`BookProof.ChapterSolidHarmonicTools.laplacian_clmPow` (ψ : E →L[ℝ] ℂ) (k : ℕ) (x : E) : (Δ fun y => (ψ y) ^ k) x = (k : ℂ) * ((k : ℂ) - 1) * (ψ x) ^ (k - 2) * ∑ i, (ψ (stdOrthonor

Proved

BookProof.ChapterSolidHarmonicTools.laplacian_clmPow

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSolidHarmonicTools.

BookProof.ChapterSolidHarmonicTools.laplacian_clmPow (ψ : E →L[ℝ] ℂ) (k : ℕ) (x : E) : (Δ fun y => (ψ y) ^ k) x = (k : ℂ) * ((k : ℂ) - 1) * (ψ x) ^ (k - 2) * ∑ i, (ψ (stdOrthonormalBasis ℝ E i)) ^ 2

Formalization note: Lean 4 identifier BookProof.ChapterSolidHarmonicTools.laplacian_clmPow.

Preamble

-- Generated from ChapterSolidHarmonicTools.lean — theorem BookProof.ChapterSolidHarmonicTools.laplacian_clmPow
import Definitions.Def_ChapterRadialLaplacian
import Definitions.Def_ChapterLaplacianProduct
import Mathlib
import Definitions.Def_ChapterSolidHarmonicTools
open BookProof.ChapterSolidHarmonicTools



open Laplacian InnerProductSpace BookProof.ChapterRadialLaplacian
open BookProof.ChapterLaplacianProduct
open scoped RealInnerProductSpace


variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]

variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]

Formal statement

theorem BookProof.ChapterSolidHarmonicTools.laplacian_clmPow (ψ : E →L[ℝ] ℂ) (k : ℕ) (x : E) :
    (Δ fun y => (ψ y) ^ k) x
      = (k : ℂ) * ((k : ℂ) - 1) * (ψ x) ^ (k - 2)
          * ∑ i, (ψ (stdOrthonormalBasis ℝ E i)) ^ 2 := by sorry

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