Holomorphic 1/z

You guys are giving me too much credit, I don't know everything!

The second one is just some miscalculation, I assume, as I get it correctly:

Let


then



Now, I looked after the last integral for some time, but could not find it. I ended up using Wolfram Alpha which is not really a satisfying answer, but it should leave you content, I hope.

http://www.wolframalpha.com/input/?i=int...D0+to+2+pi

So the second one is correct. The third one is not that much harder:

Your idea is correct, but your use of complex logarithms does not satisfy me nor the mathematics. Look it up! Also, it seems that you try to go clockwise, instead of counterclockwise. And remember, z is a function, and you can only map from one value to another, not to 4 different values. This would not affect the result here, but your professor will kill you for that, and I am against murder! (In all seriousness though, it is smart to be rigorous, at least when defining functions, as you can more easily see where the errors happen.)



Now, the contour integral will look like this:



The first integral evaluated is this:



The remaining integrals return the same value (which should not be a surprise, as this figure is symmetric!), and is up to the reader to evaluate, as I am too lazy to do it. The value of the contour integral is thus 2 pi i.

Noah

 Spoiler

Another common error is to forget to centre the area you are working with. Try, for example, to integrate the normal circle but with the centre somewhere else, say, for example, at 3 or 7. Then