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

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The germ space of a complete analytic function

Definition

Let $A (ξ_{0}, Ω)$ be a complete analytic function (The complete analytic function generated by one germ).

Its germ space is the underlying set

$R (ξ_{0}, Ω) : = A (ξ_{0}, Ω) .$

If $(f, U)$ is a function element whose germ $[f]_{z}$ lies in $R (ξ_{0}, Ω)$ for every $z \in U$ , define the associated subset

$N (f, U) : = {[f]_{z} \in R (ξ_{0}, Ω) : z \in U} .$

These sets are the intended basic neighbourhoods of the germ space.

The projection

$p : R (ξ_{0}, Ω) ⟶ Ω$

is the base-point map

$p ([f]_{z}) = z .$

When the neighbourhoods $N (f, U)$ are proved to form a compatible holomorphic atlas, this germ space is called the Riemann surface of the complete analytic function.

Depends on

Used by

Dependency tree · two levels

7 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 §1.3 (standard reference, not scraped)