Holomorphic 1/z

Wait.



∮1/z dz = ∫ [3/2 ie^(iθ - 1/2 ie^(-iθ] / [3/2 e^(iθ + 1/2 e^(-iθ] dθ, [0,2π]
= i ∫ [3 e^(iθ - e^(-iθ] / [3 e^(iθ + e^(-iθ] dθ, [0,2π]

When you reached this step, how did you go to the next one? It seems like you somehow dropped the exp(-iθ terms. Going backwards from your work:

And it seems like I forgot at 2 in the exponent, but the result of the integral did not change. I'm a lucky guy!

I don't have a professor. I'm not taking a class for this. And I thought z can map to infinitely many values, or so it seems from the Riemann surfaces resembling overlapping spirals.

That's a shame. Professors love interesting and curious students, and you do seem like one.

Well, yeah for a single theta, there can be several points. You should have some function you integrate over to distinguish the points, though. Usually, it is "time"-based.

I looked it up. I'm wondering: how do I change the base of a complex logarithm? The formula log(a) {base b} = log(b) /log(a) doesn't seem to be working.

It works fine, but you need to be careful due to branching: http://books.google.com/books?id=FUWPyHM...tAPZ5-XjCw

Noah