The process of extending the domain of a holomorphic (analytic) function beyond its original region by defining it on larger domains so that on overlaps the extension agrees with the original function; extensions are unique when connectedness and analyticity conditions hold.

Principle

Holomorphic functions are rigid: if two analytic functions agree on a set with an accumulation point within a connected domain, they agree everywhere on the connected component; analytic continuation uses local power series or monodromy to propagate values along paths while preserving analyticity.

Demonstration

Extend the Riemann zeta function from Re(s)>1 to the whole complex plane minus s=1 by expressing ζ(s) in terms of a Dirichlet series, Mellin transforms, and the functional equation; use overlapping expansions and the uniqueness of analytic continuation to obtain the global meromorphic extension.

Misapplication

Assuming analytic continuation across a natural boundary or branch essential singularity without checking obstructions can produce contradictions; attempting to continue a function across accumulation of singularities (natural boundary) is invalid.

Consequence

Analytic continuation reveals global structure (singularities, branch points, monodromy) from local data, enables analytic continuation of functional equations and transform inversions, and is central to complex-analytic proofs and classification of special functions.

Reversal

The converse is restriction: given a global analytic function, restricting to a smaller domain yields local descriptions (Taylor expansions), whereas continuation attempts to enlarge the domain starting from local information.

Boundary

Applies only to analytic (holomorphic or meromorphic) functions and along paths avoiding singularities; it cannot bypass essential singularities or cross natural boundaries, and uniqueness may fail on nonconnected domains or when only discontinuous extensions are allowed.

Semantic Tension

Tension with distributional or generalized-function extensions: analytic continuation preserves holomorphy and strong uniqueness, whereas other extension notions (e.g., as distributions) may allow extensions that are not analytic, trading rigidity for weaker existence.

Synthesis

Analytic continuation is the propagation of local analytic data to maximal domains by using power series, analytic identities, and pathwise continuation, producing the largest holomorphic (or meromorphic) extension permitted by the location of singularities and monodromy.