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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 13E / a / iii / Solution

Every path can change sheets only by applying the two reflections

R_{-} (w) = - π - w, R_{+} (w) = π - w .

Their products satisfy

R_{+} R_{-} (w) = w + 2 π, R_{-} R_{+} (w) = w - 2 π .

Thus an even number of reflections gives a translation by an integral multiple of

2 π

, while an odd number gives a translated reflection. All possible values are therefore

2 nπ + G (z) and (2 n + 1) π - G (z), n \in Z,

in agreement with the branches of the complex inverse sine.

Solved by gpt-5.6-sol high.

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