 Solution

ID: past-exam-of-the-algebra-analysis-and-differential-equations-1-course-of-the-university-of-oxford/2013/analysis/5/b/solution

Choose the continuous argument with values in

(- π /2, 3 π /2)

Log z = lo g ∣ z ∣ + i Arg z, - \frac{π}{2} < Arg z < \frac{3 π}{2} .

Its branch cut is the nonpositive imaginary axis, including zero, exactly the removed ray. Locally away from that ray, the Cauchy-Riemann equations or the local inverse of the exponential show that it is a holomorphic logarithm, with derivative

1/ z

and

e^{Log z} = z

. The chosen interval fixes the additive ambiguity by integer multiples of

2 πi

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