File: Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean

full_name: exp_neg_mul_rpow_isLittleO_exp_neg
code: theorem exp_neg_mul_rpow_isLittleO_exp_neg {p b : ℝ} (hb : 0 < b) (hp : 1 < p) :
    (fun x : ℝ => exp (- b * x ^ p)) =o[atTop] fun x : ℝ => exp (-x) 
goal: (fun x => rexp (-b * x ^ p)) =o[Filter.atTop] fun x => rexp (-x)

full_name: exp_neg_mul_sq_isLittleO_exp_neg
code: theorem exp_neg_mul_sq_isLittleO_exp_neg {b : ℝ} (hb : 0 < b) :
    (fun x : ℝ => exp (-b * x ^ 2)) =o[atTop] fun x : ℝ => exp (-x) 
goal: (fun x => rexp (-b * x ^ 2)) =o[Filter.atTop] fun x => rexp (-x)

full_name: rpow_mul_exp_neg_mul_rpow_isLittleO_exp_neg
code: theorem rpow_mul_exp_neg_mul_rpow_isLittleO_exp_neg (s : ℝ) {b p : ℝ} (hp : 1 < p) (hb : 0 < b) :
    (fun x : ℝ => x ^ s * exp (- b * x ^ p)) =o[atTop] fun x : ℝ => exp (-(1 / 2) * x) 
goal: (fun x => x ^ s * rexp (-b * x ^ p)) =o[Filter.atTop] fun x => rexp (-(1 / 2) * x)

full_name: rpow_mul_exp_neg_mul_sq_isLittleO_exp_neg
code: theorem rpow_mul_exp_neg_mul_sq_isLittleO_exp_neg {b : ℝ} (hb : 0 < b) (s : ℝ) :
    (fun x : ℝ => x ^ s * exp (-b * x ^ 2)) =o[atTop] fun x : ℝ => exp (-(1 / 2) * x) 
goal: (fun x => x ^ s * rexp (-b * x ^ 2)) =o[Filter.atTop] fun x => rexp (-(1 / 2) * x)

full_name: integrableOn_rpow_mul_exp_neg_rpow
code: theorem integrableOn_rpow_mul_exp_neg_rpow {p s : ℝ} (hs : -1 < s) (hp : 1 ≤ p) :
    IntegrableOn (fun x : ℝ => x ^ s * exp (- x ^ p)) (Ioi 0) 
goal: MeasureTheory.IntegrableOn (fun x => x ^ s * rexp (-x ^ p)) (Set.Ioi 0) MeasureTheory.volume

full_name: integrableOn_rpow_mul_exp_neg_mul_rpow
code: theorem integrableOn_rpow_mul_exp_neg_mul_rpow {p s b : ℝ} (hs : -1 < s) (hp : 1 ≤ p) (hb : 0 < b) :
    IntegrableOn (fun x : ℝ => x ^ s * exp (- b * x ^ p)) (Ioi 0) 
goal: MeasureTheory.IntegrableOn (fun x => x ^ s * rexp (-b * x ^ p)) (Set.Ioi 0) MeasureTheory.volume

full_name: integrableOn_rpow_mul_exp_neg_mul_sq
code: theorem integrableOn_rpow_mul_exp_neg_mul_sq {b : ℝ} (hb : 0 < b) {s : ℝ} (hs : -1 < s) :
    IntegrableOn (fun x : ℝ => x ^ s * exp (-b * x ^ 2)) (Ioi 0) 
goal: MeasureTheory.IntegrableOn (fun x => x ^ s * rexp (-b * x ^ 2)) (Set.Ioi 0) MeasureTheory.volume

full_name: integrable_rpow_mul_exp_neg_mul_sq
code: theorem integrable_rpow_mul_exp_neg_mul_sq {b : ℝ} (hb : 0 < b) {s : ℝ} (hs : -1 < s) :
    Integrable fun x : ℝ => x ^ s * exp (-b * x ^ 2) 
goal: MeasureTheory.Integrable (fun x => x ^ s * rexp (-b * x ^ 2)) MeasureTheory.volume

full_name: integrable_exp_neg_mul_sq
code: theorem integrable_exp_neg_mul_sq {b : ℝ} (hb : 0 < b) :
    Integrable fun x : ℝ => exp (-b * x ^ 2) 
goal: MeasureTheory.Integrable (fun x => rexp (-b * x ^ 2)) MeasureTheory.volume

full_name: integrableOn_Ioi_exp_neg_mul_sq_iff
code: theorem integrableOn_Ioi_exp_neg_mul_sq_iff {b : ℝ} :
    IntegrableOn (fun x : ℝ => exp (-b * x ^ 2)) (Ioi 0) ↔ 0 < b 
goal: MeasureTheory.IntegrableOn (fun x => rexp (-b * x ^ 2)) (Set.Ioi 0) MeasureTheory.volume ↔ 0 < b

full_name: integrable_exp_neg_mul_sq_iff
code: theorem integrable_exp_neg_mul_sq_iff {b : ℝ} :
    (Integrable fun x : ℝ => exp (-b * x ^ 2)) ↔ 0 < b 
goal: MeasureTheory.Integrable (fun x => rexp (-b * x ^ 2)) MeasureTheory.volume ↔ 0 < b

