Definition:Ramification

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Definition

In Complex Analysis

See Definition:Branch Point

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In Riemann Surfaces

Let $S$ and $S'$ be Riemann surfaces.

Let the mapping $\pi$ is complex analytic.

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$\pi$ is said to be ramified at a point $P$ in $S'$ if there exist analytic coordinates near $P$ and $\map \pi P$ such that $\pi$ takes the form:

$\map \pi z = z^n$, and $n > 1$.

An equivalent way of thinking about this is that there exists a small neighborhood $U$ of $P$ such that $\map \pi P$ has exactly one preimage in $U$, but the image of any other point in $U$ has exactly $n$ preimages in $U$.

The number $n$ is called the ramification index at $P$ and is denoted by $e_P$.