Theorems · Theorem · potential theory

InnerProductSpace.HarmonicOnNhd.circleAverage_re_herglotzRieszKernel_smul

∀ {f : ℂ → ℝ} {c w : ℂ} {R : ℝ},
  InnerProductSpace.HarmonicOnNhd f (Metric.closedBall c R) →
    w ∈ Metric.ball c R → Real.circleAverage (Complex.re ∘ herglotzRieszKernel c w • f) c R = f w

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

Defined in: Mathlib.Analysis.Complex.Harmonic.Poisson
Cited by: 2 results in Mathlib
Foundations: Depth 289 from the axioms · uses propext, Classical.choice, Quot.sound

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Cited by2

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InnerProductSpace.HarmonicContOnCl.circleAverage_re_herglotzRieszKernel_smulproof · cited by 2
InnerProductSpace.HarmonicOnNhd.circleAverage_poissonKernel_smulproof · cited by 0