I just want to know what I'm doing wrong here. So we have a singularity at $z=-i$ but this is inside the region of circle centered at $-i$ with radius 3. Hence by Cauchy Integral Formula we have $2\pi i \sin(-i)$ as our final result. Is this correct?
2026-04-09 09:25:46.1775726746
Calculate $\sin(z)/(z+i)$ using Cauchy Integral Formula on region $|z+i|=3$
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Your answer is correct, but it can be simplified further. Since $ \sin(-i) = -i\sinh(1)$, we have $2\pi i \sin(-i) = 2\pi \sinh(1)$.