Project ForgeEngineering Mathematics
Calculus
GATE 2022
Expert
The value of the integral ∮[6z/(2z⁴−3z³+7z²−3z+5)] dz evaluated over a counter-clockwise circular contour in the complex plane enclosing only the pole z=i, where i is the imaginary unit, is
- (−1+i)π
- B(1+i)π
- C2(1−i)π
- D(2+i)π
Explanation
D(z)=2z⁴−3z³+7z²−3z+5 vanishes at z=i: 2(1)−3(−i)+7(−1)−3i+5 = (2−7+5)+(3i−3i) = 0. For a simple pole, Res = N(z)/D'(z) at z=i, where D'(z)=8z³−9z²+14z−3. At z=i: D'(i)=8(−i)−9(−1)+14i−3 = 6+6i. Res = 6i/(6+6i) = i/(1+i) = (1+i)/2. By the residue theorem, the integral = 2πi×Res = 2πi(1+i)/2 = πi+πi² = π(−1+i) — option (A).
Source: GATE ME 2022 Session 1 (ME-1) (2022)
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