Hartogs's Extension Theorem¶
Extend a holomorphic function uniquely across a compact hole in a connected domain of at least two complex dimensions.
Core Idea¶
If Ω is a domain in ℂⁿ with n≥2, K is a compact subset inside Ω, and Ω\K is connected, every holomorphic function on Ω\K extends uniquely and holomorphically to all of Ω. This is Hartogs's compact-hole extension theorem.[^ref-1b9eb8138b67]
Scope of Application¶
A punctured ball in ℂ² and a connected shell around a compact inner ball in ℂ³ meet the hypotheses. Isolated singularities are therefore removable in several complex variables under the local punctured-neighborhood setup. The theorem does not automatically cover other PDE systems or arbitrary excluded sets.[^ref-9b526b64d3c0]
For explicit illustrative inputs, f(z,w)=z²+w on the punctured unit ball in ℂ² extends by the same formula with value zero at the origin. On the shell 1<|z|²+|w|²+|u|²<4 in ℂ³, g(z,w,u)=e^z+wu extends through the closed unit ball by that same formula, with value one at the origin. These easy-to-display functions illustrate the theorem; it is not limited to functions whose formulas already reveal the answer.[^ref-1b9eb8138b67]
Clarity¶
The function may be unspecified inside the hole, but its holomorphic values there are not freely choosable. Dimension matters: 1/z is holomorphic on the punctured complex plane yet cannot extend through zero when n=1.[^ref-9b526b64d3c0]
Manages Complexity¶
The same short checklist—dimension, compactness, connected complement and holomorphic input—settles extension for many hole shapes without solving each case afresh. Dropping a hypothesis removes that guarantee.
Abstract Reasoning¶
Identify Ω, K and n; verify n≥2, K compactly contained, Ω\K connected and f holomorphic there. Then infer a unique holomorphic extension. A noncompact singular hypersurface or one-variable pole needs separate analysis.[^ref-1b9eb8138b67]
Knowledge Transfer¶
The theorem applies across suitable domains of several complex variables, but broader Hartogs phenomena in other equations require their own conditions and proofs. It is a theorem about holomorphic functions, not itself an Analytic Function; a live extension-theorem genus remains unverified. Its analytic rigidity is domain-specific, not an unrestricted prime abstraction.
[^ref-1b9eb8138b67]: UC San Diego, Hartogs extension theorem lecture, exact statement. [^ref-9b526b64d3c0]: Jiří Lebl, “Several Complex Variables are Better than One”, compact-hole form and one-variable counterexample.
Neighborhood in Abstraction Space¶
Hartogs's Extension Theorem sits in a sparse region of the domain-specific corpus (75th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Complex & Holomorphic Function Theory (8 abstractions)
Nearest neighbors
- Alexander Duality — 0.84
- Four-Dimensional Chern–Simons Theory — 0.84
- Eells–Kuiper Manifold — 0.83
- Arakelian's Theorem — 0.83
- Harmonic conjugate — 0.83
Computed from structural-signature embeddings · 2026-10-08