The process of extending a holomorphic function from an initial domain to a larger domain by uniquely patching overlapping power-series or local representations, continuing along paths until an obstruction is reached.

Principle

Uniqueness follows from the identity theorem: two analytic continuations that agree on a set with accumulation points coincide on their common domain; continuation proceeds along chains of overlapping analytic neighborhoods or paths (monodromy may occur).

Demonstration

A power series for log(1+z) around z=0 can be analytically continued along paths avoiding the branch point at z=-1, producing a multi-valued logarithm on the punctured plane.

Misapplication

Assuming every singularity is removable and attempting to continue through poles or essential singularities as if no obstruction exists.

Consequence

Successful analytic continuation defines maximal analytic extensions which may form Riemann surfaces or multi-valued branches; global properties (monodromy, natural boundaries) are revealed by continuation attempts.

Reversal

Failure to continue beyond an essential singularity or a natural boundary reveals intrinsic obstructions to extension; continuation may produce multiple branches rather than a single-valued extension.

Boundary

Applies to complex-analytic (holomorphic) functions and real-analytic functions with different technicalities; requires overlapping domains with analytic agreement and excludes arbitrary distributional or merely continuous extensions.

Semantic Tension

Often confused with naive 'extension by continuity' or with real-analytic extension; analytic continuation is rigid and uniquely determined by local data, unlike many weaker extension notions.

Synthesis

Analytic continuation is the unique extension of a holomorphic function along overlapping analytic charts or paths, proceeding until genuine obstructions (singularities, monodromy, natural boundaries) prevent further extension.