`BookProof.ChapterSolidHarmonic.radialFactor_eval` (e : E) {l μ : ℕ} (hμ : μ ≤ l) {x : E} (hx : x ≠ 0) : radialFactor e l μ x = ‖x‖ ^ (l - μ) * (derivative^[μ] (legendre l)).eval (

Proved

BookProof.ChapterSolidHarmonic.radialFactor_eval

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSolidHarmonic.

BookProof.ChapterSolidHarmonic.radialFactor_eval (e : E) {l μ : ℕ} (hμ : μ ≤ l) {x : E} (hx : x ≠ 0) : radialFactor e l μ x = ‖x‖ ^ (l - μ) * (derivative^[μ] (legendre l)).eval (⟪e, x⟫_ℝ / ‖x‖)

Formalization note: Lean 4 identifier BookProof.ChapterSolidHarmonic.radialFactor_eval.

Preamble

-- Generated from ChapterSolidHarmonic.lean — theorem BookProof.ChapterSolidHarmonic.radialFactor_eval
import Definitions.Def_ChapterRadialLaplacian
import Definitions.Def_ChapterLaplacianProduct
import Definitions.Def_ChapterSolidHarmonicTools
import Mathlib
import Definitions.Def_ChapterSolidHarmonic
import Definitions.Def_ChapterLegendrePolynomial
open BookProof.ChapterLegendrePolynomial
open BookProof.ChapterSolidHarmonic



open Laplacian InnerProductSpace Polynomial
open BookProof.ChapterRadialLaplacian BookProof.ChapterLaplacianProduct
open BookProof.ChapterSolidHarmonicTools BookProof.ChapterLegendrePolynomial
open scoped RealInnerProductSpace

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

Formal statement

theorem BookProof.ChapterSolidHarmonic.radialFactor_eval (e : E) {l μ : ℕ} (hμ : μ ≤ l) {x : E} (hx : x ≠ 0) :
    radialFactor e l μ x
      = ‖x‖ ^ (l - μ) * (derivative^[μ] (legendre l)).eval (⟪e, x⟫_ℝ / ‖x‖) := by sorry

View graph