`BookProof.ChapterSolidHarmonicTools.laplacian_normSqPow` (m : ℕ) (x : E) : (Δ fun y : E => (‖y‖ ^ 2) ^ m) x = (4 * m * ((m : ℝ) - 1) + 2 * (Module.finrank ℝ E) * m) * (‖x‖ ^ 2) ^

Proved

BookProof.ChapterSolidHarmonicTools.laplacian_normSqPow

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSolidHarmonicTools.

BookProof.ChapterSolidHarmonicTools.laplacian_normSqPow (m : ℕ) (x : E) : (Δ fun y : E => (‖y‖ ^ 2) ^ m) x = (4 * m * ((m : ℝ) - 1) + 2 * (Module.finrank ℝ E) * m) * (‖x‖ ^ 2) ^ (m - 1)

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

Preamble

-- Generated from ChapterSolidHarmonicTools.lean — theorem BookProof.ChapterSolidHarmonicTools.laplacian_normSqPow
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_normSqPow (m : ℕ) (x : E) :
    (Δ fun y : E => (‖y‖ ^ 2) ^ m) x
      = (4 * m * ((m : ℝ) - 1) + 2 * (Module.finrank ℝ E) * m) * (‖x‖ ^ 2) ^ (m - 1) := by sorry

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