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latentcorpse

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from [itex]\delta[/itex] to R along the positive real axis then around the semi circle of radius -R on the upper half plane to -R on the negative real axis then along the negative real axis to [itex]-\delta[/itex] then around the semi circle of radius [itex]\delta[/itex] in the upper half plane and back to [itex]\delta[/itex] on the positive real axis.

If [itex]f(z)=\frac{1-e^{iz}}{z^2}[/itex], explain why [itex]\int_{\Gamma} f(z) dz=0[/itex]?

I was thinking of finding a domain,U, for f that was also a semi-annulus in the upper half plane but with outer radius much greater than R (say 1000R) and inner radius infinitesimaly small. Then U is star shaped if we pick the star centre at (0,1000R). [itex]f:U \rightarrow \mathbb{C}[/itex] will be holomorphic and so we can use Cauchy's Theorem to give the answer - but it doesn't seem like a very rigorous definition of U.

can anyone advise me?