I've had some issues with Cauchy's Thorem for a little while now. Namely, when does a closed contour integral nrcessarily equal 0.
The statement I have learned is:
If $f $ is analytic on a simply-connected domain, $E $, and $C $ is a piecewise smooth closed curve on $E $ then: $$\oint_C f(z)dz = 0$$
My trouble comes up with the simply-connected criteron, so I was hoping that someone could give me an answer to this:
If $f $ is analytic on a connected domain $E $ with a hole centered at $z_0$, and $C$ is a closed curve which contains $z_0$, but does not intersect it, can we still say that $$\oint_C f (z)dz = 0$$
Personally, I see no reason to think this is untrue. It seems to me that Cauchy's Theorem requires simply connected domains just so that you can say every closed curve vanishes. But I suspect that a closed contour integral should still equal $0$ if the integrand is analytic and the curve stays completely within the domain. Is this true, or am I looking at things incorrectly?
You are looking at things incorrectly. The integral $$\int_{\{z:|z|=1\}}\frac{1}{z}$$ does not vanish
(Aassume $f(z) =\frac{1}{z}$ to be defined on $\mathbb{C}\backslash \{0\}$).