GATE 2017 EC (EC2) – Question 37
An integral over a counterclockwise circle C is $I=\oint_C[(z^2-1)/(z^2+1)]e^z\,dz$. If C is defined by $|z|=3$, its value is
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Correct answer: (D) $-4\pi i\sin(1)$
Explanation
The simple poles at z=i and z=−i both lie inside the circle. Their residues are $ie^i$ and $-ie^{-i}$, respectively, using the derivative $2z$ of the denominator.
Their sum is $i(e^i-e^{-i})=-2\sin1$. The contour is counterclockwise, so the residue theorem multiplies this by $2\pi i$.
Thus $I=-4\pi i\sin1$, which is **D**.