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GATE 2018 EE – Question 45

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

If $C$ is a circle $|z|=4$ and $f(z)=\dfrac{z^2}{(z^2-3z+2)^2}$, then $\oint_Cf(z)\,dz$ is

  1. 1
  2. 0
  3. $-1$
  4. $-2$

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Correct answer: (B) 0

Explanation

All the poles ($z=1$ and $z=2$, both double) lie inside $|z|=4$. For large $|z|$, $f(z)\sim\frac{1}{z^2}$, so the residue at infinity is zero and the sum of all the residues is zero. The integral is $0$.