Analytic continuations are unique in the following sense: if V is the connected domain of two analytic functions F1 and F2 such that U is contained in V and for all z in U

F1(z) = F2(z) = f(z),

then

F1 = F2

on all of V. This is because F1 – F2 is an analytic function which vanishes on the open, connected domain U of f and hence must vanish on its entire domain. This follows directly from the identity theorem for holomorphic functions.

— Wikipedia on Analytic continuation

2010.02.12 Friday ACHK