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GATE 2023 EC – Question 14

Engineering Mathematics · Complex Analysis · 1 mark · Multiple choice

The value of the contour integral $\oint_C\left(\dfrac{z+2}{z^2+2z+2}\right)dz$, where the contour $C$ is $\left\{z:\left|z+1-\frac32j\right|=1\right\}$, taken in the counter clockwise direction, is

  1. $-\pi(1+j)$
  2. $\pi(1+j)$
  3. $\pi(1-j)$
  4. $-\pi(1-j)$

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Correct answer: (B) $\pi(1+j)$

Explanation

The poles are $z=-1\pm j$. The contour is a unit circle centred at $-1+1.5j$. The pole $-1+j$ is at distance 0.5 from the centre (inside); $-1-j$ is outside. The residue at $z=-1+j$ is $\dfrac{z+2}{z-(-1-j)}\Big|_{z=-1+j}=\dfrac{1+j}{2j}=\dfrac{1-j}{2}$. The integral is $2\pi j\cdot\dfrac{1-j}{2}=\pi j(1-j)=\pi(1+j)$.