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 Incoming links: Angle multiplication at a holomorphic critical point

Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2012 / S1 / 2 / b / Solution 2026-10-25

Put

z = 1 + ζ

. Expanding the holomorphic function near its critical point gives

f (1 + ζ) = 2 + ζ^{2} - ζ^{3} + O (ζ^{4}) .

For a small ray

ζ = r e^{i θ}

, the leading image displacement is

r^{2} e^{2 i θ}

. Thus oriented ray angles are doubled modulo $2 π$ at

z = 1

. This is angle multiplication at a holomorphic critical point of multiplicity two; an unoriented angle between full tangent lines must also be interpreted modulo

π

. The map is not locally one-to-one there.

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