analysis.special_functions.improper_integrals ⟷ Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: David Loeffler
 -/
 import Analysis.SpecialFunctions.Integrals
-import MeasureTheory.Group.Integration
+import MeasureTheory.Group.Integral
 import MeasureTheory.Integral.ExpDecay
 import MeasureTheory.Integral.IntegralEqImproper
 import MeasureTheory.Measure.Lebesgue.Integral

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -36,7 +36,7 @@ open scoped Topology
 theorem integrableOn_exp_Iic (c : ℝ) : IntegrableOn exp (Iic c) :=
   by
   refine'
-    integrable_on_Iic_of_interval_integral_norm_bounded (exp c) c
+    integrable_on_Iic_of_interval_integral_norm_bounded (NormedSpace.exp c) c
       (fun y => interval_integrable_exp.1) tendsto_id
       (eventually_of_mem (Iic_mem_at_bot 0) fun y hy => _)
   simp_rw [norm_of_nonneg (exp_pos _).le, integral_exp, sub_le_self_iff]
@@ -50,7 +50,7 @@ theorem integral_exp_Iic (c : ℝ) : ∫ x : ℝ in Iic c, exp x = exp c :=
   refine'
     tendsto_nhds_unique
       (interval_integral_tendsto_integral_Iic _ (integrableOn_exp_Iic _) tendsto_id) _
-  simp_rw [integral_exp, show 𝓝 (exp c) = 𝓝 (exp c - 0) by rw [sub_zero]]
+  simp_rw [integral_exp, show 𝓝 (NormedSpace.exp c) = 𝓝 (NormedSpace.exp c - 0) by rw [sub_zero]]
   exact tendsto_exp_at_bot.const_sub _
 #align integral_exp_Iic integral_exp_Iic
 -/
@@ -69,7 +69,7 @@ theorem integral_exp_neg_Ioi (c : ℝ) : ∫ x : ℝ in Ioi c, exp (-x) = exp (-
 
 #print integral_exp_neg_Ioi_zero /-
 theorem integral_exp_neg_Ioi_zero : ∫ x : ℝ in Ioi 0, exp (-x) = 1 := by
-  simpa only [neg_zero, exp_zero] using integral_exp_neg_Ioi 0
+  simpa only [neg_zero, NormedSpace.exp_zero] using integral_exp_neg_Ioi 0
 #align integral_exp_neg_Ioi_zero integral_exp_neg_Ioi_zero
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,11 +3,11 @@ Copyright (c) 2023 David Loeffler. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: David Loeffler
 -/
-import Mathbin.Analysis.SpecialFunctions.Integrals
-import Mathbin.MeasureTheory.Group.Integration
-import Mathbin.MeasureTheory.Integral.ExpDecay
-import Mathbin.MeasureTheory.Integral.IntegralEqImproper
-import Mathbin.MeasureTheory.Measure.Lebesgue.Integral
+import Analysis.SpecialFunctions.Integrals
+import MeasureTheory.Group.Integration
+import MeasureTheory.Integral.ExpDecay
+import MeasureTheory.Integral.IntegralEqImproper
+import MeasureTheory.Measure.Lebesgue.Integral
 
 #align_import analysis.special_functions.improper_integrals from "leanprover-community/mathlib"@"fd4551cfe4b7484b81c2c9ba3405edae27659676"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,11 +2,6 @@
 Copyright (c) 2023 David Loeffler. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: David Loeffler
-
-! This file was ported from Lean 3 source module analysis.special_functions.improper_integrals
-! leanprover-community/mathlib commit fd4551cfe4b7484b81c2c9ba3405edae27659676
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.SpecialFunctions.Integrals
 import Mathbin.MeasureTheory.Group.Integration
@@ -14,6 +9,8 @@ import Mathbin.MeasureTheory.Integral.ExpDecay
 import Mathbin.MeasureTheory.Integral.IntegralEqImproper
 import Mathbin.MeasureTheory.Measure.Lebesgue.Integral
 
+#align_import analysis.special_functions.improper_integrals from "leanprover-community/mathlib"@"fd4551cfe4b7484b81c2c9ba3405edae27659676"
+
 /-!
 # Evaluation of specific improper integrals
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -64,9 +64,11 @@ theorem integral_exp_Iic_zero : ∫ x : ℝ in Iic 0, exp x = 1 :=
 #align integral_exp_Iic_zero integral_exp_Iic_zero
 -/
 
+#print integral_exp_neg_Ioi /-
 theorem integral_exp_neg_Ioi (c : ℝ) : ∫ x : ℝ in Ioi c, exp (-x) = exp (-c) := by
   simpa only [integral_comp_neg_Ioi] using integral_exp_Iic (-c)
 #align integral_exp_neg_Ioi integral_exp_neg_Ioi
+-/
 
 #print integral_exp_neg_Ioi_zero /-
 theorem integral_exp_neg_Ioi_zero : ∫ x : ℝ in Ioi 0, exp (-x) = 1 := by
@@ -74,6 +76,7 @@ theorem integral_exp_neg_Ioi_zero : ∫ x : ℝ in Ioi 0, exp (-x) = 1 := by
 #align integral_exp_neg_Ioi_zero integral_exp_neg_Ioi_zero
 -/
 
+#print integrableOn_Ioi_rpow_of_lt /-
 /-- If `0 < c`, then `(λ 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) :=
@@ -90,7 +93,9 @@ theorem integrableOn_Ioi_rpow_of_lt {a : ℝ} (ha : a < -1) {c : ℝ} (hc : 0 <
   exact
     integrable_on_Ioi_deriv_of_nonneg' hd (fun t ht => rpow_nonneg_of_nonneg (hc.trans ht).le a) ht
 #align integrable_on_Ioi_rpow_of_lt integrableOn_Ioi_rpow_of_lt
+-/
 
