Project-declaredLean 4.31.0 · mathlib@fabf563a7c95

D real of real

D_real_of_real

sphere packing Fourier analysis modular forms derivative

Plain-language statement

If F is real on the imaginary axis and MDifferentiable, then D F is also real on the imaginary axis.

Exact Lean statement

@[fun_prop]
theorem D_real_of_real {F : ℍ → ℂ} (hF_real : ResToImagAxis.Real F)
    (hF_diff : MDiff F) : ResToImagAxis.Real (D F)

Formal artifact

Lean source

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Full Lean sourceLean 4

@[fun_prop]theorem D_real_of_real {F : ℍ → ℂ} (hF_real : ResToImagAxis.Real F)    (hF_diff : MDiff F) : ResToImagAxis.Real (D F) := fun t ht => by  have him : ∀ s, (F.resToImagAxis s).im = 0 := fun s => by    by_cases hs : 0 < s    · exact hF_real s hs    · simp [ResToImagAxis, hs]  have h_im_deriv :=    im_deriv_eq_zero_of_im_eq_zero (ResToImagAxis.Differentiable F hF_diff t ht) him  have h_im_eq : (deriv F.resToImagAxis t).im = -2 * π * ((D F).resToImagAxis t).im := by    simpa [mul_assoc, ofReal_mul] using congrArg Complex.im (deriv_resToImagAxis_eq F hF_diff ht)  exact (mul_eq_zero.mp (h_im_deriv ▸ h_im_eq).symm).resolve_left    (mul_ne_zero (by norm_num) Real.pi_ne_zero)

Project: Sphere Packing in Dimension 8
License: Apache-2.0
Commit: acfc6204e65a
Source: SpherePacking/ModularForms/Derivative.lean:766-778

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Project-declaredLean 4.31.0

Anti Der Pos

antiDerPos

Plain-language statement

If $F$ is a modular form where $F(it)$ is positive for sufficiently large $t$ (i.e. constant term is positive) and the derivative is positive, then $F$ is also positive.

sphere packingFourier analysismodular forms

Source project: Sphere Packing in Dimension 8

Person-level attribution pending.

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Project-declaredLean 4.31.0

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.

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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.

sphere packingFourier analysismodular forms

Source project: Sphere Packing in Dimension 8

Person-level attribution pending.

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Anti Der Pos

Anti Serre Der Pos

Closed Ball center subset upper Half Plane