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GATE 2019 EE – Question 37

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

The closed loop line integral

$$\oint_{|z|=5}\frac{z^3+z^2+8}{z+2}\,dz$$

evaluated counter-clockwise, is

  1. $+8j\pi$
  2. $-8j\pi$
  3. $-4j\pi$
  4. $+4j\pi$

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Show answer and explanation

Correct answer: (A) $+8j\pi$

Explanation

The pole $z=-2$ lies inside $|z|=5$. The residue is the numerator at $z=-2$: $(-8)+4+8=4$. So the integral is $2\pi j\times4=8j\pi$.