 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2004/paper-2/3/c/i/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

On the unit circle use

z = e^{i θ}

with

0 < θ < 2 π

. The specified branch has

Log z = i θ

there and

d z = i e^{i θ} d θ

. Therefore

I = - \int_{0}^{2 π} θ e^{i θ} d θ = - [(1 - i θ) e^{i θ}]_{0}^{2 π} = 2 πi .

This is the circle integral of a logarithm with a radial branch cut, using the one-sided values at the cut crossing. The logarithm is discontinuous at that crossing, so the Cauchy integral theorem for a function holomorphic around a closed contour does not apply.

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