Past exam of the mathematics course of the University of Cambridge / 2020 / ia / Paper 1 / 1C / v

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- Solution v

Solution

All values of the complex logarithm are

w = lo g ∣ z ∣ + i (ar g z + 2 πn), n \in Z .

Because

x > 0

, one may take the complex argument

ar g z = arctan (y / x)

. Consequently

Re w = \frac{1}{2} lo g (x^{2} + y^{2}), Im w = arctan \frac{y}{x} + 2 πn, q u a d n \in Z .

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