I'm learning some basic complex analysis and came across this integral $$\int_{-i}^{i} \frac{dz}{z}.$$
First of all, Wolfram can't calculate it, but it might be because he treats $i$ like a real parameter (BTW, is there a way to tell Wolfram how to calculate contour integrals along a specified contour?).
Since the function is analytic on any domain that doesn't contain the origin, the integral doesn't depend on path choice there. But there's a problem with the antiderivative, $\log z$, which is multi-valued, and I don't know exactly how to deal with that. So I'm not sure if I can use the fundamental thm. of calculus there. Evaluating by direct parametrization, for instance choosing an anti-clockwise circular path from $-i$ to $i$ gives the answer $i \pi$. But, unless I made an error, the same integral along the clockwise circular path gives the answer $-i \pi$...?
The excercise says to use Cauchy's integral formula, which gives me that ($C$ being a clockwise circular contour around the origin) $$2 \pi i = \oint _C \frac{dz}{z} $$
but this only gives me the same thing I got via parametrization... Is the excercise posed like this on purpose to get you thinking, or am I doing a mistake somewhere?