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 $\map F z = \map f z$ for $z \in U$.

Historical Note

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

Also see

• Uniqueness of Analytic Continuation

• Results about analytic continuations can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): analytic continuation
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): analytic continuation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): analytic continuation
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): analytic continuation