 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ii/paper-2/13b/b/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

On the cut plane choose

0 < θ < 2 π

. For

z = x + i y

u (x, y) = \frac{1}{2} lo g (x^{2} + y^{2}), v (x, y) = ar g (x + i y) .

u_{x} = \frac{x}{x ^{2} + y ^{2}} = v_{y}, u_{y} = \frac{y}{x ^{2} + y ^{2}} = - v_{x} .

The Cauchy-Riemann equations hold, so this branch is holomorphic. Its derivative is

\frac{d w}{d z} = u_{x} + i v_{x} = \frac{x - i y}{x ^{2} + y ^{2}} = \frac{1}{z} .

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