Dist Laplacian dist Of Function norm zpow
Space.distLaplacian_distOfFunction_norm_zpow
Project documentation
A power series which is differentiable everywhere, and in the limit as n → ∞ tends to ‖x‖. -/ def normPowerSeries {d} : ℕ → Space d → ℝ := fun n x => √(‖x‖ ^ 2 + 1/(n + 1)) lemma normPowerSeries_eq (n : ℕ) : normPowerSeries (d := d) n = fun x => √(‖x‖ ^ 2 + 1/(n + 1)) := rfl lemma normPowerSeries_eq_rpow {d} (n : ℕ) : normPowerSeries (d := d) n = fun...
Exact Lean statement
lemma distLaplacian_distOfFunction_norm_zpow {d : ℕ} [NeZero d] (m : ℤ)
(hdiv : 0 < m - 2 + (d : ℤ)) :
Δᵈ (distOfFunction (fun x : Space d => ‖x‖ ^ m)
(IsDistBounded.pow m (by omega))) =
(((m : ℝ) * (((m - 2 + d : ℤ) : ℝ))) •
distOfFunction (fun x : Space d => ‖x‖ ^ (m - 2))
(IsDistBounded.pow (m - 2) (by omega)))Formal artifact
Lean source
lemma distLaplacian_distOfFunction_norm_zpow {d : ℕ} [NeZero d] (m : ℤ) (hdiv : 0 < m - 2 + (d : ℤ)) : Δᵈ (distOfFunction (fun x : Space d => ‖x‖ ^ m) (IsDistBounded.pow m (by omega))) = (((m : ℝ) * (((m - 2 + d : ℤ) : ℝ))) • distOfFunction (fun x : Space d => ‖x‖ ^ (m - 2)) (IsDistBounded.pow (m - 2) (by omega))) := by rw [distLaplacian, LinearMap.comp_apply, distGrad_distOfFunction_norm_zpow m (by omega)] have hdist : distOfFunction (fun x : Space d => (m * ‖x‖ ^ (m - 2)) • basis.repr x) (by simp [← smul_smul] refine IsDistBounded.const_fun_smul ?_ ↑m apply IsDistBounded.zpow_smul_repr_self omega) = (m : ℝ) • distOfFunction (fun x : Space d => ‖x‖ ^ (m - 2) • basis.repr x) (IsDistBounded.zpow_smul_repr_self (m - 2) (by omega)) := by convert distOfFunction_smul_fun (fun x : Space d => ‖x‖ ^ (m - 2) • basis.repr x) (IsDistBounded.zpow_smul_repr_self (m - 2) (by omega)) (m : ℝ) using 1 ext x simp [smul_smul] rw [hdist, map_smul, distDiv_norm_zpow_smul_repr_self_eq_smul (m - 2) hdiv, smul_smul]- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/SpaceAndTime/Space/Norm/Basic.lean:916-939
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