Analytic continuation

Definition

Let $U$ and $V$ be complex domains, and let $C$ be a connected component of $U\cap V$ . A holomorphic function $g:V\to\mathbb C$ is an analytic continuation of a holomorphic function $f:U\to\mathbb C$ through $C$ if $f=g$ on $C$ . By the identity theorem, it is enough to require equality on any nonempty open subset of $C$ .

Specifying the component matters when $U\cap V$ is disconnected: agreement on one component does not force agreement on the others. When the overlap is connected, one simply says that $g$ is an analytic continuation of $f$ through $U\cap V$ .

Along a path

Continuation along a path is described by a chain of overlapping discs and holomorphic functions that agree successively. Its endpoint value can depend on the path when the ambient domain is not simply connected. The monodromy theorem supplies path independence under suitable simple-connectivity hypotheses.

Germ viewpoint

A holomorphic germ at $a$ is an equivalence class of holomorphic functions defined near $a$ , where two representatives agree on some smaller neighborhood. Analytic continuation transports such germs. Singularities and nontrivial monodromy may obstruct a single-valued global continuation even when continuation is possible along every short segment.

References

Otto Forster, Lectures on Riemann Surfaces, Springer, 1981. Publisher record. Relevant: Chapter 1, §§5–7.