= Ramification of a power map of the Riemann sphere
{title2=$f(z)=z^d,\quad R_f=2d-2$}
For an integer $d\geq2$, extend $f(z)=z^d$ to a <holomorphic map> from the <Riemann sphere> to itself with $f(\infty)=\infty$. A nonzero finite <regular value> has $d$ distinct preimages, so the <degree of a holomorphic map> is $d$. At zero the <analytic ramification index> is $d$. At infinity, use <local coordinates> $u=1/z$ upstairs and $v=1/f(z)$ downstairs: the same <holomorphic map> becomes $v=u^d$, so $e_\infty=d$ too. Everywhere else the <derivative> $dz^{d-1}$ is nonzero.
\Image[/riemann-sphere-power-map-local-degree.png]
{title=Schematic of the sphere power map and its local degree: zero and infinity both have ramification index d}
There are two distinct <ramification points> and two distinct <branch values of a holomorphic map>, but
$$
R_f=(d-1)+(d-1)=2d-2,\qquad -2=d(-2)+2d-2.
$$
For $d=3$, the <total ramification of a holomorphic map> is four, not two. The local-circle panel shows three nearby preimages converging to the single <ramification point>. For $d=1$ the <holomorphic map> is the identity and neither zero nor infinity is ramified.
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