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 $2\pi i$.

Explanations

L1 human

A formula such as a power series often defines a function only on a small region. Analytic continuation extends the function to a larger region in the only possible holomorphic way.