Bochner–Martinelli formula
E613405
The Bochner–Martinelli formula is a fundamental integral representation in several complex variables that generalizes the Cauchy integral formula to higher dimensions.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Bochner–Martinelli formula canonical | 2 |
| Bochner–Martinelli kernel | 1 |
| Cauchy integral formula in several complex variables | 1 |
| Cauchy integral formulas in several variables | 1 |
| Cauchy–Green formula | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6716263 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Bochner–Martinelli formula Context triple: [Salomon Bochner, notableFor, Bochner–Martinelli formula]
-
A.
Szegő kernel
The Szegő kernel is a fundamental reproducing kernel in complex analysis and operator theory, associated with Hardy spaces on the boundary of a domain and central to the study of orthogonal polynomials and boundary behavior of analytic functions.
-
B.
Fefferman metric in several complex variables
The Fefferman metric in several complex variables is a canonical Lorentz–Kähler-type metric associated with strictly pseudoconvex domains, fundamental in the study of CR geometry and the boundary behavior of holomorphic functions.
-
C.
Mittag-Leffler theorem
The Mittag-Leffler theorem is a fundamental result in complex analysis that characterizes meromorphic functions by allowing the construction of such functions with prescribed principal parts at given poles.
-
D.
Lempert function on convex domains
The Lempert function on convex domains is a complex-analytic invariant that coincides with the Kobayashi distance and provides an extremal characterization of holomorphic mappings between convex domains in several complex variables.
-
E.
Fubini–Study form
The Fubini–Study form is the canonical Kähler form on complex projective space, encoding its standard Hermitian and symplectic geometry.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Bochner–Martinelli formula Target entity description: The Bochner–Martinelli formula is a fundamental integral representation in several complex variables that generalizes the Cauchy integral formula to higher dimensions.
-
A.
Szegő kernel
The Szegő kernel is a fundamental reproducing kernel in complex analysis and operator theory, associated with Hardy spaces on the boundary of a domain and central to the study of orthogonal polynomials and boundary behavior of analytic functions.
-
B.
Fefferman metric in several complex variables
The Fefferman metric in several complex variables is a canonical Lorentz–Kähler-type metric associated with strictly pseudoconvex domains, fundamental in the study of CR geometry and the boundary behavior of holomorphic functions.
-
C.
Mittag-Leffler theorem
The Mittag-Leffler theorem is a fundamental result in complex analysis that characterizes meromorphic functions by allowing the construction of such functions with prescribed principal parts at given poles.
-
D.
Lempert function on convex domains
The Lempert function on convex domains is a complex-analytic invariant that coincides with the Kobayashi distance and provides an extremal characterization of holomorphic mappings between convex domains in several complex variables.
-
E.
Fubini–Study form
The Fubini–Study form is the canonical Kähler form on complex projective space, encoding its standard Hermitian and symplectic geometry.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
integral representation formula
ⓘ
mathematical formula ⓘ result in several complex variables ⓘ |
| appearsIn | advanced textbooks on several complex variables ⓘ |
| appliesTo |
domains in \mathbb{C}^n
ⓘ
holomorphic functions of several complex variables ⓘ |
| assumptionOnFunction | holomorphic in the interior of the domain ⓘ |
| category | theorem in complex analysis ⓘ |
| component |
d\sigma(\zeta) surface measure element
ⓘ
factor (n-1)! / (2\pi i)^n ⓘ |
| dimension | n \ge 1 ⓘ |
| domainCondition | bounded domain in \mathbb{C}^n with sufficiently smooth boundary ⓘ |
| expresses | function value f(z) as an integral over the boundary involving f(\zeta) ⓘ |
| field |
complex analysis
ⓘ
several complex variables ⓘ |
| generalizes | Cauchy integral formula ⓘ |
| gives |
boundary integral representation
ⓘ
integral representation of holomorphic functions ⓘ |
| hasPrerequisite |
Cauchy integral formula
ⓘ
Stokes' theorem ⓘ differential forms on \mathbb{C}^n ⓘ |
| historicalPeriod | 20th century mathematics ⓘ |
| implies |
Cauchy integral formula when n = 1
ⓘ
linked to:
Cauchy integral formula
|
| integralType | surface integral ⓘ |
| integratesOver | boundary of the domain ⓘ |
| kernelProperty |
(n,n-1)-form
ⓘ
Cauchy–Fantappiè type kernel ⓘ |
| kernelType |
Bochner–Martinelli kernel
ⓘ
linked to:
Bochner–Martinelli formula
|
| mathematicalArea |
analysis
ⓘ
complex geometry ⓘ |
| namedAfter |
Enzo Martinelli
ⓘ
Salomon Bochner ⓘ |
| property | invariant under biholomorphic changes of coordinates (up to natural factors) ⓘ |
| relatedConcept |
Cauchy–Fantappiè formula
ⓘ
linked to:
Cauchy–Pompeiu formula
Henkin kernel ⓘ
linked to:
Bergman kernel
integral representation in complex manifolds ⓘ |
| relates | values of a holomorphic function inside a domain to its values on the boundary ⓘ |
| requires | orientation of the boundary of the domain ⓘ |
| requiresFunctionClass | C^1 functions on the closure of the domain ⓘ |
| role | fundamental tool in higher-dimensional complex analysis ⓘ |
| topicOf | research in several complex variables ⓘ |
| usedFor |
deriving estimates for holomorphic functions
ⓘ
extension problems in several complex variables ⓘ representation of solutions to boundary value problems ⓘ solving \bar{\partial}-equations ⓘ |
| variable |
\zeta \in \mathbb{C}^n
ⓘ
z \in \mathbb{C}^n ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Bochner–Martinelli formula Description of subject: The Bochner–Martinelli formula is a fundamental integral representation in several complex variables that generalizes the Cauchy integral formula to higher dimensions.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Bochner–Martinelli formula
Cauchy integral formula
→
hasGeneralization
→
Cauchy integral formula in several complex variables
ⓘ
linked to: Bochner–Martinelli formula
subject linked to:
Bochner
linked to: Bochner–Martinelli formula
linked to: Bochner–Martinelli formula