G functional equation
G_functional_equation'
Plain-language statement
Functional equation of restricted to the imaginary axis.
Exact Lean statement
theorem G_functional_equation' {t : ℝ} (ht : 0 < t) :
GReal (1 / t) = t ^ 10 * H₄.resToImagAxis t ^ 3
* (2 * H₄.resToImagAxis t ^ 2 + 5 * H₂.resToImagAxis t * H₄.resToImagAxis t
+ 5 * H₂.resToImagAxis t ^ 2)Formal artifact
Lean source
theorem G_functional_equation' {t : ℝ} (ht : 0 < t) : GReal (1 / t) = t ^ 10 * H₄.resToImagAxis t ^ 3 * (2 * H₄.resToImagAxis t ^ 2 + 5 * H₂.resToImagAxis t * H₄.resToImagAxis t + 5 * H₂.resToImagAxis t ^ 2) := by let z : ℍ := ⟨I * t, by simp [ht]⟩ rw [← G_eq_GReal (1 / t), ResToImagAxis.one_div_eq_S_smul G ht, G_functional_equation z, show (z : ℂ) = I * t from rfl, I_mul_npow] simp only [Nat.reduceMod, I_sq, z, ResToImagAxis.I_mul_t_eq H₂ t ht, ResToImagAxis.I_mul_t_eq H₄ t ht] ring- Project
- Sphere Packing in Dimension 8
- License
- Apache-2.0
- Commit
- acfc6204e65a
- Source
- SpherePacking/ModularForms/FG.lean:1069-1078
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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.