+#print integral_Ioi_rpow_of_lt /-
 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) :=
   by
@@ -106,7 +111,9 @@ theorem integral_Ioi_rpow_of_lt {a : ℝ} (ha : a < -1) {c : ℝ} (hc : 0 < c) :
   convert integral_Ioi_of_has_deriv_at_of_tendsto' hd (integrableOn_Ioi_rpow_of_lt ha hc) ht
   simp only [neg_div, zero_div, zero_sub]
 #align integral_Ioi_rpow_of_lt integral_Ioi_rpow_of_lt
+-/
 
+#print integrableOn_Ioi_cpow_of_lt /-
 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) :=
   by
@@ -117,7 +124,9 @@ theorem integrableOn_Ioi_cpow_of_lt {a : ℂ} (ha : a.re < -1) {c : ℝ} (hc : 0
   · refine' ContinuousOn.aestronglyMeasurable (fun t ht => _) measurableSet_Ioi
     exact (Complex.continuousAt_ofReal_cpow_const _ _ (Or.inr (hc.trans ht).ne')).ContinuousWithinAt
 #align integrable_on_Ioi_cpow_of_lt integrableOn_Ioi_cpow_of_lt
+-/
 
+#print integral_Ioi_cpow_of_lt /-
 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) :=
   by
@@ -143,4 +152,5 @@ theorem integral_Ioi_cpow_of_lt {a : ℂ} (ha : a.re < -1) {c : ℝ} (hc : 0 < c
   simp_rw [neg_neg, Complex.norm_eq_abs, Complex.abs_cpow_eq_rpow_re_of_pos hx, Complex.add_re,
     Complex.one_re]
 #align integral_Ioi_cpow_of_lt integral_Ioi_cpow_of_lt
+-/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/c471da714c044131b90c133701e51b877c246677

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: David Loeffler
 
 ! This file was ported from Lean 3 source module analysis.special_functions.improper_integrals
-! leanprover-community/mathlib commit fd5edc43dc4f10b85abfe544b88f82cf13c5f844
+! leanprover-community/mathlib commit fd4551cfe4b7484b81c2c9ba3405edae27659676
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -17,6 +17,9 @@ import Mathbin.MeasureTheory.Measure.Lebesgue.Integral
 /-!
 # Evaluation of specific improper integrals
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 This file contains some integrability results, and evaluations of integrals, over `ℝ` or over
 half-infinite intervals in `ℝ`.
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3

@@ -45,7 +45,7 @@ theorem integrableOn_exp_Iic (c : ℝ) : IntegrableOn exp (Iic c) :=
 -/
 
 #print integral_exp_Iic /-
-theorem integral_exp_Iic (c : ℝ) : (∫ x : ℝ in Iic c, exp x) = exp c :=
+theorem integral_exp_Iic (c : ℝ) : ∫ x : ℝ in Iic c, exp x = exp c :=
   by
   refine'
     tendsto_nhds_unique
@@ -56,17 +56,17 @@ theorem integral_exp_Iic (c : ℝ) : (∫ x : ℝ in Iic c, exp x) = exp c :=
 -/
 
 #print integral_exp_Iic_zero /-
-theorem integral_exp_Iic_zero : (∫ x : ℝ in Iic 0, exp x) = 1 :=
+theorem integral_exp_Iic_zero : ∫ x : ℝ in Iic 0, exp x = 1 :=
   exp_zero ▸ integral_exp_Iic 0
 #align integral_exp_Iic_zero integral_exp_Iic_zero
 -/
 
-theorem integral_exp_neg_Ioi (c : ℝ) : (∫ x : ℝ in Ioi c, exp (-x)) = exp (-c) := by
+theorem integral_exp_neg_Ioi (c : ℝ) : ∫ x : ℝ in Ioi c, exp (-x) = exp (-c) := by
   simpa only [integral_comp_neg_Ioi] using integral_exp_Iic (-c)
 #align integral_exp_neg_Ioi integral_exp_neg_Ioi
 
 #print integral_exp_neg_Ioi_zero /-
-theorem integral_exp_neg_Ioi_zero : (∫ x : ℝ in Ioi 0, exp (-x)) = 1 := by
+theorem integral_exp_neg_Ioi_zero : ∫ x : ℝ in Ioi 0, exp (-x) = 1 := by
   simpa only [neg_zero, exp_zero] using integral_exp_neg_Ioi 0
 #align integral_exp_neg_Ioi_zero integral_exp_neg_Ioi_zero
 -/
@@ -89,7 +89,7 @@ theorem integrableOn_Ioi_rpow_of_lt {a : ℝ} (ha : a < -1) {c : ℝ} (hc : 0 <
 #align integrable_on_Ioi_rpow_of_lt integrableOn_Ioi_rpow_of_lt
 
 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) :=
+    ∫ t : ℝ in Ioi c, t ^ a = -c ^ (a + 1) / (a + 1) :=
   by
   have hd : ∀ (x : ℝ) (hx : x ∈ Ici c), HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x :=
     by
@@ -116,7 +116,7 @@ theorem integrableOn_Ioi_cpow_of_lt {a : ℂ} (ha : a.re < -1) {c : ℝ} (hc : 0
 #align integrable_on_Ioi_cpow_of_lt integrableOn_Ioi_cpow_of_lt
 
 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) :=
+    ∫ t : ℝ in Ioi c, (t : ℂ) ^ a = -(c : ℂ) ^ (a + 1) / (a + 1) :=
   by
   refine'
     tendsto_nhds_unique

