Project ForgeEngineering Mathematics
Calculus
GATE 2018
Hard
Let z be a complex variable. For a counter-clockwise integration around a unit circle C, centred at origin, ∮C [1/(5z−4)] dz = Aπi, the value of A is
- 2/5
- B1/2
- C2
- D4/5
Explanation
The integrand 1/(5z−4) = (1/5)/(z−4/5) has a simple pole at z=4/5=0.8, which lies inside the unit circle. By the residue theorem: ∮C f(z)dz = 2πi×Res(f, 4/5) = 2πi×(1/5) = (2/5)πi. So A=2/5.
Source: GATE ME 2018 Session 2 (Set 2) (2018)
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