GATE 2021 EE – Question 38

Engineering Mathematics · Complex Variables: Cauchy's integral theorem · 2 marks · Multiple choice

Let $(-1-j)$, $(3-j)$, $(3+j)$ and $(-1+j)$ be the vertices of a rectangle $C$ in the complex plane. Assuming that $C$ is traversed in counter-clockwise direction, the value of the contour integral $\oint_C\dfrac{dz}{z^2(z-4)}$ is

$j\pi/2$
0
$-j\pi/8$
$j\pi/16$

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Correct answer: (C) $-j\pi/8$

Explanation

The rectangle spans $-1\le x\le3$ and $-1\le y\le1$, so it encloses the double pole at $z=0$ but not the pole at $z=4$. The residue at $0$ is $\frac{d}{dz}\frac{1}{z-4}\Big|_{0}=-\frac1{16}$, so the integral is $2\pi j\left(-\frac1{16}\right)=-\frac{j\pi}{8}$.