Analytic continuation

Analytic continuation extends a holomorphic function beyond its initial domain by matching its local values on overlapping regions. The extension is unique wherever such regions connect.

Holomorphic function: A complex-valued function that is complex differentiable throughout an open domain. Continuation extends these functions by matching their values locally.
Riemann zeta function: A complex function initially defined by a Dirichlet series and continued meromorphically to the complex plane, except at one pole. Its continuation extends a convergent series into regions where that series diverges.
Natural boundary: A boundary across which a function cannot be analytically continued at any point. It marks a genuine limit to continuation, unlike a mere series convergence boundary.
Analytic number theory: The study of number-theoretic questions using methods from complex analysis. Continued zeta and L-functions connect complex analysis to prime-number estimates.
Augustin-Louis Cauchy: A French mathematician who established foundational results in complex analysis. His integral theorem underlies the local structure of holomorphic functions being continued.
Identity theorem: A theorem stating that holomorphic functions agreeing on a set with an accumulation point in a connected domain agree everywhere there. It guarantees uniqueness when two proposed extensions overlap.
Gamma function: A meromorphic function extending the factorial to complex arguments through the relation Γ(z+1)=zΓ(z). Its integral definition continues beyond the half-plane where the integral converges.
Branch point: A point around which analytic continuation can permute the values of a multivalued function. It makes continuation path-dependent even when local extensions exist.
Functional equation: An equation relating values of a function at different arguments, often reflecting symmetry. For zeta and related functions, it relates continued values across the complex plane.
Karl Weierstrass: A German mathematician known for rigorous foundations of analysis and the theory of holomorphic functions. His power-series approach made continuation a precise local-to-global procedure.

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