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

References

{{Citation | last1=Forster | first1=Otto | author1-link= Otto Forster | title=Lectures on Riemann surfaces | publisher=Springer Verlag | location=New-York | series=Graduate Text in Mathematics | isbn=0-387-90617-7 | year=1981 | volume=81 | page=6}}

Category:Theorems in complex analysis

Category:Riemann surfaces