Theorems · Theorem · complex analysis

DiffContOnCl.circleAverage_re_herglotzRieszKernel_smul

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] {f : ℂ → E} {R : ℝ} {w : ℂ} [CompleteSpace E]
  {c : ℂ},
  DiffContOnCl ℂ f (Metric.ball c R) →
    w ∈ Metric.ball c R → Real.circleAverage (Complex.re ∘ herglotzRieszKernel c w • f) c R = f w

Poisson integral formula for ℂ-differentiable 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.Poisson

Cited by

2 results in Mathlib

Foundations

Depth 287 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceCompleteSpace