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 (1)
| Label | Occurrences |
|---|---|
| Poisson integral canonical | 2 |
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
NERFINISHED
ⓘ
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 NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
Dirichlet problem on more general domains via conformal mapping ⓘ Fourier series NERFINISHED ⓘ Fourier transform NERFINISHED ⓘ Green function methods ⓘ Hardy spaces NERFINISHED ⓘ Poisson kernel on the real line NERFINISHED ⓘ 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
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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 (2)
Full triples — surface form annotated when it differs from this entity's canonical label.