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Suppose M , N {\displaystyle M,N} are Riemann surfaces. A map f : M → N {\displaystyle f:M\to N} is termed an analytic map or analytic mapping if given any coordinate charts φ : U 1 → V 1 ⊂ C {\displaystyle \varphi :U_{1}\to V_{1}\subset \mathbb {C} } and ψ : U 2 → V 2 ⊂ C {\displaystyle \psi :U_{2}\to V_{2}\subset \mathbb {C} } on M {\displaystyle M} and N {\displaystyle N} respectively, the map:
ψ ∘ f ∘ φ − 1 : U 1 ∩ f − 1 ( U 2 ) → f ( U 1 ∩ f − 1 ( U 2 ) ) ⊂ C {\displaystyle \psi \circ f\circ \varphi ^{-1}:U_{1}\cap f^{-1}(U_{2})\to f(U_{1}\cap f^{-1}(U_{2}))\subset \mathbb {C} }
is a holomorphic function.
