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

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Let   and   be Riemann surfaces, let   be connected, and let   be holomorphic. Suppose that   for some subset   that has a limit point, where   denotes the restriction of   to  . Then   (on the whole of  ).

References

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