Project-declaredLean 4.31.0 · mathlib@fabf563a7c95

D qexp term

D_qexp_term

sphere packing Fourier analysis modular forms derivative

Plain-language statement

Lemma 6.45 (Blueprint): The normalized derivative $D$ acts as $q \frac{d}{dq}$ on $q$ -series. For a single q-power term: D(a·qⁿ) = n·a·qⁿ where q = exp(2πiz) and n ∈ ℤ. The key calculation: - d/dz(exp(2πinz)) = 2πin·exp(2πinz) - D(exp(2πinz)) = (2πi)⁻¹·(2πin·exp(2πinz)) = n·exp(2πinz)

Exact Lean statement

theorem D_qexp_term (n : ℤ) (a : ℂ) (z : ℍ) :
    D (fun w => a * cexp (2 * π * I * n * w)) z =
      n * a * cexp (2 * π * I * n * z)

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

theorem D_qexp_term (n : ℤ) (a : ℂ) (z : ℍ) :    D (fun w => a * cexp (2 * π * I * n * w)) z =      n * a * cexp (2 * π * I * n * z) := by  simp only [D]  have h_agree : ((fun w : ℍ => a * cexp (2 * π * I * n * w)) ∘ ofComplex) =ᶠ[nhds (z : ℂ)]      (fun w : ℂ => a * cexp (2 * π * I * n * w)) := by    filter_upwards [isOpen_upperHalfPlaneSet.mem_nhds z.2] with w hw    simp only [Function.comp_apply, ofComplex_apply_of_im_pos hw, UpperHalfPlane.coe_mk]  rw [h_agree.deriv_eq, (hasDerivAt_qexp a n z).deriv]  field_simp [two_pi_I_ne_zero]

Project: Sphere Packing in Dimension 8
License: Apache-2.0
Commit: acfc6204e65a
Source: SpherePacking/ModularForms/Derivative.lean:261-270

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Project-declaredLean 4.31.0

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sphere packingFourier analysismodular forms

Source project: Sphere Packing in Dimension 8

Person-level attribution pending.

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Project-declaredLean 4.31.0

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Let $F : \mathbb{H} \to \mathbb{C}$ be a holomorphic function where $F(it)$ is real for all $t > 0$ . Assume that Serre derivative $\partial_k F$ is positive on the imaginary axis. If $F(it)$ is positive for sufficiently large $t$ , then $F(it)$ is positive for all $t > 0$ .

sphere packingFourier analysismodular forms

Source project: Sphere Packing in Dimension 8

Person-level attribution pending.

View proof record

Project-declaredLean 4.31.0

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.

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Source project: Sphere Packing in Dimension 8

Person-level attribution pending.

View proof record