Angle multiplication at a holomorphic critical point ($f(z)-f(z_0)=a(z-z_0{)}^m+O((z-z_0{)}^{m+1})$)

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 $m=2$.