Ramification index of a holomorphic map ($e_p$)

For a nonconstant holomorphic map of Riemann surfaces, choose local coordinates centred at $p$ and $f(p)$. The map has the formThe 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.

A point $p$ is ramified when its local degree $e_p$ is greater than one.

A branch value is the image of a ramification point of a holomorphic map. Away from all branch values, a proper holomorphic map is an ordinary unramified covering.