Identity theorem for Riemann surfaces

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

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