File: Mathlib/Analysis/SpecialFunctions/Gamma/Deligne.lean

full_name: Complex.Gammaℝ
code: /-- Deligne's archimedean Gamma factor for a real infinite place.

See "Valeurs de fonctions L et periodes d'integrales" § 5.3. Note that this is not the same as
`Real.Gamma`; in particular it is a function `ℂ → ℂ`. -/
noncomputable def Gammaℝ (s : ℂ) 
goal: ℂ

full_name: Complex.Gammaℝ_def
code: lemma Gammaℝ_def (s : ℂ) : Gammaℝ s = π ^ (-s / 2) * Gamma (s / 2) 
goal: s.Gammaℝ = ↑π ^ (-s / 2) * Complex.Gamma (s / 2)

full_name: Complex.Gammaℂ
code: /-- Deligne's archimedean Gamma factor for a complex infinite place.

See "Valeurs de fonctions L et periodes d'integrales" § 5.3. (Some authors omit the factor of 2).
Note that this is not the same as `Complex.Gamma`. -/
noncomputable def Gammaℂ (s : ℂ) 
goal: ℂ

full_name: Complex.Gammaℂ_def
code: lemma Gammaℂ_def (s : ℂ) : Gammaℂ s = 2 * (2 * π) ^ (-s) * Gamma s 
goal: s.Gammaℂ = 2 * (2 * ↑π) ^ (-s) * Complex.Gamma s

full_name: Complex.Gammaℝ_add_two
code: lemma Gammaℝ_add_two {s : ℂ} (hs : s ≠ 0) : Gammaℝ (s + 2) = Gammaℝ s * s / 2 / π 
goal: (s + 2).Gammaℝ = s.Gammaℝ * s / 2 / ↑π

full_name: Complex.Gammaℂ_add_one
code: lemma Gammaℂ_add_one {s : ℂ} (hs : s ≠ 0) : Gammaℂ (s + 1) = Gammaℂ s * s / 2 / π 
goal: (s + 1).Gammaℂ = s.Gammaℂ * s / 2 / ↑π

full_name: Complex.Gammaℝ_ne_zero_of_re_pos
code: lemma Gammaℝ_ne_zero_of_re_pos {s : ℂ} (hs : 0 < re s) : Gammaℝ s ≠ 0 
goal: s.Gammaℝ ≠ 0

full_name: Complex.Gammaℝ_eq_zero_iff
code: lemma Gammaℝ_eq_zero_iff {s : ℂ} : Gammaℝ s = 0 ↔ ∃ n : ℕ, s = -(2 * n) 
goal: s.Gammaℝ = 0 ↔ ∃ n, s = -(2 * ↑n)

full_name: Complex.Gammaℝ_one
code: @[simp]
lemma Gammaℝ_one : Gammaℝ 1 = 1 
goal: Complex.Gammaℝ 1 = 1

full_name: Complex.Gammaℂ_one
code: @[simp]
lemma Gammaℂ_one : Gammaℂ 1 = 1 / π 
goal: Complex.Gammaℂ 1 = 1 / ↑π

full_name: Complex.differentiable_Gammaℝ_inv
code: lemma differentiable_Gammaℝ_inv : Differentiable ℂ (fun s ↦ (Gammaℝ s)⁻¹) 
goal: Differentiable ℂ fun s => s.Gammaℝ⁻¹

full_name: Complex.Gammaℝ_residue_zero
code: lemma Gammaℝ_residue_zero : Tendsto (fun s ↦ s * Gammaℝ s) (𝓝[≠] 0) (𝓝 2) 
goal: Filter.Tendsto (fun s => s * s.Gammaℝ) (𝓝[≠] 0) (𝓝 2)

full_name: Complex.Gammaℝ_mul_Gammaℝ_add_one
code: /-- Reformulation of the doubling formula in terms of `Gammaℝ`. -/
lemma Gammaℝ_mul_Gammaℝ_add_one (s : ℂ) : Gammaℝ s * Gammaℝ (s + 1) = Gammaℂ s 
goal: s.Gammaℝ * (s + 1).Gammaℝ = s.Gammaℂ

full_name: Complex.Gammaℝ_one_sub_mul_Gammaℝ_one_add
code: /-- Reformulation of the reflection formula in terms of `Gammaℝ`. -/
lemma Gammaℝ_one_sub_mul_Gammaℝ_one_add (s : ℂ) :
    Gammaℝ (1 - s) * Gammaℝ (1 + s) = (cos (π * s / 2))⁻¹ 
goal: (1 - s).Gammaℝ * (1 + s).Gammaℝ = (Complex.cos (↑π * s / 2))⁻¹

full_name: Complex.Gammaℝ_div_Gammaℝ_one_sub
code: /-- Another formulation of the reflection formula in terms of `Gammaℝ`. -/
lemma Gammaℝ_div_Gammaℝ_one_sub {s : ℂ} (hs : ∀ (n : ℕ), s ≠ -(2 * n + 1)) :
    Gammaℝ s / Gammaℝ (1 - s) = Gammaℂ s * cos (π * s / 2) 
goal: s.Gammaℝ / (1 - s).Gammaℝ = s.Gammaℂ * Complex.cos (↑π * s / 2)

full_name: Complex.inv_Gammaℝ_one_sub
code: /-- Formulation of reflection formula tailored to functional equations of L-functions of even
Dirichlet characters (including Riemann zeta). -/
lemma inv_Gammaℝ_one_sub {s : ℂ} (hs : ∀ (n : ℕ), s ≠ -n) :
    (Gammaℝ (1 - s))⁻¹ = Gammaℂ s * cos (π * s / 2) * (Gammaℝ s)⁻¹ 
goal: (1 - s).Gammaℝ⁻¹ = s.Gammaℂ * Complex.cos (↑π * s / 2) * s.Gammaℝ⁻¹

full_name: Complex.inv_Gammaℝ_two_sub
code: /-- Formulation of reflection formula tailored to functional equations of L-functions of odd
Dirichlet characters. -/
lemma inv_Gammaℝ_two_sub {s : ℂ} (hs : ∀ (n : ℕ), s ≠ -n) :
    (Gammaℝ (2 - s))⁻¹ = Gammaℂ s * sin (π * s / 2) * (Gammaℝ (s + 1))⁻¹ 
goal: (2 - s).Gammaℝ⁻¹ = s.Gammaℂ * Complex.sin (↑π * s / 2) * (s + 1).Gammaℝ⁻¹