Plain-language statement
Derivative of the Möbius transformation: d/dz[(az+b)/(cz+d)] = 1/(cz+d)². Uses det(γ) = 1: a(cz+d) - c(az+b) = ad - bc = 1.
Exact Lean statement
lemma deriv_moebius (z : ℍ) :
deriv (fun w => num γ w / denom γ w) z = 1 / (denom γ z) ^ 2Formal artifact
Lean source
lemma deriv_moebius (z : ℍ) : deriv (fun w => num γ w / denom γ w) z = 1 / (denom γ z) ^ 2 := by have hz : denom γ z ≠ 0 := UpperHalfPlane.denom_ne_zero γ z have hdet : ((γ : Matrix (Fin 2) (Fin 2) ℤ) 0 0 : ℂ) * (γ 1 1) - ((γ : Matrix (Fin 2) (Fin 2) ℤ) 0 1 : ℂ) * (γ 1 0) = 1 := by have := Matrix.SpecialLinearGroup.det_coe γ simp only [Matrix.det_fin_two, ← Int.cast_mul, ← Int.cast_sub] at this ⊢ exact_mod_cast this rw [deriv_fun_div (differentiableAt_num γ z) (differentiableAt_denom γ z) hz, deriv_num, deriv_denom] simp only [denom_apply, num, Matrix.SpecialLinearGroup.coe_GL_coe_matrix, Matrix.SpecialLinearGroup.map_apply_coe, RingHom.mapMatrix_apply, Int.coe_castRingHom, Matrix.map_apply, ofReal_intCast] at * have hnum_eq : ((γ 0 0 : ℤ) : ℂ) * ((γ 1 0 : ℤ) * z + (γ 1 1 : ℤ)) - ((γ 0 0 : ℤ) * z + (γ 0 1 : ℤ)) * (γ 1 0 : ℤ) = 1 := by linear_combination hdet simp only [hnum_eq, one_div]- Project
- Sphere Packing in Dimension 8
- License
- Apache-2.0
- Commit
- acfc6204e65a
- Source
- SpherePacking/ModularForms/Derivative.lean:495-510
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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.