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Observe that: $\log(\frac{1+z}{1-z}) = -2\int\frac{dz}{1-z^2}$. (Not precisely true but read on)

Supposedly this function is analytic on the domain $\mathbb{C}-[-1,1]$, despite the fact that it's not decomposable into two analytic functions in the usual way, and thus shouldn't be thought of as a composition of functions.

Now for the closed curve $\alpha (t)=2e^{it}$ for $0\leq t\leq 2\pi$, $\int_{\alpha}\frac{-2dz}{1-z^2} = \int_{\alpha}\frac{-2dz}{(1-z)(1+z)} = 2\pi i$ by the Residue Theorem. Yet by the Cauchy Integral Theorem $\int\frac{-2dz}{1-z^2}$ having a primitive on $\mathbb{C}-[-1,1]$ implies that $\int_{\alpha}\frac{-2dz}{1-z^2} = 0$ for any closed curve on $\mathbb{C}-[-1,1]$.

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    Actually I believe $log(\frac{1-z}{1+z})$ IS a primitive of my function on the given domain, however it is not the composition of two analytic functions in the usual way. Observe that $\int\frac{-2dz}{1-z^2}$ is zero along any piecewise smooth closed curve on our domain, hence it has a primitive in said domain. We define this primitive by $log(\frac{1-z}{1+z}) := \int\frac{-2dz}{1-z^2}$2012-05-09

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The residue is $0$, since $ \frac1{(1-z)(1+z)} = \frac12\left(\frac1{z+1}-\frac1{z-1}\right). $