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:

i - 2i /(1+3exp(iθ)
= i - 2i /(3exp(iθ+1)
= i (3exp(iθ+1) /(3exp(iθ+1) - 2i /(3exp(iθ+1)
= (3i*exp(iθ +i -2i) /(3exp(iθ+1)
= (3i*exp(iθ -i) /(3exp(iθ+1)
?

Also...

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.)

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.

Also, Wolfram Alpha is partly correct, but look up complex logarithms! Some of them have infinitely many solutions.

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.