Poisson integral
E559805
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Poisson integral canonical | 5 |
| Poisson integral formula | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5973628 — 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: Poisson integral Context triple: [Siméon Denis Poisson, notableWork, Poisson integral]
-
A.
Dirichlet problem
The Dirichlet problem is a fundamental boundary value problem in potential theory and partial differential equations, asking for a function that solves a specified PDE inside a domain while taking prescribed values on the domain’s boundary.
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B.
Cauchy principal value
The Cauchy principal value is a method in mathematical analysis for assigning finite values to certain improper or divergent integrals and series by symmetrically balancing their singularities.
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C.
Mittag-Leffler function
The Mittag-Leffler function is a complex function that generalizes the exponential function and plays a central role in fractional calculus and the theory of differential and integral equations.
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D.
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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E.
Riemann–Liouville integral
The Riemann–Liouville integral is a fundamental operator in fractional calculus that generalizes the concept of an n-fold repeated integral to non-integer (fractional) orders.
- 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: Poisson integral Target entity description: 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.
-
A.
Dirichlet problem
The Dirichlet problem is a fundamental boundary value problem in potential theory and partial differential equations, asking for a function that solves a specified PDE inside a domain while taking prescribed values on the domain’s boundary.
-
B.
Cauchy principal value
The Cauchy principal value is a method in mathematical analysis for assigning finite values to certain improper or divergent integrals and series by symmetrically balancing their singularities.
-
C.
Mittag-Leffler function
The Mittag-Leffler function is a complex function that generalizes the exponential function and plays a central role in fractional calculus and the theory of differential and integral equations.
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D.
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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E.
Riemann–Liouville integral
The Riemann–Liouville integral is a fundamental operator in fractional calculus that generalizes the concept of an n-fold repeated integral to non-integer (fractional) orders.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
integral transform
ⓘ
mathematical concept ⓘ |
| appliesTo |
continuous boundary data
ⓘ
integrable boundary data ⓘ |
| assumption |
Poisson kernel is positive and integrates to 1
ⓘ
linked to:
Poisson kernel
boundary function defined almost everywhere ⓘ |
| context |
classical theory of the unit disk
ⓘ
classical theory of the upper half-plane ⓘ |
| convergesTo |
boundary function almost everywhere under mild conditions
ⓘ
boundary function in L^p for 1 < p < ∞ ⓘ |
| domain |
unit disk
ⓘ
upper half-plane ⓘ |
| ensures |
harmonicity in the interior of the domain
ⓘ
smoothness of interior values for rough boundary data ⓘ |
| field |
complex analysis
ⓘ
harmonic analysis ⓘ potential theory ⓘ |
| generalizationOf | mean value property for harmonic functions ⓘ |
| halfPlaneFormula | u(x+iy) = \frac{1}{\pi}\int_{-\infty}^{\infty} \frac{y}{(x-t)^2 + y^2} f(t)\,dt ⓘ |
| input | boundary function ⓘ |
| kernelFormula | P_r(\theta) = \frac{1-r^2}{1-2r\cos\theta + r^2} ⓘ |
| namedAfter | Siméon Denis Poisson ⓘ |
| output | harmonic function ⓘ |
| property |
Poisson kernel acts as an approximate identity
ⓘ
gives harmonic extension of boundary data ⓘ preserves harmonicity under conformal maps (up to change of domain) ⓘ solution of Dirichlet problem in the disk ⓘ solution of Dirichlet problem in the half-plane ⓘ |
| purpose | reconstruct harmonic functions from boundary values ⓘ |
| relatedTo |
Cauchy integral formula
ⓘ
Dirichlet problem on more general domains via conformal mapping ⓘ Fourier series ⓘ Fourier transform ⓘ Green function methods ⓘ Hardy spaces ⓘ
linked to:
Hardy space
Poisson kernel on the real line ⓘ
linked to:
Poisson kernel
Poisson kernel on the unit circle ⓘ Riesz representation for harmonic functions ⓘ boundary values of analytic functions ⓘ harmonic measure ⓘ |
| representationFormula | u(re^{i\theta}) = \frac{1}{2\pi}\int_{0}^{2\pi} P_r(\theta-t) f(e^{it})\,dt ⓘ |
| satisfies |
Laplace equation in the interior
ⓘ
maximum principle for harmonic functions ⓘ |
| type | convolution with Poisson kernel ⓘ |
| usedFor |
constructing harmonic conjugates
ⓘ
solving classical boundary value problems ⓘ studying boundary behavior of harmonic functions ⓘ |
| uses | Poisson kernel ⓘ |
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: Poisson integral Description of subject: 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.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Poisson
subject linked to:
Siméon
subject linked to:
Functions of One Complex Variable by John B. Conway
linked to: Poisson integral