GATE 2020 EE – Question 15
The value of the following complex integral, with $C$ representing the unit circle centered at origin in the counterclockwise sense, is:
$$\oint_C\frac{z^2+1}{z^2-2z}\,dz$$
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Show answer and explanation
Correct answer: (C) $-\pi i$
Explanation
The poles are at $z=0$ and $z=2$, and only $z=0$ lies inside the unit circle. The residue there is $\frac{0+1}{0-2}=-\frac12$, so the integral is $2\pi i\left(-\frac12\right)=-\pi i$.