talked to students who took that examination from Kobe on earlier occasions. And then I also talk- read a very nice book by Bieberbach on the theory of functions, which I greatly enjoyed. And, at the proper day, I went to see Kobe. And he started by asking me one or two trivial questions, not much more difficult in the theory of functions, how much is two and two? That, I knew. Then, Kobe says- Now, talk about whatever you like. And for that, I was prepared. Because I found out by talking with others who took the examination, that he almost always asked that question. I know precisely what I talked about. The theory of functions really works in the complex plane, where a number is given. For instance, like 5 + 7 x i, where i is the square root of -1, which in a common sense of the word does not exist, and is defined like i- as i. And what a function in the complex plane does is to transform the complex plane, the points in the complex plane, from one configuration to the other, all right? I talk of a function, f(x) . Then x has a position in the complex plane, and f(x) another position. And in this way, a region in the complex plane, by the function, is transformed into another region. Now, I set out to prove a theorem that if you have any simply connected region in the complex plane, essentially one without holes in it, then you can find the function - an analytic function - a function that satisfies certain simple criteria, which tram-transforms that comple- complex system without a hole in it. Transforms it into the interior of the unit- unit circle. Beautiful proof. I talked and I talked and I talked, for much more than half an hour. And Kobe sat there and listened. And he asked- Where did you get that proof? I said- I got it from the book of Bieberbach, the common textbook. - Ah, you got it from Bieberbach. I suppose you know that I was the first one to prove that theorem? I said- Yes sir, I know that. I did not add- and that's why I talked about it. I won - I got my good grade.
