Tendsto rpow mul of is Big O exp
tendsto_rpow_mul_of_isBigO_exp
Project documentation
The analytic kernel: if g : ℝ → ℂ is eventually bounded by C * exp(-b * t) for some b > 0, then t^s * g(t) → 0 as t → ∞ for any real power s. This follows from the fact that t^s * exp(-b * t) → 0 (mathlib's tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero) combined with the big-O transfer lemma.
Exact Lean statement
lemma tendsto_rpow_mul_of_isBigO_exp {g : ℝ → ℂ} {s b : ℝ} (hb : 0 < b)
(hg : g =O[atTop] fun t => rexp (-b * t)) :
Tendsto (fun t : ℝ => (t : ℂ) ^ (s : ℂ) * g t) atTop (𝓝 0)Formal artifact
Lean source
lemma tendsto_rpow_mul_of_isBigO_exp {g : ℝ → ℂ} {s b : ℝ} (hb : 0 < b) (hg : g =O[atTop] fun t => rexp (-b * t)) : Tendsto (fun t : ℝ => (t : ℂ) ^ (s : ℂ) * g t) atTop (𝓝 0) := by refine ((isBigO_refl _ _).mul (Complex.isBigO_ofReal_right.mpr hg)).trans_tendsto ?_ refine (tendsto_ofReal_iff.mpr (tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero s b hb)).congr' ?_ filter_upwards [eventually_gt_atTop 0] with t ht rw [Complex.ofReal_mul, Complex.ofReal_cpow (le_of_lt ht)]- Project
- Sphere Packing in Dimension 8
- License
- Apache-2.0
- Commit
- acfc6204e65a
- Source
- SpherePacking/ModularForms/ResToImagAxis.lean:405-411
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Person-level attribution pending.