Functions of a complex variable Series expansions Concept

Analytic continuation

Summary

Let $U \subseteq V$ be open subsets of $\mathbb{C}$ , with $U$ nonempty and $V$ a domain, and let $f$ be holomorphic on $U$ . An analytic continuation of $f$ to $V$ is a function $F$ that is holomorphic on $V$ with $F(z) = f(z)$ for each $z \in U$ . By the identity theorem, $f$ has at most one analytic continuation to $V$ .

Remarks

The geometric series $\sum_{n \ge 0} z^n$ is holomorphic on the unit disc, and $1/(1 - z)$ is its analytic continuation to $\mathbb{C} \setminus \{1\}$ . The Riemann zeta function $\sum_{n \ge 1} n^{-s}$ on $\operatorname{Re} s > 1$ has an analytic continuation to $\mathbb{C} \setminus \{1\}$ . Continuation along a path is a more general process. Its result can depend on the path: the logarithm continued once around $0$ changes by $2\pi i$ .

Tag	`0M3YQJ81NB`
URI	`m2://0M3YQJ81NB`
Aliases	`holomorphic-extension`
Kind	concept
Writing	W1 stated
MSC	30B40
Same as	Wikidata Q485287 Wikipedia Analytic continuation

Analytic continuation

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