`BookProof.ChapterBesselHarmonic.reduced_radial_sbessel` (l : ℕ) {p r : ℝ} (hp : p ≠ 0) (hr : r ≠ 0) : deriv (deriv fun s => sbessel l (p * s) / s ^ l) r + ((2 + 2 * (l : ℝ)) / r)

Proved

BookProof.ChapterBesselHarmonic.reduced_radial_sbessel

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterBesselHarmonic.

BookProof.ChapterBesselHarmonic.reduced_radial_sbessel (l : ℕ) {p r : ℝ} (hp : p ≠ 0) (hr : r ≠ 0) : deriv (deriv fun s => sbessel l (p * s) / s ^ l) r + ((2 + 2 * (l : ℝ)) / r) * deriv (fun s => sbessel l (p * s) / s ^ l) r = -(p ^ 2) * (sbessel l (p * r) / r ^ l)

Formalization note: Lean 4 identifier BookProof.ChapterBesselHarmonic.reduced_radial_sbessel.

Preamble

-- Generated from ChapterBesselHarmonic.lean — theorem BookProof.ChapterBesselHarmonic.reduced_radial_sbessel
import Definitions.Def_ChapterSphericalBesselODE
import Definitions.Def_ChapterRadialLaplacian
import Definitions.Def_ChapterLaplacianProduct
import Mathlib
import Definitions.Def_ChapterBesselHarmonic
import Definitions.Def_ChapterSphericalBessel
open BookProof.ChapterSphericalBessel
open BookProof.ChapterBesselHarmonic



open Filter Laplacian InnerProductSpace
open BookProof.ChapterSphericalBessel BookProof.ChapterSphericalBesselODE
open BookProof.ChapterRadialLaplacian BookProof.ChapterLaplacianProduct
open scoped InnerProductSpace RealInnerProductSpace

Formal statement

theorem BookProof.ChapterBesselHarmonic.reduced_radial_sbessel (l : ℕ) {p r : ℝ} (hp : p ≠ 0) (hr : r ≠ 0) :
    deriv (deriv fun s => sbessel l (p * s) / s ^ l) r
        + ((2 + 2 * (l : ℝ)) / r) * deriv (fun s => sbessel l (p * s) / s ^ l) r
      = -(p ^ 2) * (sbessel l (p * r) / r ^ l) := by sorry

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