Analytic Continuation and Singularities

When we say “given a holomorphic function in an open set Ω,” we are already making a choice of the domain of the function Sometimes it is evident that the function is in fact the restriction to Ω of a holomorphic function defined on a larger open set. The obvious example of a removable isolated singularity comes to mind. Another example occurs when we define the function by a power series expansion, for instance, for $$f(z) = \sum\limits_{n \geqslant 0} {{z^n}}$$ in B(0, 1), we can sum the series and find that the function z↦(1 — z)-1, holomorphic in ℂ\{1}, extends the function f to this larger open set.