F functional equation
F_functional_equation
Project documentation
Reduce a power of I * w via I ^ 4 = 1. -/ lemma I_mul_npow (w : ℂ) (n : ℕ) : (I * w) ^ n = I ^ (n % 4) * w ^ n := by rw [mul_pow, I_pow_eq_pow_mod] /- Functional equation of
Exact Lean statement
theorem F_functional_equation (z : ℍ) :
F (S • z) = z ^ 12 * F z - 12 * I * π ^ (-1 : ℤ) * z ^ 11 * (F₁ * E₄.toFun) z
- 36 * π ^ (-2 : ℤ) * z ^ 10 * (E₄.toFun z) ^ 2Formal artifact
Lean source
theorem F_functional_equation (z : ℍ) : F (S • z) = z ^ 12 * F z - 12 * I * π ^ (-1 : ℤ) * z ^ 11 * (F₁ * E₄.toFun) z - 36 * π ^ (-2 : ℤ) * z ^ 10 * (E₄.toFun z) ^ 2 := by simp only [F, F₁, Pi.pow_apply, Pi.mul_apply, Pi.sub_apply, ModularForm.toFun_eq_coe, E₂_S_transform, E₄_S_transform, E₆_S_transform, zpow_neg, zpow_one] field_simp [ne_zero z] linear_combination (12 * I * (z : ℂ) * π * (E₂ z * E₄ z - E₆ z) * E₄ z + 36 * E₄ z ^ 2) * I_sq- Project
- Sphere Packing in Dimension 8
- License
- Apache-2.0
- Commit
- acfc6204e65a
- Source
- SpherePacking/ModularForms/FG.lean:1034-1040
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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.
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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.
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closedBall_center_subset_upperHalfPlane
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Source project: Sphere Packing in Dimension 8
Person-level attribution pending.