Let ${\displaystyle X}$  and ${\displaystyle Y}$  be Riemann surfaces, and let ${\displaystyle f:X\to Y}$  be a non-constant holomorphic map. Fix a point ${\displaystyle a\in X}$  and set ${\displaystyle b:=f(a)\in Y}$ . Then there exist ${\displaystyle k\in \mathbb {N} }$  and charts ${\displaystyle \psi _{1}:U_{1}\to V_{1}}$  on ${\displaystyle X}$  and ${\displaystyle \psi _{2}:U_{2}\to V_{2}}$  on ${\displaystyle Y}$  such that

• ${\displaystyle \psi _{1}(a)=\psi _{2}(b)=0}$ ; and
• ${\displaystyle \psi _{2}\circ f\circ \psi _{1}^{-1}:V_{1}\to V_{2}}$  is ${\displaystyle z\mapsto z^{k}.}$ 

This theorem gives rise to several definitions:

• We call ${\displaystyle k}$  the multiplicity of ${\displaystyle f}$  at ${\displaystyle a}$ . Some authors denote this ${\displaystyle \nu (f,a)}$ .
• If ${\displaystyle k>1}$ , the point ${\displaystyle a}$  is called a branch point of ${\displaystyle f}$ .
• If ${\displaystyle f}$  has no branch points, it is called unbranched. See also unramified morphism.

• Ahlfors, Lars (1953), Complex analysis (3rd ed.), McGraw Hill (published 1979), ISBN 0-07-000657-1.