= Solution
Put $z=1+\zeta$. Expanding the <holomorphic function> near its <critical point> gives
$$
f(1+\zeta)=2+\zeta^2-\zeta^3+O(\zeta^4).
$$
For a small ray $\zeta=re^{i\theta}$, the leading image displacement is $r^2e^{2i\theta}$. Thus \b[oriented ray angles are doubled modulo $2\pi$] 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 $\pi$. The <map> is not locally one-to-one there.
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