full_name: integrable_mul_exp_neg_mul_sq
code: theorem integrable_mul_exp_neg_mul_sq {b : ℝ} (hb : 0 < b) :
    Integrable fun x : ℝ => x * exp (-b * x ^ 2) 
goal: MeasureTheory.Integrable (fun x => x * rexp (-b * x ^ 2)) MeasureTheory.volume

full_name: norm_cexp_neg_mul_sq
code: theorem norm_cexp_neg_mul_sq (b : ℂ) (x : ℝ) :
    ‖Complex.exp (-b * (x : ℂ) ^ 2)‖ = exp (-b.re * x ^ 2) 
goal: ‖cexp (-b * ↑x ^ 2)‖ = rexp (-b.re * x ^ 2)

full_name: integrable_cexp_neg_mul_sq
code: theorem integrable_cexp_neg_mul_sq {b : ℂ} (hb : 0 < b.re) :
    Integrable fun x : ℝ => cexp (-b * (x : ℂ) ^ 2) 
goal: MeasureTheory.Integrable (fun x => cexp (-b * ↑x ^ 2)) MeasureTheory.volume

full_name: integrable_mul_cexp_neg_mul_sq
code: theorem integrable_mul_cexp_neg_mul_sq {b : ℂ} (hb : 0 < b.re) :
    Integrable fun x : ℝ => ↑x * cexp (-b * (x : ℂ) ^ 2) 
goal: MeasureTheory.Integrable (fun x => ↑x * cexp (-b * ↑x ^ 2)) MeasureTheory.volume

full_name: integral_mul_cexp_neg_mul_sq
code: theorem integral_mul_cexp_neg_mul_sq {b : ℂ} (hb : 0 < b.re) :
    ∫ r : ℝ in Ioi 0, (r : ℂ) * cexp (-b * (r : ℂ) ^ 2) = (2 * b)⁻¹ 
goal: ∫ (r : ℝ) in Set.Ioi 0, ↑r * cexp (-b * ↑r ^ 2) = (2 * b)⁻¹

full_name: integral_gaussian_sq_complex
code: /-- The *square* of the Gaussian integral `∫ x:ℝ, exp (-b * x^2)` is equal to `π / b`. -/
theorem integral_gaussian_sq_complex {b : ℂ} (hb : 0 < b.re) :
    (∫ x : ℝ, cexp (-b * (x : ℂ) ^ 2)) ^ 2 = π / b 
goal: (∫ (x : ℝ), cexp (-b * ↑x ^ 2)) ^ 2 = ↑π / b

full_name: integral_gaussian
code: theorem integral_gaussian (b : ℝ) : ∫ x : ℝ, exp (-b * x ^ 2) = √(π / b) 
goal: ∫ (x : ℝ), rexp (-b * x ^ 2) = √(π / b)

full_name: continuousAt_gaussian_integral
code: theorem continuousAt_gaussian_integral (b : ℂ) (hb : 0 < re b) :
    ContinuousAt (fun c : ℂ => ∫ x : ℝ, cexp (-c * (x : ℂ) ^ 2)) b 
goal: ContinuousAt (fun c => ∫ (x : ℝ), cexp (-c * ↑x ^ 2)) b

full_name: integral_gaussian_complex
code: theorem integral_gaussian_complex {b : ℂ} (hb : 0 < re b) :
    ∫ x : ℝ, cexp (-b * (x : ℂ) ^ 2) = (π / b) ^ (1 / 2 : ℂ) 
goal: ∫ (x : ℝ), cexp (-b * ↑x ^ 2) = (↑π / b) ^ (1 / 2)

full_name: integral_gaussian_complex_Ioi
code: theorem integral_gaussian_complex_Ioi {b : ℂ} (hb : 0 < re b) :
    ∫ x : ℝ in Ioi 0, cexp (-b * (x : ℂ) ^ 2) = (π / b) ^ (1 / 2 : ℂ) / 2 
goal: ∫ (x : ℝ) in Set.Ioi 0, cexp (-b * ↑x ^ 2) = (↑π / b) ^ (1 / 2) / 2

full_name: integral_gaussian_Ioi
code: theorem integral_gaussian_Ioi (b : ℝ) :
    ∫ x in Ioi (0 : ℝ), exp (-b * x ^ 2) = √(π / b) / 2 
goal: ∫ (x : ℝ) in Set.Ioi 0, rexp (-b * x ^ 2) = √(π / b) / 2

full_name: Real.Gamma_one_half_eq
code: /-- The special-value formula `Γ(1/2) = √π`, which is equivalent to the Gaussian integral. -/
theorem Real.Gamma_one_half_eq : Real.Gamma (1 / 2) = √π 
goal: Real.Gamma (1 / 2) = √π

full_name: Complex.Gamma_one_half_eq
code: /-- The special-value formula `Γ(1/2) = √π`, which is equivalent to the Gaussian integral. -/
theorem Complex.Gamma_one_half_eq : Complex.Gamma (1 / 2) = (π : ℂ) ^ (1 / 2 : ℂ) 
goal: Complex.Gamma (1 / 2) = ↑π ^ (1 / 2)