I’m done with the exams!

No more exams! I had my last exam last Tuesday: Calculus. It went fantastic — I got the grade 13 which is the top grade in the Danish school-system!

The exam was an aural exam with 30 minutes preparation. I was so lucky that I got “The Complex Exponential Function” as my question. That was probably the easiest question among the 15 that I had to choose from. The other questions were about things like the length of a curve in Rn, Fourier Series and so on — much harder questions. I don’t think I would have gotten a 13 if I had gotten one of the other questions… perhaps 10 or 11 instead (there’s no 12 in the Danish grade system).

But that doesn’t matter: I got the question and after waiting 30 minutes I told them everything they wanted to know about it. They then asked me some questions about other things, such a Fourier Series. That also went well — It’s not that I bad at handling Fourier Series, it’s just that most proofs are terribly long and boring. In the end the censor got curious: he wanted to know if the Exponential Function had an inverse function, e.g. he wanted me to talk about the complex logarithm. We haven’t seen this in our books, but I did manage to find some of the formula for z given w = ez:

*e**x + iy* = *u + iv ⇒*
*x + iy* = ln(|*u + iv*|) + *i* arg(*u + iv*)

The formula is found by taking advantage of, that |ez| = |w| and arg(ez) = arg(w). Here we have that |ez| = ex and that arg(ez) = y and the result follows easily from that.

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