The integral (3z-5)/[(z-1)(z-2)]\,dz=4 i, where z=x+iy and i=-1. Which closed path Γ is correct?

GATE 2016 ME (ME2) – Question 37

Engineering Mathematics · Complex Variables: Analytic functions, Cauchy-Riemann equations, integral theorem, Taylor and Laurent series · 2 marks · Multiple choice

The integral $\oint_\Gamma(3z-5)/[(z-1)(z-2)]\,dz=4\pi i$, where $z=x+iy$ and $i=\sqrt{-1}$. Which closed path Γ is correct?

Diagram for GATE 2016 ME (ME2) question 37

Path A
Path B
Path C
Path D

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Show answer and explanation

Correct answer: (B) Path B

Explanation

Residues of(3z−5)/[(z−1)(z−2)] are2 at z1 and1 at z2. A positively oriented contour enclosing only z1 gives2πi·2=4πi. Graph **B** has that enclosure and orientation. Enclosing both poles would give6πi.