HarmonicContOnCl.circleAverage_eq

Deprecated since 2026-08-04Use InnerProductSpace.HarmonicContOnCl.circleAverage_eq instead.

∀ {F : Type u_1} [inst : NormedAddCommGroup F] [inst_1 : NormedSpace ℝ F] [CompleteSpace F] {f : ℂ → F} {c : ℂ} {R : ℝ},
  InnerProductSpace.HarmonicContOnCl f (Metric.ball c |R|) → Real.circleAverage f c R = f c

Alias of InnerProductSpace.HarmonicContOnCl.circleAverage_eq. The Mean Value Property of harmonic functions: If f : ℂ → F is harmonic on a disc of radius |R| and center c and continuous on its closure, then the circle average circleAverage f c R equals f c.

Defined in: Mathlib.Analysis.Complex.Harmonic.MeanValue
Cited by: 0 results in Mathlib
Foundations: Depth 290 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: NormedAddCommGroup NormedSpace CompleteSpace

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Realstatement · cited by 25,697
NormedAddCommGroupstatement · cited by 15,752
NormedSpacestatement · cited by 12,499
Complexstatement · cited by 5,565
CompleteSpacestatement · cited by 2,532
absstatement · cited by 1,814
Metric.ballstatement · cited by 735
Real.circleAveragestatement · cited by 106
InnerProductSpace.HarmonicContOnClstatement · cited by 32
InnerProductSpace.HarmonicContOnCl.circleAverage_eqproof · cited by 1

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