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3

@@ -112,8 +112,7 @@ theorem integrableOn_Ioi_cpow_of_lt {a : ℂ} (ha : a.re < -1) {c : ℝ} (hc : 0
   · dsimp only
     rw [Complex.norm_eq_abs, Complex.abs_cpow_eq_rpow_re_of_pos (hc.trans hx)]
   · refine' ContinuousOn.aestronglyMeasurable (fun t ht => _) measurableSet_Ioi
-    exact
-      (Complex.continuousAt_of_real_cpow_const _ _ (Or.inr (hc.trans ht).ne')).ContinuousWithinAt
+    exact (Complex.continuousAt_ofReal_cpow_const _ _ (Or.inr (hc.trans ht).ne')).ContinuousWithinAt
 #align integrable_on_Ioi_cpow_of_lt integrableOn_Ioi_cpow_of_lt
 
 theorem integral_Ioi_cpow_of_lt {a : ℂ} (ha : a.re < -1) {c : ℝ} (hc : 0 < c) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0

@@ -32,6 +32,7 @@ open Real Set Filter MeasureTheory intervalIntegral
 
 open scoped Topology
 
+#print integrableOn_exp_Iic /-
 theorem integrableOn_exp_Iic (c : ℝ) : IntegrableOn exp (Iic c) :=
   by
   refine'
@@ -41,7 +42,9 @@ theorem integrableOn_exp_Iic (c : ℝ) : IntegrableOn exp (Iic c) :=
   simp_rw [norm_of_nonneg (exp_pos _).le, integral_exp, sub_le_self_iff]
   exact (exp_pos _).le
 #align integrable_on_exp_Iic integrableOn_exp_Iic
+-/
 
+#print integral_exp_Iic /-
 theorem integral_exp_Iic (c : ℝ) : (∫ x : ℝ in Iic c, exp x) = exp c :=
   by
   refine'
@@ -50,18 +53,23 @@ theorem integral_exp_Iic (c : ℝ) : (∫ x : ℝ in Iic c, exp x) = exp c :=
   simp_rw [integral_exp, show 𝓝 (exp c) = 𝓝 (exp c - 0) by rw [sub_zero]]
   exact tendsto_exp_at_bot.const_sub _
 #align integral_exp_Iic integral_exp_Iic
+-/
 
+#print integral_exp_Iic_zero /-
 theorem integral_exp_Iic_zero : (∫ x : ℝ in Iic 0, exp x) = 1 :=
   exp_zero ▸ integral_exp_Iic 0
 #align integral_exp_Iic_zero integral_exp_Iic_zero
+-/
 
 theorem integral_exp_neg_Ioi (c : ℝ) : (∫ x : ℝ in Ioi c, exp (-x)) = exp (-c) := by
   simpa only [integral_comp_neg_Ioi] using integral_exp_Iic (-c)
 #align integral_exp_neg_Ioi integral_exp_neg_Ioi
 
+#print integral_exp_neg_Ioi_zero /-
 theorem integral_exp_neg_Ioi_zero : (∫ x : ℝ in Ioi 0, exp (-x)) = 1 := by
   simpa only [neg_zero, exp_zero] using integral_exp_neg_Ioi 0
 #align integral_exp_neg_Ioi_zero integral_exp_neg_Ioi_zero
+-/
 
 /-- If `0 < c`, then `(λ 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) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

@@ -70,7 +70,7 @@ theorem integrableOn_Ioi_rpow_of_lt {a : ℝ} (ha : a < -1) {c : ℝ} (hc : 0 <
   have hd : ∀ (x : ℝ) (hx : x ∈ Ici c), HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x :=
     by
     intro x hx
-    convert(has_deriv_at_rpow_const (Or.inl (hc.trans_le hx).ne')).div_const _
+    convert (has_deriv_at_rpow_const (Or.inl (hc.trans_le hx).ne')).div_const _
     field_simp [show a + 1 ≠ 0 from ne_of_lt (by linarith), mul_comm]
   have ht : tendsto (fun t => t ^ (a + 1) / (a + 1)) at_top (𝓝 (0 / (a + 1))) :=
     by
@@ -86,7 +86,7 @@ theorem integral_Ioi_rpow_of_lt {a : ℝ} (ha : a < -1) {c : ℝ} (hc : 0 < c) :
   have hd : ∀ (x : ℝ) (hx : x ∈ Ici c), HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x :=
     by
     intro x hx
-    convert(has_deriv_at_rpow_const (Or.inl (hc.trans_le hx).ne')).div_const _
+    convert (has_deriv_at_rpow_const (Or.inl (hc.trans_le hx).ne')).div_const _
     field_simp [show a + 1 ≠ 0 from ne_of_lt (by linarith), mul_comm]
   have ht : tendsto (fun t => t ^ (a + 1) / (a + 1)) at_top (𝓝 (0 / (a + 1))) :=
     by

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -103,7 +103,7 @@ theorem integrableOn_Ioi_cpow_of_lt {a : ℂ} (ha : a.re < -1) {c : ℝ} (hc : 0
   refine' (integrableOn_Ioi_rpow_of_lt ha hc).congr_fun (fun x hx => _) measurableSet_Ioi
   · dsimp only
     rw [Complex.norm_eq_abs, Complex.abs_cpow_eq_rpow_re_of_pos (hc.trans hx)]
-  · refine' ContinuousOn.aEStronglyMeasurable (fun t ht => _) measurableSet_Ioi
+  · refine' ContinuousOn.aestronglyMeasurable (fun t ht => _) measurableSet_Ioi
     exact
       (Complex.continuousAt_of_real_cpow_const _ _ (Or.inr (hc.trans ht).ne')).ContinuousWithinAt
 #align integrable_on_Ioi_cpow_of_lt integrableOn_Ioi_cpow_of_lt

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -30,7 +30,7 @@ half-infinite intervals in `ℝ`.
 
