Plain-language statement
Derivative of denom^(-k): d/dz[(cz+d)^(-k)] = -k * c * (cz+d)^(-k-1).
Exact Lean statement
lemma deriv_denom_zpow (k : ℤ) (z : ℍ) :
deriv (fun w => (denom γ w) ^ (-k)) z =
(-k : ℂ) * ((γ : Matrix (Fin 2) (Fin 2) ℤ) 1 0 : ℂ) * (denom γ z) ^ (-k - 1)Formal artifact
Lean source
lemma deriv_denom_zpow (k : ℤ) (z : ℍ) : deriv (fun w => (denom γ w) ^ (-k)) z = (-k : ℂ) * ((γ : Matrix (Fin 2) (Fin 2) ℤ) 1 0 : ℂ) * (denom γ z) ^ (-k - 1) := by have hz : denom γ z ≠ 0 := UpperHalfPlane.denom_ne_zero γ z have hdiff := differentiableAt_denom γ (z : ℂ) have hderiv_zpow := hasDerivAt_zpow (-k) (denom γ z) (Or.inl hz) have hderiv_denom : HasDerivAt (fun w => denom γ w) ((γ : Matrix (Fin 2) (Fin 2) ℤ) 1 0 : ℂ) (z : ℂ) := by rw [← deriv_denom]; exact hdiff.hasDerivAt have hcomp := hderiv_zpow.comp (z : ℂ) hderiv_denom have heq : (fun w => w ^ (-k)) ∘ (fun w => denom γ w) = (fun w => (denom γ w) ^ (-k)) := rfl rw [← heq, hcomp.deriv]; simp only [Int.cast_neg]; ring- Project
- Sphere Packing in Dimension 8
- License
- Apache-2.0
- Commit
- acfc6204e65a
- Source
- SpherePacking/ModularForms/Derivative.lean:513-524
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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.