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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 and Y be Riemann surfaces, let X be connected, and let \( f:X\to Y \) be holomorphic. Suppose that \( {\displaystyle f|_{A}=g|_{A}} \) for some subset \( A\subseteq X \) that has a limit point, where \( {\displaystyle f|_{A}:A\to Y} denotes the restriction of f to A. Then f = g (on the whole of X).


Forster, Otto (1981), Lectures on Riemann surfaces, Graduate Text in Mathematics, 81, New-York: Springer Verlag, p. 6, ISBN 0-387-90617-7

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