 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ia/paper-1/1c/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

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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