The process of extending a holomorphic (complex-analytic) function from its original domain to a larger connected region by using the unique continuation properties of analytic functions, typically by continuing power series or values along paths.

Principle

The organizing rule is the identity theorem and monodromy: holomorphic functions determined on any set with a limit point extend uniquely along paths until encountering singularities or obstructions; continuation is governed by analytic continuation along chains of overlapping discs or analytic continuation across branch cuts when multivaluedness appears.

Demonstration

Start with a power series centered at z0 that converges on a disc. If on the boundary a point z1 has a convergent local power series representation agreeing with the original on the overlap, then extend to a disc around z1 and repeat. Concrete example: the series for log(1+z) around 0 extends along paths avoiding the branch point at z=-1, producing branches of the logarithm.

Misapplication

Assuming analytic continuation always passes through any boundary point; attempting to extend across a natural boundary or essential singularity without acknowledging branching or divergence leads to incorrect conclusions.

Consequence

When applicable, analytic continuation yields a unique holomorphic extension on each simply connected region of continuation and exposes singularities and branch structure; it allows global analytic information (poles, branch cuts) to be deduced from local data.

Reversal

The inverse viewpoint is restriction: given a maximal analytic function, taking its restriction to a smaller open set removes information about monodromy and singularities and may conceal multivalued behavior that only appears when continuing.

Boundary

Applies to holomorphic (complex-analytic) functions on connected open sets; it does not apply to arbitrary continuous, differentiable, or distributional functions without complex analyticity. Excludes formal algebraic prolongations that lack convergence or analytic validity.

Semantic Tension

Tension exists between analytic continuation and formal or algebraic continuation: a formal power series extension need not converge; similarly, distributional or weak extensions differ from holomorphic continuation because they do not preserve complex differentiability.

Synthesis

Analytic continuation is the operation of following local holomorphic representations outward, using the identity theorem and overlapping expansions to produce the maximal holomorphic extension on reachable domains while respecting singularities and monodromy.