 open Real Set Filter MeasureTheory intervalIntegral
 
-open Topology
+open scoped Topology
 
 theorem integrableOn_exp_Iic (c : ℝ) : IntegrableOn exp (Iic c) :=
   by

mathlib commit https://github.com/leanprover-community/mathlib/commit/75e7fca56381d056096ce5d05e938f63a6567828

@@ -103,7 +103,7 @@ theorem integrableOn_Ioi_cpow_of_lt {a : ℂ} (ha : a.re < -1) {c : ℝ} (hc : 0
   refine' (integrableOn_Ioi_rpow_of_lt ha hc).congr_fun (fun x hx => _) measurableSet_Ioi
   · dsimp only
     rw [Complex.norm_eq_abs, Complex.abs_cpow_eq_rpow_re_of_pos (hc.trans hx)]
-  · refine' ContinuousOn.aeStronglyMeasurable (fun t ht => _) measurableSet_Ioi
+  · refine' ContinuousOn.aEStronglyMeasurable (fun t ht => _) measurableSet_Ioi
     exact
       (Complex.continuousAt_of_real_cpow_const _ _ (Or.inr (hc.trans ht).ne')).ContinuousWithinAt
 #align integrable_on_Ioi_cpow_of_lt integrableOn_Ioi_cpow_of_lt

mathlib commit https://github.com/leanprover-community/mathlib/commit/f51de8769c34652d82d1c8e5f8f18f8374782bed

@@ -4,14 +4,15 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: David Loeffler
 
 ! This file was ported from Lean 3 source module analysis.special_functions.improper_integrals
-! leanprover-community/mathlib commit ec4528061e02f0acc848ed06eb22573645602c7e
+! leanprover-community/mathlib commit fd5edc43dc4f10b85abfe544b88f82cf13c5f844
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
-import Mathbin.MeasureTheory.Integral.IntegralEqImproper
+import Mathbin.Analysis.SpecialFunctions.Integrals
 import Mathbin.MeasureTheory.Group.Integration
 import Mathbin.MeasureTheory.Integral.ExpDecay
-import Mathbin.Analysis.SpecialFunctions.Integrals
+import Mathbin.MeasureTheory.Integral.IntegralEqImproper
+import Mathbin.MeasureTheory.Measure.Lebesgue.Integral
 
 /-!
 # Evaluation of specific improper integrals

mathlib commit https://github.com/leanprover-community/mathlib/commit/92c69b77c5a7dc0f7eeddb552508633305157caa

@@ -31,7 +31,7 @@ open Real Set Filter MeasureTheory intervalIntegral
 
 open Topology
 
-theorem integrableOnExpIic (c : ℝ) : IntegrableOn exp (Iic c) :=
+theorem integrableOn_exp_Iic (c : ℝ) : IntegrableOn exp (Iic c) :=
   by
   refine'
     integrable_on_Iic_of_interval_integral_norm_bounded (exp c) c
@@ -39,13 +39,13 @@ theorem integrableOnExpIic (c : ℝ) : IntegrableOn exp (Iic c) :=
       (eventually_of_mem (Iic_mem_at_bot 0) fun y hy => _)
   simp_rw [norm_of_nonneg (exp_pos _).le, integral_exp, sub_le_self_iff]
   exact (exp_pos _).le
-#align integrable_on_exp_Iic integrableOnExpIic
+#align integrable_on_exp_Iic integrableOn_exp_Iic
 
 theorem integral_exp_Iic (c : ℝ) : (∫ x : ℝ in Iic c, exp x) = exp c :=
   by
   refine'
-    tendsto_nhds_unique (interval_integral_tendsto_integral_Iic _ (integrableOnExpIic _) tendsto_id)
-      _
+    tendsto_nhds_unique
+      (interval_integral_tendsto_integral_Iic _ (integrableOn_exp_Iic _) tendsto_id) _
   simp_rw [integral_exp, show 𝓝 (exp c) = 𝓝 (exp c - 0) by rw [sub_zero]]
   exact tendsto_exp_at_bot.const_sub _
 #align integral_exp_Iic integral_exp_Iic
@@ -63,7 +63,7 @@ theorem integral_exp_neg_Ioi_zero : (∫ x : ℝ in Ioi 0, exp (-x)) = 1 := by
 #align integral_exp_neg_Ioi_zero integral_exp_neg_Ioi_zero
 
 /-- If `0 < c`, then `(λ t : ℝ, t ^ a)` is integrable on `(c, ∞)` for all `a < -1`. -/
-theorem integrableOnIoiRpowOfLt {a : ℝ} (ha : a < -1) {c : ℝ} (hc : 0 < c) :
+theorem integrableOn_Ioi_rpow_of_lt {a : ℝ} (ha : a < -1) {c : ℝ} (hc : 0 < c) :
     IntegrableOn (fun t : ℝ => t ^ a) (Ioi c) :=
   by
   have hd : ∀ (x : ℝ) (hx : x ∈ Ici c), HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x :=
@@ -77,7 +77,7 @@ theorem integrableOnIoiRpowOfLt {a : ℝ} (ha : a < -1) {c : ℝ} (hc : 0 < c) :
     simpa only [neg_neg] using tendsto_rpow_neg_atTop (by linarith : 0 < -(a + 1))
   exact
     integrable_on_Ioi_deriv_of_nonneg' hd (fun t ht => rpow_nonneg_of_nonneg (hc.trans ht).le a) ht
-#align integrable_on_Ioi_rpow_of_lt integrableOnIoiRpowOfLt
+#align integrable_on_Ioi_rpow_of_lt integrableOn_Ioi_rpow_of_lt
 
