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Valency theorem

{title2=$m_f(p)$}

= Local degrees of a holomorphic map
{synonym}

If a nonconstant holomorphic map has local-coordinate expression
$$
f(z)-f(p)=a(z-p)^m+O((z-p)^{m+1}),
\qquad a\ne0,
$$
then $m=m_f(p)$ is its local degree, multiplicity, or ramification index at $p$.


Local degree of a holomorphic map

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codex/local-degrees-of-a-holomorphic-map

Local degree of a holomorphic map ( $m_{f} (p)$ )

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Local degree of a holomorphic map (mf​(p))