 Solution

ID: past-exam-of-the-metric-spaces-and-complex-analysis-course-of-the-university-of-oxford/2017/3/b/iv/solution

Along

z = e^{i θ}

with

0 < θ < 2 π

, the chosen branch has

L (z) - iπ = i (θ - π)

. Use part (iii) and the circular estimate in part (a):

\int_{0}^{1} f (x) d x \leq \frac{1}{2 π} \int_{0}^{2 π} ∣ f (e^{i θ}) ∣ ∣ θ - π ∣ d θ \leq \frac{1}{2} \int_{0}^{2 π} ∣ f (e^{i θ}) ∣ d θ .

∣ θ - π ∣ \leq π

. The single branch-cut point does not affect the integral. Keeping that weight gives the sharper intermediate bound.

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