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GATE 2018 EE – Question 23

Engineering Mathematics · Complex Variables: Cauchy's integral theorem · 1 mark · Multiple choice

The value of the integral $\oint_C\dfrac{z+1}{z^2-4}\,dz$ in counter clockwise direction around a circle $C$ of radius 1 with center at the point $z=-2$ is

  1. $\frac{\pi i}{2}$
  2. $2\pi i$
  3. $-\frac{\pi i}{2}$
  4. $-2\pi i$

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Show answer and explanation

Correct answer: (A) $\frac{\pi i}{2}$

Explanation

Only the pole at $z=-2$ is inside the circle. The residue is $\frac{z+1}{z-2}\Big|_{z=-2}=\frac{-1}{-4}=\frac14$, so the integral is $2\pi i\times\frac14=\frac{\pi i}2$.