## 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
**R**^{n},
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 = e*^{z}:

*e*^{*x + iy*} | = | *u + iv ⇒* |

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

The formula is found by taking advantage of, that *|e ^{z}| = |w|*
and arg(

*e*) = arg(

^{z}*w*). Here we have that

*|e*and that arg(

^{z}| = e^{x}*e*) =

^{z}*y*and the result follows easily from that.

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