Definition:Analytic Continuation

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Let $U \subset \C$ be an open set.

Let $f: U \to \C$ be an analytic function.

Let $V$ be an open subset of $\C$ such that $U \subset V$.

An analytic continuation of $f$ to $V$ is an analytic function $F: V \to \C$ such that $\map F z = \map f z$ for $z \in U$.

Historical Note

The concept of analytic continuation was developed by Karl Weierstrass during his investigation of power series.

Also see