File: Mathlib/NumberTheory/Harmonic/GammaDeriv.lean

full_name: _aux_Mathlib_NumberTheory_Harmonic_GammaDeriv___unexpand_Real_eulerMascheroniConstant_1
code: local notation "γ" => Real.eulerMascheroniConstant
goal: Lean.PrettyPrinter.Unexpander

full_name: _aux_Mathlib_NumberTheory_Harmonic_GammaDeriv___macroRules_termγ_1
code: local notation "γ" => Real.eulerMascheroniConstant
goal: Lean.Macro

full_name: Real.deriv_Gamma_nat
code: /-- Explicit formula for the derivative of the Gamma function at positive integers, in terms of
harmonic numbers and the Euler-Mascheroni constant `γ`. -/
lemma deriv_Gamma_nat (n : ℕ) :
    deriv Gamma (n + 1) = n ! * (-γ + harmonic n) 
goal: deriv Real.Gamma (↑n + 1) = ↑n ! * (-γ + ↑(harmonic n))

full_name: Real.hasDerivAt_Gamma_nat
code: lemma hasDerivAt_Gamma_nat (n : ℕ) :
    HasDerivAt Gamma (n ! * (-γ + harmonic n)) (n + 1) 
goal: HasDerivAt Real.Gamma (↑n ! * (-γ + ↑(harmonic n))) (↑n + 1)

full_name: Real.eulerMascheroniConstant_eq_neg_deriv
code: lemma eulerMascheroniConstant_eq_neg_deriv : γ = -deriv Gamma 1 
goal: γ = -deriv Real.Gamma 1

full_name: Real.hasDerivAt_Gamma_one
code: lemma hasDerivAt_Gamma_one : HasDerivAt Gamma (-γ) 1 
goal: HasDerivAt Real.Gamma (-γ) 1

full_name: Real.hasDerivAt_Gamma_one_half
code: lemma hasDerivAt_Gamma_one_half : HasDerivAt Gamma (-√π * (γ + 2 * log 2)) (1 / 2) 
goal: HasDerivAt Real.Gamma (-√Real.pi * (γ + 2 * Real.log 2)) (1 / 2)

full_name: _private.Mathlib.NumberTheory.Harmonic.GammaDeriv.0.Complex.HasDerivAt.complex_of_real
code: private lemma HasDerivAt.complex_of_real {f : ℂ → ℂ} {g : ℝ → ℝ} {g' s : ℝ}
    (hf : DifferentiableAt ℂ f s) (hg : HasDerivAt g g' s) (hfg : ∀ s : ℝ, f ↑s = ↑(g s)) :
    HasDerivAt f ↑g' s 
goal: HasDerivAt f ↑g' ↑s

full_name: Complex.differentiable_at_Gamma_nat_add_one
code: lemma differentiable_at_Gamma_nat_add_one (n : ℕ) :
    DifferentiableAt ℂ Gamma (n + 1) 
goal: DifferentiableAt ℂ Complex.Gamma (↑n + 1)

full_name: Complex.hasDerivAt_Gamma_nat
code: lemma hasDerivAt_Gamma_nat (n : ℕ) :
    HasDerivAt Gamma (n ! * (-γ + harmonic n)) (n + 1) 
goal: HasDerivAt Complex.Gamma (↑n ! * (-↑γ + ↑(harmonic n))) (↑n + 1)

full_name: Complex.deriv_Gamma_nat
code: /-- Explicit formula for the derivative of the complex Gamma function at positive integers, in
terms of harmonic numbers and the Euler-Mascheroni constant `γ`. -/
lemma deriv_Gamma_nat (n : ℕ) :
    deriv Gamma (n + 1) = n ! * (-γ + harmonic n) 
goal: deriv Complex.Gamma (↑n + 1) = ↑n ! * (-↑γ + ↑(harmonic n))

full_name: Complex.hasDerivAt_Gamma_one
code: lemma hasDerivAt_Gamma_one : HasDerivAt Gamma (-γ) 1 
goal: HasDerivAt Complex.Gamma (-↑γ) 1

full_name: Complex.hasDerivAt_Gamma_one_half
code: lemma hasDerivAt_Gamma_one_half : HasDerivAt Gamma (-√π * (γ + 2 * log 2)) (1 / 2) 
goal: HasDerivAt Complex.Gamma (-↑√Real.pi * (↑γ + 2 * Complex.log 2)) (1 / 2)

full_name: Complex.hasDerivAt_Gammaℂ_one
code: lemma hasDerivAt_Gammaℂ_one : HasDerivAt Gammaℂ (-(γ + log (2 * π)) / π) 1 
goal: HasDerivAt Complex.Gammaℂ (-(↑γ + Complex.log (2 * ↑Real.pi)) / ↑Real.pi) 1

full_name: Complex.hasDerivAt_Gammaℝ_one
code: lemma hasDerivAt_Gammaℝ_one : HasDerivAt Gammaℝ (-(γ + log (4 * π)) / 2) 1 
goal: HasDerivAt Complex.Gammaℝ (-(↑γ + Complex.log (4 * ↑Real.pi)) / 2) 1