Identity theorem for Riemann surfaces

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

References

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