= Angle multiplication at a holomorphic critical point
{title2=$f(z)-f(z_0)=a(z-z_0)^m+O((z-z_0)^{m+1})$}
If a nonconstant <holomorphic function> has its first nonzero local Taylor coefficient at order $m$, small rays from $z_0$ have image arguments $\arg a+m\arg(z-z_0)$ to leading order. Their oriented angles are therefore multiplied by $m$ modulo $2\pi$. The case $m=1$ is locally <conformal>; $m>1$ gives a <critical point>, and the local inverse has $m$ branches. For the <Joukowski map> at $z_0=1$, $f(1+\zeta)=2+\zeta^2+O(\zeta^3)$, so $m=2$.
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