In mathematics, the identity theorem for Riemann surfaces is a theorem that states that a holomorphic function is completely determined by its values on any subset of its domain that has a limit point.
Statement of the theorem Let X {\displaystyle X} and Y {\displaystyle Y} be Riemann surfaces, let X {\displaystyle X} be connected, and let f , g : X → Y {\displaystyle f,g:X\to Y} be holomorphic. Suppose that f | A = g | A {\displaystyle f|_{A}=g|_{A}} for some subset A ⊆ X {\displaystyle A\subseteq X} that has a limit point, where f | A : A → Y {\displaystyle f|_{A}:A\to Y} denotes the restriction of f {\displaystyle f} to A {\displaystyle A} . Then f = g {\displaystyle f=g} (on the whole of X {\displaystyle X} ).
References Forster, Otto (1981), Lectures on Riemann surfaces, Graduate Text in Mathematics, vol. 81, New-York: Springer Verlag, p. 6, ISBN 0-387-90617-7