 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) :=
@@ -91,28 +91,28 @@ theorem integral_Ioi_rpow_of_lt {a : ℝ} (ha : a < -1) {c : ℝ} (hc : 0 < c) :
     by
     apply tendsto.div_const
     simpa only [neg_neg] using tendsto_rpow_neg_atTop (by linarith : 0 < -(a + 1))
-  convert integral_Ioi_of_has_deriv_at_of_tendsto' hd (integrableOnIoiRpowOfLt ha hc) ht
+  convert integral_Ioi_of_has_deriv_at_of_tendsto' hd (integrableOn_Ioi_rpow_of_lt ha hc) ht
   simp only [neg_div, zero_div, zero_sub]
 #align integral_Ioi_rpow_of_lt integral_Ioi_rpow_of_lt
 
-theorem integrableOnIoiCpowOfLt {a : ℂ} (ha : a.re < -1) {c : ℝ} (hc : 0 < c) :
+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) :=
   by
   rw [integrable_on, ← integrable_norm_iff, ← integrable_on]
-  refine' (integrableOnIoiRpowOfLt ha hc).congr_fun (fun x hx => _) measurableSet_Ioi
+  refine' (integrableOn_Ioi_rpow_of_lt ha hc).congr_fun (fun x hx => _) measurableSet_Ioi
   · dsimp only
     rw [Complex.norm_eq_abs, Complex.abs_cpow_eq_rpow_re_of_pos (hc.trans hx)]
   · refine' ContinuousOn.aeStronglyMeasurable (fun t ht => _) measurableSet_Ioi
     exact
       (Complex.continuousAt_of_real_cpow_const _ _ (Or.inr (hc.trans ht).ne')).ContinuousWithinAt
-#align integrable_on_Ioi_cpow_of_lt integrableOnIoiCpowOfLt
+#align integrable_on_Ioi_cpow_of_lt integrableOn_Ioi_cpow_of_lt
 
 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) :=
   by
   refine'
     tendsto_nhds_unique
-      (interval_integral_tendsto_integral_Ioi c (integrableOnIoiCpowOfLt ha hc) tendsto_id) _
+      (interval_integral_tendsto_integral_Ioi c (integrableOn_Ioi_cpow_of_lt ha hc) tendsto_id) _
   suffices
     tendsto (fun x : ℝ => ((x : ℂ) ^ (a + 1) - (c : ℂ) ^ (a + 1)) / (a + 1)) at_top
       (𝓝 <| -c ^ (a + 1) / (a + 1))

mathlib commit https://github.com/leanprover-community/mathlib/commit/92c69b77c5a7dc0f7eeddb552508633305157caa

A PR analogous to #12338: reformatting proofs following the multiple goals linter of #12339.

@@ -119,9 +119,9 @@ theorem integral_Ioi_rpow_of_lt {a : ℝ} (ha : a < -1) {c : ℝ} (hc : 0 < c) :
 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) := by
   rw [IntegrableOn, ← integrable_norm_iff, ← IntegrableOn]
-  refine' (integrableOn_Ioi_rpow_of_lt ha hc).congr_fun (fun x hx => _) measurableSet_Ioi
-  · dsimp only
-    rw [Complex.norm_eq_abs, Complex.abs_cpow_eq_rpow_re_of_pos (hc.trans hx)]
+  · refine' (integrableOn_Ioi_rpow_of_lt ha hc).congr_fun (fun x hx => _) measurableSet_Ioi
+    · dsimp only
+      rw [Complex.norm_eq_abs, Complex.abs_cpow_eq_rpow_re_of_pos (hc.trans hx)]
   · refine' ContinuousOn.aestronglyMeasurable (fun t ht => _) measurableSet_Ioi
     exact
       (Complex.continuousAt_ofReal_cpow_const _ _ (Or.inr (hc.trans ht).ne')).continuousWithinAt

Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com>

@@ -164,7 +164,7 @@ theorem integral_Ioi_cpow_of_lt {a : ℂ} (ha : a.re < -1) {c : ℝ} (hc : 0 < c
       (𝓝 <| -c ^ (a + 1) / (a + 1)) by
     refine' this.congr' ((eventually_gt_atTop 0).mp (eventually_of_forall fun x hx => _))
     dsimp only
-    rw [integral_cpow, id.def]
+    rw [integral_cpow, id]
     refine' Or.inr ⟨_, not_mem_uIcc_of_lt hc hx⟩
     apply_fun Complex.re
     rw [Complex.neg_re, Complex.one_re]

I loogled for every occurrence of "cast", Nat and "natCast" and where the casted nat was n, and made sure there were corresponding @[simp] lemmas for 0, 1, and OfNat.ofNat n. This is necessary in general for simp confluence. Example:

import Mathlib

variable {α : Type*} [LinearOrderedRing α] (m n : ℕ) [m.AtLeastTwo] [n.AtLeastTwo]

example : ((OfNat.ofNat m : ℕ) : α) ≤ ((OfNat.ofNat n : ℕ) : α) ↔ (OfNat.ofNat m : ℕ) ≤ (OfNat.ofNat n : ℕ) := by
  simp only [Nat.cast_le] -- this `@[simp]` lemma can apply

example : ((OfNat.ofNat m : ℕ) : α) ≤ ((OfNat.ofNat n : ℕ) : α) ↔ (OfNat.ofNat m : α) ≤ (OfNat.ofNat n : α) := by
  simp only [Nat.cast_ofNat] -- and so can this one

example : (OfNat.ofNat m : α) ≤ (OfNat.ofNat n : α) ↔ (OfNat.ofNat m : ℕ) ≤ (OfNat.ofNat n : ℕ) := by
  simp -- fails! `simp` doesn't have a lemma to bridge their results. confluence issue.

As far as I know, the only file this PR leaves with ofNat gaps is PartENat.lean. #8002 is addressing that file in parallel.

Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -181,7 +181,7 @@ theorem integral_Ioi_cpow_of_lt {a : ℂ} (ha : a.re < -1) {c : ℝ} (hc : 0 < c
 
 theorem integrable_inv_one_add_sq : Integrable fun (x : ℝ) ↦ (1 + x ^ 2)⁻¹ := by
   suffices Integrable fun (x : ℝ) ↦ (1 + ‖x‖ ^ 2) ^ ((-2 : ℝ) / 2) by simpa [rpow_neg_one]
-  exact integrable_rpow_neg_one_add_norm_sq (by simpa using by norm_num)
+  exact integrable_rpow_neg_one_add_norm_sq (by simp)
 
 @[simp]
 theorem integral_Iic_inv_one_add_sq {i : ℝ} :

Use Lean 4 syntax fun x ↦ instead, matching the style guide. This is close to exhaustive for doc comments; mathlib has about 460 remaining uses of λ (not all in Lean 3 syntax).

@@ -58,7 +58,7 @@ theorem integral_exp_neg_Ioi_zero : (∫ x : ℝ in Ioi 0, exp (-x)) = 1 := by
   simpa only [neg_zero, exp_zero] using integral_exp_neg_Ioi 0
 #align integral_exp_neg_Ioi_zero integral_exp_neg_Ioi_zero
 
-/-- If `0 < c`, then `(λ t : ℝ, t ^ a)` is integrable on `(c, ∞)` for all `a < -1`. -/
+/-- 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) := by
   have hd : ∀ x ∈ Ici c, HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x := by

Co-authored-by: L Lllvvuu <git@llllvvuu.dev>

@@ -3,6 +3,7 @@ Copyright (c) 2023 David Loeffler. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: David Loeffler
 -/
+import Mathlib.Analysis.SpecialFunctions.JapaneseBracket
 import Mathlib.Analysis.SpecialFunctions.Integrals
 import Mathlib.MeasureTheory.Group.Integral
 import Mathlib.MeasureTheory.Integral.IntegralEqImproper
@@ -177,3 +178,26 @@ theorem integral_Ioi_cpow_of_lt {a : ℂ} (ha : a.re < -1) {c : ℝ} (hc : 0 < c
   simp_rw [neg_neg, Complex.norm_eq_abs, Complex.abs_cpow_eq_rpow_re_of_pos hx, Complex.add_re,
     Complex.one_re]
 #align integral_Ioi_cpow_of_lt integral_Ioi_cpow_of_lt
+
+theorem integrable_inv_one_add_sq : Integrable fun (x : ℝ) ↦ (1 + x ^ 2)⁻¹ := by
+  suffices Integrable fun (x : ℝ) ↦ (1 + ‖x‖ ^ 2) ^ ((-2 : ℝ) / 2) by simpa [rpow_neg_one]
+  exact integrable_rpow_neg_one_add_norm_sq (by simpa using by norm_num)
+
+@[simp]
+theorem integral_Iic_inv_one_add_sq {i : ℝ} :
+    ∫ (x : ℝ) in Set.Iic i, (1 + x ^ 2)⁻¹ = arctan i + (π / 2) :=
+  integral_Iic_of_hasDerivAt_of_tendsto' (fun x _ => hasDerivAt_arctan' x)
+    integrable_inv_one_add_sq.integrableOn (tendsto_nhds_of_tendsto_nhdsWithin tendsto_arctan_atBot)
+    |>.trans (sub_neg_eq_add _ _)
+
+@[simp]
+theorem integral_Ioi_inv_one_add_sq {i : ℝ} :
+    ∫ (x : ℝ) in Set.Ioi i, (1 + x ^ 2)⁻¹ = (π / 2) - arctan i :=
+  integral_Ioi_of_hasDerivAt_of_tendsto' (fun x _ => hasDerivAt_arctan' x)
+    integrable_inv_one_add_sq.integrableOn (tendsto_nhds_of_tendsto_nhdsWithin tendsto_arctan_atTop)
+
+@[simp]
+theorem integral_univ_inv_one_add_sq : ∫ (x : ℝ), (1 + x ^ 2)⁻¹ = π :=
+  (by ring : π = (π / 2) - (-(π / 2))) ▸ integral_of_hasDerivAt_of_tendsto hasDerivAt_arctan'
+    integrable_inv_one_add_sq (tendsto_nhds_of_tendsto_nhdsWithin tendsto_arctan_atBot)
+    (tendsto_nhds_of_tendsto_nhdsWithin tendsto_arctan_atTop)

This uses the improved shake script from #9772 to reduce imports across mathlib. The corresponding noshake.json file has been added to #9772.

Co-authored-by: Mario Carneiro <di.gama@gmail.com>

@@ -5,7 +5,6 @@ Authors: David Loeffler
 -/
 import Mathlib.Analysis.SpecialFunctions.Integrals
 import Mathlib.MeasureTheory.Group.Integral
-import Mathlib.MeasureTheory.Integral.ExpDecay
 import Mathlib.MeasureTheory.Integral.IntegralEqImproper
 import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
 

This better matches other lemma names.

From LeanAPAP

@@ -70,7 +70,7 @@ theorem integrableOn_Ioi_rpow_of_lt {a : ℝ} (ha : a < -1) {c : ℝ} (hc : 0 <
     apply Tendsto.div_const
     simpa only [neg_neg] using tendsto_rpow_neg_atTop (by linarith : 0 < -(a + 1))
   exact
-    integrableOn_Ioi_deriv_of_nonneg' hd (fun t ht => rpow_nonneg_of_nonneg (hc.trans ht).le a) ht
+    integrableOn_Ioi_deriv_of_nonneg' hd (fun t ht => rpow_nonneg (hc.trans ht).le a) ht
 #align integrable_on_Ioi_rpow_of_lt integrableOn_Ioi_rpow_of_lt
 
 theorem integrableOn_Ioi_rpow_iff {s t : ℝ} (ht : 0 < t) :

