 Solution

ID: 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

Put

z = 1 + ζ

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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