File: Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean

full_name: integrableOn_exp_Iic
code: theorem integrableOn_exp_Iic (c : ℝ) : IntegrableOn exp (Iic c) 
goal: MeasureTheory.IntegrableOn rexp (Set.Iic c) MeasureTheory.volume

full_name: integral_exp_Iic
code: theorem integral_exp_Iic (c : ℝ) : ∫ x : ℝ in Iic c, exp x = exp c 
goal: ∫ (x : ℝ) in Set.Iic c, rexp x = rexp c

full_name: integral_exp_Iic_zero
code: theorem integral_exp_Iic_zero : ∫ x : ℝ in Iic 0, exp x = 1 
goal: ∫ (x : ℝ) in Set.Iic 0, rexp x = 1

full_name: integral_exp_neg_Ioi
code: theorem integral_exp_neg_Ioi (c : ℝ) : (∫ x : ℝ in Ioi c, exp (-x)) = exp (-c) 
goal: ∫ (x : ℝ) in Set.Ioi c, rexp (-x) = rexp (-c)

full_name: integral_exp_neg_Ioi_zero
code: theorem integral_exp_neg_Ioi_zero : (∫ x : ℝ in Ioi 0, exp (-x)) = 1 
goal: ∫ (x : ℝ) in Set.Ioi 0, rexp (-x) = 1

full_name: integrableOn_Ioi_rpow_of_lt
code: /-- If `0 < c`, then `(fun t : ℝ ↦ t ^ a)` is integrable on `(c, ∞)` for all `a < -1`. -/
theorem integrableOn_Ioi_rpow_of_lt {a : ℝ} (ha : a < -1) {c : ℝ} (hc : 0 < c) :
    IntegrableOn (fun t : ℝ => t ^ a) (Ioi c) 
goal: MeasureTheory.IntegrableOn (fun t => t ^ a) (Set.Ioi c) MeasureTheory.volume

full_name: integrableOn_Ioi_rpow_iff
code: theorem integrableOn_Ioi_rpow_iff {s t : ℝ} (ht : 0 < t) :
    IntegrableOn (fun x ↦ x ^ s) (Ioi t) ↔ s < -1 
goal: MeasureTheory.IntegrableOn (fun x => x ^ s) (Set.Ioi t) MeasureTheory.volume ↔ s < -1

full_name: not_integrableOn_Ioi_rpow
code: /-- The real power function with any exponent is not integrable on `(0, +∞)`. -/
theorem not_integrableOn_Ioi_rpow (s : ℝ) : ¬ IntegrableOn (fun x ↦ x ^ s) (Ioi (0 : ℝ)) 
goal: ¬MeasureTheory.IntegrableOn (fun x => x ^ s) (Set.Ioi 0) MeasureTheory.volume

full_name: setIntegral_Ioi_zero_rpow
code: theorem setIntegral_Ioi_zero_rpow (s : ℝ) : ∫ x in Ioi (0 : ℝ), x ^ s = 0 
goal: ∫ (x : ℝ) in Set.Ioi 0, x ^ s = 0

full_name: integral_Ioi_rpow_of_lt
code: theorem integral_Ioi_rpow_of_lt {a : ℝ} (ha : a < -1) {c : ℝ} (hc : 0 < c) :
    ∫ t : ℝ in Ioi c, t ^ a = -c ^ (a + 1) / (a + 1) 
goal: ∫ (t : ℝ) in Set.Ioi c, t ^ a = -c ^ (a + 1) / (a + 1)

full_name: integrableOn_Ioi_cpow_of_lt
code: theorem integrableOn_Ioi_cpow_of_lt {a : ℂ} (ha : a.re < -1) {c : ℝ} (hc : 0 < c) :
    IntegrableOn (fun t : ℝ => (t : ℂ) ^ a) (Ioi c) 
goal: MeasureTheory.IntegrableOn (fun t => ↑t ^ a) (Set.Ioi c) MeasureTheory.volume

full_name: integrableOn_Ioi_cpow_iff
code: theorem integrableOn_Ioi_cpow_iff {s : ℂ} {t : ℝ} (ht : 0 < t) :
    IntegrableOn (fun x : ℝ ↦ (x : ℂ) ^ s) (Ioi t) ↔ s.re < -1 
goal: MeasureTheory.IntegrableOn (fun x => ↑x ^ s) (Set.Ioi t) MeasureTheory.volume ↔ s.re < -1

full_name: not_integrableOn_Ioi_cpow
code: /-- The complex power function with any exponent is not integrable on `(0, +∞)`. -/
theorem not_integrableOn_Ioi_cpow (s : ℂ) :
    ¬ IntegrableOn (fun x : ℝ ↦ (x : ℂ) ^ s) (Ioi (0 : ℝ)) 
goal: ¬MeasureTheory.IntegrableOn (fun x => ↑x ^ s) (Set.Ioi 0) MeasureTheory.volume

full_name: setIntegral_Ioi_zero_cpow
code: theorem setIntegral_Ioi_zero_cpow (s : ℂ) : ∫ x in Ioi (0 : ℝ), (x : ℂ) ^ s = 0 
goal: ∫ (x : ℝ) in Set.Ioi 0, ↑x ^ s = 0

full_name: integral_Ioi_cpow_of_lt
code: theorem integral_Ioi_cpow_of_lt {a : ℂ} (ha : a.re < -1) {c : ℝ} (hc : 0 < c) :
    (∫ t : ℝ in Ioi c, (t : ℂ) ^ a) = -(c : ℂ) ^ (a + 1) / (a + 1) 
goal: ∫ (t : ℝ) in Set.Ioi c, ↑t ^ a = -↑c ^ (a + 1) / (a + 1)

full_name: integrable_inv_one_add_sq
code: theorem integrable_inv_one_add_sq : Integrable fun (x : ℝ) ↦ (1 + x ^ 2)⁻¹ 
goal: MeasureTheory.Integrable (fun x => (1 + x ^ 2)⁻¹) MeasureTheory.volume

full_name: integral_Iic_inv_one_add_sq
code: @[simp]
theorem integral_Iic_inv_one_add_sq {i : ℝ} :
    ∫ (x : ℝ) in Set.Iic i, (1 + x ^ 2)⁻¹ = arctan i + (π / 2) 
goal: ∫ (x : ℝ) in Set.Iic i, (1 + x ^ 2)⁻¹ = Real.arctan i + π / 2

full_name: integral_Ioi_inv_one_add_sq
code: @[simp]
theorem integral_Ioi_inv_one_add_sq {i : ℝ} :
    ∫ (x : ℝ) in Set.Ioi i, (1 + x ^ 2)⁻¹ = (π / 2) - arctan i 
goal: ∫ (x : ℝ) in Set.Ioi i, (1 + x ^ 2)⁻¹ = π / 2 - Real.arctan i

full_name: integral_univ_inv_one_add_sq
code: @[simp]
theorem integral_univ_inv_one_add_sq : ∫ (x : ℝ), (1 + x ^ 2)⁻¹ = π 
goal: ∫ (x : ℝ), (1 + x ^ 2)⁻¹ = π