Plain-language statement
G / q^(3/2) → 20480 as im(z) → ∞. Here q^(3/2) = exp(2πi · (3/2) · z).
Exact Lean statement
theorem G_vanishing_order :
Tendsto (fun z : ℍ ↦ G z / cexp (2 * π * I * (3/2) * z)) atImInfty (nhds 20480)Formal artifact
Lean source
theorem G_vanishing_order : Tendsto (fun z : ℍ ↦ G z / cexp (2 * π * I * (3/2) * z)) atImInfty (nhds 20480) := by simp only [show ∀ z : ℍ, cexp (2 * π * I * (3 / 2) * z) = cexp (3 * π * I * z) from fun z => by ring_nf] have h_exp_pow : ∀ z : ℍ, cexp (π * I * z) ^ 3 = cexp (3 * π * I * z) := fun z => by simp only [← Complex.exp_nat_mul]; ring_nf have h_eq : ∀ z : ℍ, G z / cexp (3 * π * I * z) = (H₂ z / cexp (π * I * z)) ^ 3 * (2 * H₂ z ^ 2 + 5 * H₂ z * H₄ z + 5 * H₄ z ^ 2) := fun z => by simp only [G, Pi.mul_apply, Pi.pow_apply, Pi.add_apply, Pi.smul_apply, Complex.real_smul, div_pow, h_exp_pow] push_cast field_simp [Complex.exp_ne_zero] simp_rw [h_eq] have h_poly : Filter.Tendsto (fun z : ℍ ↦ 2 * H₂ z ^ 2 + 5 * H₂ z * H₄ z + 5 * H₄ z ^ 2) atImInfty (nhds 5) := by tendsto_cont [H₂_tendsto_atImInfty, H₄_tendsto_atImInfty] convert (H₂_div_exp_tendsto.pow 3).mul h_poly norm_num- Project
- Sphere Packing in Dimension 8
- License
- Apache-2.0
- Commit
- acfc6204e65a
- Source
- SpherePacking/ModularForms/FG.lean:885-902
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Related declarations
Anti Der Pos
antiDerPos
Plain-language statement
If is a modular form where is positive for sufficiently large (i.e. constant term is positive) and the derivative is positive, then is also positive.
Source project: Sphere Packing in Dimension 8
Person-level attribution pending.
Anti Serre Der Pos
antiSerreDerPos
Plain-language statement
Let be a holomorphic function where is real for all . Assume that Serre derivative is positive on the imaginary axis. If is positive for sufficiently large , then is positive for all .
Source project: Sphere Packing in Dimension 8
Person-level attribution pending.
Closed Ball center subset upper Half Plane
closedBall_center_subset_upperHalfPlane
Plain-language statement
Closed ball centered at z with radius z.im/2 is contained in the upper half plane.
Source project: Sphere Packing in Dimension 8
Person-level attribution pending.