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I have been using a textbook which shows that ∫cos z/z^2 around the circle |z|=1 is zero by doing a Laurent expansion and finding the residue is zero.

I was under the impression that only analytic functions have a integral of zero around a closed surface. ( the Cauchy-Goursat Theorem ). The above function is not analytic at z=0 so can non-analytic functions have a closed integral of zero ?

Thanks