 Solution

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

L (z) = lo g ∣ z ∣ + i Arg z, - π < Arg z < π .

The excluded nonpositive real ray allows a continuous choice of argument, and

L (1) = 0

. For

z = x + i y

, its real part is

u = \frac{1}{2} lo g (x^{2} + y^{2})

. Differentiating the chosen local argument gives

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

L^{'} (z) = u_{x} + i v_{x} = \frac{x - i y}{x ^{2} + y ^{2}} = \frac{1}{z} .

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