C is the closed path z =3. Evaluate Cz2-z+4jz+2j\,dz.

GATE 2014 EC (EC1) – Question 13

Engineering Mathematics · Complex Analysis · 1 mark · Multiple choice

C is the closed path $|z|=3$. Evaluate $\oint_C\frac{z^2-z+4j}{z+2j}\,dz$.

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

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Correct answer: (C) $-4\pi(3+2j)$

Explanation

For the conventional positive orientation, the pole z=−2j lies inside |z|=3. Its residue is z²−z+4j at−2j, namely−4+6j. The residue theorem gives2πj(−4+6j)=−12π−8πj=**−4π(3+2j),C**. Reversing the contour orientation would reverse the sign.