 Solution

ID: past-exam-of-the-metric-spaces-and-complex-analysis-course-of-the-university-of-oxford/2014/3/b/ii/solution

P (z) = z L (z) - z

, since the product rule and

L^{'} = 1/ z

give

P^{'} = L

\int_{[1, i]} L (z) d z = P (i) - P (1) = (i \frac{πi}{2} - i) - (- 1) .

Hence

\int_{[1, i]} L (z) d z = 1 - \frac{π}{2} - i .

The value

L (i) = πi /2

is fixed by the chosen branch.

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