Statements (23)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
|
| gptkbp:appliesTo |
gptkb:unit_disk
upper half-plane |
| gptkbp:category |
mathematical analysis
|
| gptkbp:date_introduced |
19th century
|
| gptkbp:expressedIn |
harmonic function inside disk in terms of boundary values
|
| gptkbp:field |
gptkb:mathematics
complex analysis harmonic analysis |
| gptkbp:formula_(unit_disk) |
u(r,θ) = (1/2π) ∫₀^{2π} P_r(θ−φ) f(φ) dφ
|
| gptkbp:formula_(upper_half-plane) |
u(x,y) = (1/π) ∫_{−∞}^{∞} y f(t)/[(x−t)² + y²] dt
|
| gptkbp:generalizes |
mean value property of harmonic functions
|
| https://www.w3.org/2000/01/rdf-schema#label |
Poisson integral formula
|
| gptkbp:involves |
gptkb:Poisson_kernel
|
| gptkbp:namedAfter |
gptkb:Siméon_Denis_Poisson
|
| gptkbp:Poisson_kernel_(unit_disk) |
P_r(θ) = (1−r²)/(1−2r cos θ + r²)
|
| gptkbp:relatedTo |
gptkb:Laplace_equation
potential theory |
| gptkbp:used_in |
Fourier analysis
boundary value problems |
| gptkbp:usedFor |
solving Dirichlet problem
|
| gptkbp:bfsParent |
gptkb:Herglotz-Riesz_representation_theorem
|
| gptkbp:bfsLayer |
6
|