TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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A locally bounded holomorphic function extends across a coordinate hyperplane

Statement

Let $m \geq 2$ , let $Ω \subseteq C^{m}$ be a domain, and let

H : = Ω \cap {z_{m} = 0} .

If $f : Ω ∖ H \to C$ is holomorphic and locally bounded near $H$ , then there exists a unique holomorphic $F : Ω \to C$ such that $F = f$ on $Ω ∖ H$ .

Facts & Assumptions

Given: A domain $Ω \subseteq C^{m}$ , a holomorphic function $f : Ω ∖ H \to C$ , and local boundedness near the coordinate hyperplane $H$ .

[L1]

A bounded punctured slice extends holomorphically, and the missing value depends holomorphically on the remaining parameters (A locally bounded punctured slice has a holomorphic parameter extension).

[L2]

A holomorphic function on a connected open set is determined by its values on a nonempty open subset (A holomorphic function vanishing on a nonempty open subset of a domain vanishes identically).

[L3]

Holomorphic extension means agreement on some nonempty open overlap (Holomorphic extension and domains of holomorphy in several variables).

Proof

technique · direct

1.1

Let $p = (p^{'}, 0) \in H$ . By local boundedness, choose a product neighborhood $U \times {∣ w ∣ < R} \subseteq Ω$ of $p$ on which $f$ is bounded whenever $0 < ∣ w ∣ < R$ , after translating coordinates so that $p_{m} = 0$ . Applying [L1] on this product neighborhood gives a holomorphic function $F_{p} : U \times {∣ w ∣ < R} \to C$ extending $f$ across the slice $U \times {0}$ .

givenL1

2.1

The functions $F_{p}$ and $f$ agree on the nonempty open overlap $U \times {0 < ∣ w ∣ < R}$ , so each $F_{p}$ is a local holomorphic extension in the sense of [L3].

step 1.1L3

3.1

If two such neighborhoods overlap, their local extensions agree on the nonempty open subset of the overlap where $w \neq 0$ , because both equal $f$ there. The overlap is connected after shrinking if necessary, so [L2] makes the two local extensions equal on the whole overlap.

step 2.1L2

4.1

The local extensions therefore glue to a single holomorphic function on $Ω$ that agrees with $f$ off $H$ . Uniqueness follows from [L2], since two global extensions agree on the nonempty open set $Ω ∖ H$ .

step 3.1L2∎

Depends on

Used by

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Dependency tree · two levels

13 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

J. Lebl, Tasty Bits of Several Complex Variables, Theorem 1.6.1 (standard reference, not scraped)