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GATE 2023 BM – Question 48

Engineering Mathematics · Complex Variables · 2 marks · Multiple select

For f(z)=1/(z−1), which counterclockwise circular contours yield contour integral 2πj?

  1. $|z-2|=0.01$
  2. $|z-1|=0.1$
  3. $|z-3|=5$
  4. $|z|=2$

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Correct answer: (B) $|z-1|=0.1$; (C) $|z-3|=5$; (D) $|z|=2$

Explanation

The only singularity is z=1 with residue one. A circle gives 2πj precisely when it encloses this pole. A has centre two and radius .01 and excludes it. B encloses its own centre at the pole, C has distance two less than radius five, and D contains one inside radius two. Thus B,C,D.

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