Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 13B / b / i

Table of contents
- Solution i

Solution

Write

z = r e^{i θ}

with

r > 0

and

0 \leq θ < 2 π

, and write

w = u + i v

. Since

e^{w} = e^{u} e^{i v} = r e^{i θ},

we have

u = lo g r, v = θ + 2 πn, n \in Z .

Thus the inverse of the exponential is the infinitely valued complex logarithm

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

For the angle convention in the question, its principal value is

Log z = lo g r + i θ, 0 \leq θ < 2 π .

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