Project-declaredLean 4.31.0 · mathlib@fabf563a7c95

F₄ tendsto at Im Infty

f₄_tendsto_atImInfty

sphere packing Fourier analysis modular forms derivative

Plain-language statement

f₄ tends to 0 at infinity. Proof: f₄ = serre_D 2 H₄ + (1/6)H₄(2H₂ + H₄) serre_D 2 H₄ = D H₄ - (1/6)E₂ H₄ → 0 - (1/6)11 = -1/6 (since H₄ → 1, E₂ → 1) H₄(2H₂ + H₄) → 1*(0 + 1) = 1 So f₄ → -1/6 + (1/6)*1 = 0.

Exact Lean statement

lemma f₄_tendsto_atImInfty : Tendsto f₄ atImInfty (𝓝 0)

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma f₄_tendsto_atImInfty : Tendsto f₄ atImInfty (𝓝 0) := by  have h_serre_H₄ : Tendsto (serre_D 2 H₄) atImInfty (𝓝 (-(1/6 : ℂ))) := by    simpa [show -(2 : ℂ) / 12 = -(1 / 6 : ℂ) by norm_num] using      serre_D_tendsto_neg_k_div_12 2 H₄ H₄_SIF_MDifferentiable isBoundedAtImInfty_H₄        H₄_tendsto_atImInfty  have h_scaled : Tendsto (fun z => (1/6 : ℂ) * (H₄ z * (2 * H₂ z + H₄ z)))      atImInfty (𝓝 (1/6 : ℂ)) := by    have := H₂_tendsto_atImInfty    have := H₄_tendsto_atImInfty    tendsto_cont  change Tendsto    (fun z => serre_D 2 H₄ z + (1 / 6 : ℂ) * (H₄ z * (2 * H₂ z + H₄ z)))    atImInfty (𝓝 0)  simpa using h_serre_H₄.add h_scaled

Project: Sphere Packing in Dimension 8
License: Apache-2.0
Commit: acfc6204e65a
Source: SpherePacking/ModularForms/JacobiTheta/Derivative.lean:474-487

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