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.
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 ⓘ |
Referenced by (2)
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