Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 1 / 1C / a

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- Solution a

Solution

By complex exponentiation, a value of

z^{i}

has the form

exp (i (lo g ∣ z ∣ + i (ar g z + 2 πk))) .

For this to equal one, its exponent must be

2 πin

for some

n \in Z

. Hence a logarithm of

z

must equal

2 πn

, so

z

is positive real and

z = e^{2 πn}, n \in Z .

With the principal branch this is the same list; in the multivalued convention these are precisely the

z

for which one value of

z^{i}

equals one.

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