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

## References

* 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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