GATE 2017 EE (EE2) – Question 37
Engineering Mathematics · Complex Variables: Analytic functions, Cauchy integral formula, Taylor series, Laurent series, Residue theorem · 2 marks · Multiple choice
The counterclockwise contour integral $\oint_{|z|=3}(z^3-2z+3)/(z-2)\,dz$ equals
- $-18\pi i$
- $0$
- $14\pi i$
- $48\pi i$
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Show answer and explanation
Correct answer: (C) $14\pi i$
Explanation
The only pole is z=2, inside the circle of radius 3. Its residue is the numerator evaluated there: $2^3-2(2)+3=7$.
The residue theorem gives the integral $2\pi i(7)=$ **14πi, C**. Counterclockwise orientation gives the positive sign.
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