Project-declaredLean 4.31.0 · mathlib@fabf563a7c95

Serre D tendsto of tendsto

serre_D_tendsto_of_tendsto

sphere packing Fourier analysis modular forms eisenstein asymptotics

Project documentation

General limit: if f → c at i∞ and f is holomorphic and bounded, then serre_D k f → -k*c/12. This is the continuous mapping theorem applied to serre_D k f = D f - (k/12) * E₂ * f: - D f → 0 (Cauchy estimate from boundedness) - E₂ → 1 - f → c Therefore serre_D k f → 0 - (k/12) * 1 * c = -k*c/12.

Exact Lean statement

lemma serre_D_tendsto_of_tendsto (k : ℤ) (f : ℍ → ℂ) (c : ℂ)
    (hf_holo : MDiff f) (hf_bdd : IsBoundedAtImInfty f)
    (hf_lim : Filter.Tendsto f atImInfty (nhds c)) :
    Filter.Tendsto (serre_D k f) atImInfty (nhds (-(k : ℂ) * c / 12))

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma serre_D_tendsto_of_tendsto (k : ℤ) (f : ℍ → ℂ) (c : ℂ)    (hf_holo : MDiff f) (hf_bdd : IsBoundedAtImInfty f)    (hf_lim : Filter.Tendsto f atImInfty (nhds c)) :    Filter.Tendsto (serre_D k f) atImInfty (nhds (-(k : ℂ) * c / 12)) := by  rw [show serre_D k f = fun z => D f z - (k : ℂ) * 12⁻¹ * E₂ z * f z from serre_D_eq k f]  have hD := D_tendsto_zero_of_isBoundedAtImInfty hf_holo hf_bdd  have hprod := E₂_tendsto_one_atImInfty.mul hf_lim  have hlim : (0 : ℂ) - (k : ℂ) * 12⁻¹ * 1 * c = -(k : ℂ) * c / 12 := by ring  rw [← hlim]  refine hD.sub ?_  have hconst : Filter.Tendsto (fun _ : ℍ => (k : ℂ) * 12⁻¹)      atImInfty (nhds ((k : ℂ) * 12⁻¹)) := tendsto_const_nhds  convert hconst.mul hprod using 1 <;> ring_nf

Project: Sphere Packing in Dimension 8
License: Apache-2.0
Commit: acfc6204e65a
Source: SpherePacking/ModularForms/EisensteinAsymptotics.lean:160-172

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