Theorems · Theorem · potential theory

InnerProductSpace.HarmonicContOnCl.circleAverage_poissonKernel_smul

∀ {f : ℂ → ℝ} {c w : ℂ} {R : ℝ},
  InnerProductSpace.HarmonicContOnCl f (Metric.ball c R) →
    w ∈ Metric.ball c R → Real.circleAverage (poissonKernel c w • f) c R = f w

Poisson integral formula for harmonic functions on arbitrary disks in the complex plane, formulated with the Poisson kernel of integration.

Defined in: Mathlib.Analysis.Complex.Harmonic.Poisson
Cited by: 0 results in Mathlib
Foundations: Depth 291 from the axioms · uses propext, Classical.choice, Quot.sound

Mathlib docs Source Loogle LeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Cites11

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Setstatement · cited by 53,352
Realstatement and proof · cited by 25,697
Complexstatement and proof · cited by 5,565
absproof · cited by 1,814
Metric.ballstatement and proof · cited by 735
Metric.sphereproof · cited by 371
Real.circleAveragestatement and proof · cited by 106
InnerProductSpace.HarmonicContOnClstatement and proof · cited by 32
Real.circleAverage_congr_sphereproof · cited by 9
poissonKernelstatement and proof · cited by 5
InnerProductSpace.HarmonicContOnCl.circleAverage_re_herglotzRieszKernel_smulproof · cited by 2

Cited by0

Results whose statement or proof uses this declaration.

Nothing cites this yet.