Follow-up #9184

@@ -61,7 +61,7 @@ theorem integral_exp_neg_Ioi_zero : (∫ x : ℝ in Ioi 0, exp (-x)) = 1 := by
 /-- If `0 < c`, then `(λ 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) := by
-  have hd : ∀ (x : ℝ) (_ : x ∈ Ici c), HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x := by
+  have hd : ∀ x ∈ Ici c, HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x := by
     intro x hx
     -- Porting note: helped `convert` with explicit arguments
     convert (hasDerivAt_rpow_const (p := a + 1) (Or.inl (hc.trans_le hx).ne')).div_const _ using 1
@@ -105,7 +105,7 @@ theorem setIntegral_Ioi_zero_rpow (s : ℝ) : ∫ x in Ioi (0 : ℝ), x ^ s = 0
 
 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) := by
-  have hd : ∀ (x : ℝ) (_ : x ∈ Ici c), HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x := by
+  have hd : ∀ x ∈ Ici c, HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x := by
     intro x hx
     convert (hasDerivAt_rpow_const (p := a + 1) (Or.inl (hc.trans_le hx).ne')).div_const _ using 1
     field_simp [show a + 1 ≠ 0 from ne_of_lt (by linarith), mul_comm]

We have in the library the lemma not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_filter, saying that if a function tends to infinity at a point in an interval [a, b], then its derivative is not interval-integrable on [a, b]. We generalize this result to allow for any set instead of [a, b], and apply it to half-infinite intervals.

In particular, we characterize integrability of x^s on [a, +oo), and deduce that x^s is never integrable on [0, +oo). This makes it possible to remove one assumption in Lemma mellin_comp_rpow on the Mellin transform.

@@ -73,6 +73,36 @@ theorem integrableOn_Ioi_rpow_of_lt {a : ℝ} (ha : a < -1) {c : ℝ} (hc : 0 <
     integrableOn_Ioi_deriv_of_nonneg' hd (fun t ht => rpow_nonneg_of_nonneg (hc.trans ht).le a) ht
 #align integrable_on_Ioi_rpow_of_lt integrableOn_Ioi_rpow_of_lt
 
+theorem integrableOn_Ioi_rpow_iff {s t : ℝ} (ht : 0 < t) :
+    IntegrableOn (fun x ↦ x ^ s) (Ioi t) ↔ s < -1 := by
+  refine ⟨fun h ↦ ?_, fun h ↦ integrableOn_Ioi_rpow_of_lt h ht⟩
+  contrapose! h
+  intro H
+  have H' : IntegrableOn (fun x ↦ x ^ s) (Ioi (max 1 t)) :=
+    H.mono (Set.Ioi_subset_Ioi (le_max_right _ _)) le_rfl
+  have : IntegrableOn (fun x ↦ x⁻¹) (Ioi (max 1 t)) := by
+    apply H'.mono' measurable_inv.aestronglyMeasurable
+    filter_upwards [ae_restrict_mem measurableSet_Ioi] with x hx
+    have x_one : 1 ≤ x := ((le_max_left _ _).trans_lt (mem_Ioi.1 hx)).le
+    simp only [norm_inv, Real.norm_eq_abs, abs_of_nonneg (zero_le_one.trans x_one)]
+    rw [← Real.rpow_neg_one x]
+    exact Real.rpow_le_rpow_of_exponent_le x_one h
+  exact not_IntegrableOn_Ioi_inv this
+
+/-- 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)) := by
+  intro h
+  rcases le_or_lt s (-1) with hs|hs
+  · have : IntegrableOn (fun x ↦ x ^ s) (Ioo (0 : ℝ) 1) := h.mono Ioo_subset_Ioi_self le_rfl
+    rw [integrableOn_Ioo_rpow_iff zero_lt_one] at this
+    exact hs.not_lt this
+  · have : IntegrableOn (fun x ↦ x ^ s) (Ioi 1) := h.mono (Ioi_subset_Ioi zero_le_one) le_rfl
+    rw [integrableOn_Ioi_rpow_iff zero_lt_one] at this
+    exact hs.not_lt this
+
+theorem setIntegral_Ioi_zero_rpow (s : ℝ) : ∫ x in Ioi (0 : ℝ), x ^ s = 0 :=
+  MeasureTheory.integral_undef (not_integrableOn_Ioi_rpow s)
+
 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) := by
   have hd : ∀ (x : ℝ) (_ : x ∈ Ici c), HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x := by
@@ -97,6 +127,33 @@ theorem integrableOn_Ioi_cpow_of_lt {a : ℂ} (ha : a.re < -1) {c : ℝ} (hc : 0
       (Complex.continuousAt_ofReal_cpow_const _ _ (Or.inr (hc.trans ht).ne')).continuousWithinAt
 #align integrable_on_Ioi_cpow_of_lt integrableOn_Ioi_cpow_of_lt
 
+theorem integrableOn_Ioi_cpow_iff {s : ℂ} {t : ℝ} (ht : 0 < t) :
+    IntegrableOn (fun x : ℝ ↦ (x : ℂ) ^ s) (Ioi t) ↔ s.re < -1 := by
+  refine ⟨fun h ↦ ?_, fun h ↦ integrableOn_Ioi_cpow_of_lt h ht⟩
+  have B : IntegrableOn (fun a ↦ a ^ s.re) (Ioi t) := by
+    apply (integrableOn_congr_fun _ measurableSet_Ioi).1 h.norm
+    intro a ha
+    have : 0 < a := ht.trans ha
+    simp [Complex.abs_cpow_eq_rpow_re_of_pos this]
+  rwa [integrableOn_Ioi_rpow_iff ht] at B
+
+/-- 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)) := by
+  intro h
+  rcases le_or_lt s.re (-1) with hs|hs
+  · have : IntegrableOn (fun x : ℝ ↦ (x : ℂ) ^ s) (Ioo (0 : ℝ) 1) :=
+      h.mono Ioo_subset_Ioi_self le_rfl
+    rw [integrableOn_Ioo_cpow_iff zero_lt_one] at this
+    exact hs.not_lt this
+  · have : IntegrableOn (fun x : ℝ ↦ (x : ℂ) ^ s) (Ioi 1) :=
+      h.mono (Ioi_subset_Ioi zero_le_one) le_rfl
+    rw [integrableOn_Ioi_cpow_iff zero_lt_one] at this
+    exact hs.not_lt this
+
+theorem setIntegral_Ioi_zero_cpow (s : ℂ) : ∫ x in Ioi (0 : ℝ), (x : ℂ) ^ s = 0 :=
+  MeasureTheory.integral_undef (not_integrableOn_Ioi_cpow s)
+
 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) := by
   refine'

