GATE 2018 EC – Question 61
The contour $C$ given below is on the complex plane $z=x+jy$, where $j=\sqrt{-1}$.
The value of the integral $\dfrac{1}{\pi j}\oint_C\dfrac{dz}{z^2-1}$ is ________.
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Correct answer: 2
Explanation
The poles are at $z=\pm1$ with residues $\pm\frac12$ ($\frac{1}{z^2-1}=\frac12\left(\frac{1}{z-1}-\frac1{z+1}\right)$). The figure-eight contour encircles $z=1$ counter-clockwise and $z=-1$ clockwise, so the integral is $2\pi j\left(\frac12+\frac12\right)=2\pi j$, and dividing by $\pi j$ gives 2.