GATE 2026 EE – Question 62
The magnitude of the contour integral
$$\oint_C\left(\frac{(z+1)^2}{(z-i)(z-2)}\right)dz$$
over the contour $C:\ |z-2-i|=3/2$ is ______.
(Round off to two decimal places)
Note: $z$ is a complex variable and $i=\sqrt{-1}$
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Show answer and explanation
Correct answer: 25.16 to 25.42
Explanation
The contour is a circle of radius 1.5 centred at $2+i$. The pole $z=2$ is at distance $|i|=1<1.5$ from the centre, so it is inside; the pole $z=i$ is at distance 2, outside. The residue at $z=2$ is $\dfrac{(2+1)^2}{2-i}=\dfrac{9}{2-i}$. The integral is $2\pi i\cdot\dfrac9{2-i}$, with magnitude $\dfrac{18\pi}{\sqrt5}=25.29$.