Project ForgeEngineering Mathematics
Complex Variables
GATE 2012
Hard
If i=√−1, the value of the integral ∮c [7z+i]/[z(z²+1)] dz, |z|<2, using the Cauchy residue theorem is
- A2πi
- 0
- C−6π
- D6π
Explanation
Poles at z=0,i,−i, all inside |z|<2. Res at z=0: i/((−i)(i))=i/1=i. Res at z=i: (8i)/(i·2i)=8i/(−2)=−4i. Res at z=−i: (−6i)/((−i)(−2i))=−6i/(−2)=3i. Sum of residues=i−4i+3i=0. Integral=2πi×0=0.
Source: GATE CH 2012 (2012)
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