Home › Previous year questions › GATE 2020 PH › Question 32GATE 2020 PH – Question 32 Mathematical Physics · Complex analysis: Cauchy-Riemann conditions, Cauchy's theorem, singularities, residue theorem and applications · 1 mark · Numerical answer
For a counterclockwise contour $C:|z|=1$, evaluate $$\frac1{2\pi i}\oint_C\left(z-\frac2z+\frac3{z^2}\right)dz.$$
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Show answer and explanation Correct answer: -2 to -2
Explanation Only the coefficient of $z^{-1}$ contributes to this contour integral. The analytic term z has zero integral, and $3/z^2$ has zero residue. The residue at the origin is −2, so the integral is $2\pi i(-2)$. Dividing by $2\pi i$ gives −2.
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