Using Cauchy's residue theorem, evaluate the contour integral ∮_C [z^2 / (z - 2)] dz counterclockwise around the circle C: |z| = 3.
Answer and explanation
Correct answer: D. 8π i
- Step 1: The integrand has a simple pole at z = 2.
- Step 2: The pole z = 2 lies inside the contour |z| = 3 since |2| < 3.
- Step 3: Residue at z = 2: Res(f, 2) = lim_(z -> 2) (z - 2) * [z^2 / (z - 2)] = 2^2 = 4.
- Step 4: By the residue theorem: ∮_C f(z) dz = 2π i Res(f, 2) = 2π i 4 = 8π i.
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