And fix some names in comments where this revealed issues

@@ -19,9 +19,9 @@ half-infinite intervals in `ℝ`.
 
 ## See also
 
-- `analysis.special_functions.integrals` -- integrals over finite intervals
-- `analysis.special_functions.gaussian` -- integral of `exp (-x ^ 2)`
-- `analysis.special_functions.japanese_bracket`-- integrability of `(1+‖x‖)^(-r)`.
+- `Mathlib.Analysis.SpecialFunctions.Integrals` -- integrals over finite intervals
+- `Mathlib.Analysis.SpecialFunctions.Gaussian` -- integral of `exp (-x ^ 2)`
+- `Mathlib.Analysis.SpecialFunctions.JapaneseBracket`-- integrability of `(1+‖x‖)^(-r)`.
 -/
 
 

I want to use the lemma lintegral_add_right_eq_self in a file that doesn't import Bochner integration.

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: David Loeffler
 -/
 import Mathlib.Analysis.SpecialFunctions.Integrals
-import Mathlib.MeasureTheory.Group.Integration
+import Mathlib.MeasureTheory.Group.Integral
 import Mathlib.MeasureTheory.Integral.ExpDecay
 import Mathlib.MeasureTheory.Integral.IntegralEqImproper
 import Mathlib.MeasureTheory.Measure.Lebesgue.Integral

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,11 +2,6 @@
 Copyright (c) 2023 David Loeffler. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: David Loeffler
-
-! This file was ported from Lean 3 source module analysis.special_functions.improper_integrals
-! leanprover-community/mathlib commit fd5edc43dc4f10b85abfe544b88f82cf13c5f844
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.SpecialFunctions.Integrals
 import Mathlib.MeasureTheory.Group.Integration
@@ -14,6 +9,8 @@ import Mathlib.MeasureTheory.Integral.ExpDecay
 import Mathlib.MeasureTheory.Integral.IntegralEqImproper
 import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
 
+#align_import analysis.special_functions.improper_integrals from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
+
 /-!
 # Evaluation of specific improper integrals
 

@@ -41,7 +41,7 @@ theorem integrableOn_exp_Iic (c : ℝ) : IntegrableOn exp (Iic c) := by
   exact (exp_pos _).le
 #align integrable_on_exp_Iic integrableOn_exp_Iic
 
-theorem integral_exp_Iic (c : ℝ) : (∫ x : ℝ in Iic c, exp x) = exp c := by
+theorem integral_exp_Iic (c : ℝ) : ∫ x : ℝ in Iic c, exp x = exp c := by
   refine'
     tendsto_nhds_unique
       (intervalIntegral_tendsto_integral_Iic _ (integrableOn_exp_Iic _) tendsto_id) _
@@ -49,7 +49,7 @@ theorem integral_exp_Iic (c : ℝ) : (∫ x : ℝ in Iic c, exp x) = exp c := by
   exact tendsto_exp_atBot.const_sub _
 #align integral_exp_Iic integral_exp_Iic
 
-theorem integral_exp_Iic_zero : (∫ x : ℝ in Iic 0, exp x) = 1 :=
+theorem integral_exp_Iic_zero : ∫ x : ℝ in Iic 0, exp x = 1 :=
   exp_zero ▸ integral_exp_Iic 0
 #align integral_exp_Iic_zero integral_exp_Iic_zero
 
@@ -77,7 +77,7 @@ theorem integrableOn_Ioi_rpow_of_lt {a : ℝ} (ha : a < -1) {c : ℝ} (hc : 0 <
 #align integrable_on_Ioi_rpow_of_lt integrableOn_Ioi_rpow_of_lt
 
 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) := by
+    ∫ t : ℝ in Ioi c, t ^ a = -c ^ (a + 1) / (a + 1) := by
   have hd : ∀ (x : ℝ) (_ : x ∈ Ici c), HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x := by
     intro x hx
     convert (hasDerivAt_rpow_const (p := a + 1) (Or.inl (hc.trans_le hx).ne')).div_const _ using 1

@@ -97,7 +97,7 @@ theorem integrableOn_Ioi_cpow_of_lt {a : ℂ} (ha : a.re < -1) {c : ℝ} (hc : 0
     rw [Complex.norm_eq_abs, Complex.abs_cpow_eq_rpow_re_of_pos (hc.trans hx)]
   · refine' ContinuousOn.aestronglyMeasurable (fun t ht => _) measurableSet_Ioi
     exact
-      (Complex.continuousAt_of_real_cpow_const _ _ (Or.inr (hc.trans ht).ne')).continuousWithinAt
+      (Complex.continuousAt_ofReal_cpow_const _ _ (Or.inr (hc.trans ht).ne')).continuousWithinAt
 #align integrable_on_Ioi_cpow_of_lt integrableOn_Ioi_cpow_of_lt
 
 theorem integral_Ioi_cpow_of_lt {a : ℂ} (ha : a.re < -1) {c : ℝ} (hc : 0 < c) :