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> Using residue theory, show that $$\oint\_C\frac{e^{z/2}}{1+e^z}dz = 2\pi$$
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I've been attempted this problem using residue theory and the Cauchy integral formula that over a closed contour containing a pole (root at $\pi i$),

We should be able to confirm $2\pi$ using the equation $f(i\pi)\cdot 2\pi i$, but when I do this I am getting $-2\pi$, can anyone assist me?