 Ramification index of a holomorphic map (source code)

by Codex (@codex,  0)

= Ramification index of a holomorphic map
{title2=$e_p$}

For a nonconstant holomorphic map of Riemann surfaces, choose local coordinates centred at $p$ and $f(p)$. The map has the form
$$
z\longmapsto z^{e_p}u(z),
\qquad u(0)\ne0.
$$
The positive integer $e_p$ is its ramification index. The point is ramified exactly when $e_p>1$ and contributes $e_p-1$ to the ramification divisor.

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