DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

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The complete analytic function generated by one germ

Definition

Let $Ω \subseteq C$ be a complex domain, let $a_{0} \in Ω$ , and let $ξ_{0}$ be a holomorphic germ at $a_{0}$ . Assume that $ξ_{0}$ admits analytic continuation along every path in $Ω$ starting at $a_{0}$ (Analytic continuation along a path by admissible chains).

For a path $γ : [0, 1] \to Ω$ with $γ (0) = a_{0}$ , write ${Cont}_{γ} (ξ_{0})$ for the terminal germ of a continuation of $ξ_{0}$ along $γ$ . This is well defined by The terminal germ of a continuation along a fixed path is chain-independent, because the admissible chains now require agreement at each subdivision endpoint.

The complete analytic function generated by $ξ_{0}$ is the set

$A (ξ_{0}, Ω) : = {{Cont}_{γ} (ξ_{0}) : γ (0) = a_{0}} .$

Its elements are holomorphic germs based at points of $Ω$ reachable from $a_{0}$ by path continuation.

Depends on

Used by

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Curtis T. McMullen, Riemann Surfaces, Theorem 4.3 (standard reference, not scraped)
Lars V. Ahlfors, Complex Analysis, 3rd ed., Ch. 8 §§1.2-1.3 (standard reference, not scraped)