Project ForgeEngineering Mathematics
Calculus
GATE 2022
Hard
Given z=x+iy, i=√−1. C is a circle of radius 2 with the centre at the origin. If the contour C is traversed anticlockwise, then the value of the integral (1/2π)∫_C [1/((z−i)(z+4i))] dz is _______ (round off to one decimal place).
0.2(± 0.02 accepted)
Explanation
The integrand has simple poles at z=i (|i|=1, inside |z|=2) and z=−4i (|−4i|=4, outside). Only z=i contributes. Residue at z=i: 1/(z+4i) evaluated at z=i = 1/(5i) = −i/5. By the residue theorem, ∫_C f(z)dz = 2πi×(−i/5) = 2π/5. Dividing by 2π gives 1/5 = 0.2.
Source: GATE ME 2022 Session 2 (ME-2) (2022)
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