Cauchy–Pompeiu formula
E854243
The Cauchy–Pompeiu formula is a fundamental result in complex analysis that extends the Cauchy integral formula to functions that are not necessarily holomorphic by expressing them via both boundary and area integrals.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Cauchy–Fantappiè formula | 1 |
| Cauchy–Pompeiu formula canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10304990 — 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: Cauchy–Pompeiu formula Context triple: [Cauchy integral formula, hasGeneralization, Cauchy–Pompeiu formula]
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A.
Bochner–Martinelli formula
The Bochner–Martinelli formula is a fundamental integral representation in several complex variables that generalizes the Cauchy integral formula to higher dimensions.
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B.
Cauchy integral formula
The Cauchy integral formula is a fundamental result in complex analysis that expresses the value of a holomorphic function inside a disk in terms of a contour integral of the function around the disk’s boundary.
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C.
Cauchy integral theorem
The Cauchy integral theorem is a fundamental result in complex analysis stating that the integral of a holomorphic function over any closed contour in a simply connected domain is zero.
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D.
Poisson integral
The Poisson integral is a fundamental formula in harmonic analysis that reconstructs harmonic functions inside a disk (or half-plane) from their boundary values using the Poisson kernel.
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E.
Cauchy–Riemann equations
The Cauchy–Riemann equations are fundamental conditions in complex analysis that characterize when a complex-valued function is holomorphic (complex differentiable).
- 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: Cauchy–Pompeiu formula Target entity description: The Cauchy–Pompeiu formula is a fundamental result in complex analysis that extends the Cauchy integral formula to functions that are not necessarily holomorphic by expressing them via both boundary and area integrals.
-
A.
Bochner–Martinelli formula
The Bochner–Martinelli formula is a fundamental integral representation in several complex variables that generalizes the Cauchy integral formula to higher dimensions.
-
B.
Cauchy integral formula
The Cauchy integral formula is a fundamental result in complex analysis that expresses the value of a holomorphic function inside a disk in terms of a contour integral of the function around the disk’s boundary.
-
C.
Cauchy integral theorem
The Cauchy integral theorem is a fundamental result in complex analysis stating that the integral of a holomorphic function over any closed contour in a simply connected domain is zero.
-
D.
Poisson integral
The Poisson integral is a fundamental formula in harmonic analysis that reconstructs harmonic functions inside a disk (or half-plane) from their boundary values using the Poisson kernel.
-
E.
Cauchy–Riemann equations
The Cauchy–Riemann equations are fundamental conditions in complex analysis that characterize when a complex-valued function is holomorphic (complex differentiable).
- F. None of above. chosen
Statements (31)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical formula
ⓘ
theorem in complex analysis ⓘ |
| appliesTo |
complex-valued functions
ⓘ
functions not necessarily holomorphic ⓘ |
| assumes | function is continuously differentiable on the closure of the domain ⓘ |
| category | integral representation formula ⓘ |
| context | functions defined on subsets of the complex plane ⓘ |
| domain | planar domains in the complex plane ⓘ |
| expressedIn | complex coordinates z and \bar{z} ⓘ |
| expresses | value of a function at an interior point ⓘ |
| field | complex analysis ⓘ |
| generalizes | Cauchy integral formula ⓘ |
| implies | Cauchy integral formula when the function is holomorphic ⓘ |
| involves |
area integrals
ⓘ
boundary integrals ⓘ |
| namedAfter |
Augustin-Louis Cauchy
ⓘ
Dimitrie Pompeiu ⓘ |
| relatedTo |
Cauchy transform
ⓘ
Cauchy–Riemann equations ⓘ Green's identities ⓘ |
| relates | function values to its \/bar{∂} derivative ⓘ |
| requires | piecewise smooth boundary of the domain ⓘ |
| toolFor |
representation of non-holomorphic functions
ⓘ
solving inhomogeneous Cauchy–Riemann equations ⓘ |
| type | integral identity ⓘ |
| usedFor | deriving regularity properties of solutions to \bar{∂} problems ⓘ |
| usedIn |
boundary value problems in complex analysis
ⓘ
potential theory ⓘ theory of several complex variables ⓘ |
| uses |
Cauchy kernel
ⓘ
linked to:
Cauchy transform
Green-type representation ⓘ |
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: Cauchy–Pompeiu formula Description of subject: The Cauchy–Pompeiu formula is a fundamental result in complex analysis that extends the Cauchy integral formula to functions that are not necessarily holomorphic by expressing them via both boundary and area integrals.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Cauchy–Pompeiu formula