# Definition:Analytic Continuation

## Definition

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 $F \left({z}\right) = f \left({z}\right)$ for $z \in U$.

## Historical Note

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