state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
s : ℂ
n : ℕ
hn : n ≠ 0
| ↑n / (↑n + 1 + s) * GammaSeq s n | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [GammaSeq, Finset.prod_range_succ', Nat.cast_zero, add_zero, div_mul_div_comm, ← mul_assoc,
← mul_assoc, mul_comm _ (Finset.prod _ _)] | theorem GammaSeq_add_one_left (s : ℂ) {n : ℕ} (hn : n ≠ 0) :
GammaSeq (s + 1) n / s = n / (n + 1 + s) * GammaSeq s n := by
conv_lhs => rw [GammaSeq, Finset.prod_range_succ, div_div]
conv_rhs =>
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.251_0.in2QiCFW52coQT2 | theorem GammaSeq_add_one_left (s : ℂ) {n : ℕ} (hn : n ≠ 0) :
GammaSeq (s + 1) n / s = n / (n + 1 + s) * GammaSeq s n | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
n : ℕ
hn : n ≠ 0
| ↑n / (↑n + 1 + s) * GammaSeq s n | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [GammaSeq, Finset.prod_range_succ', Nat.cast_zero, add_zero, div_mul_div_comm, ← mul_assoc,
← mul_assoc, mul_comm _ (Finset.prod _ _)] | theorem GammaSeq_add_one_left (s : ℂ) {n : ℕ} (hn : n ≠ 0) :
GammaSeq (s + 1) n / s = n / (n + 1 + s) * GammaSeq s n := by
conv_lhs => rw [GammaSeq, Finset.prod_range_succ, div_div]
conv_rhs =>
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.251_0.in2QiCFW52coQT2 | theorem GammaSeq_add_one_left (s : ℂ) {n : ℕ} (hn : n ≠ 0) :
GammaSeq (s + 1) n / s = n / (n + 1 + s) * GammaSeq s n | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
n : ℕ
hn : n ≠ 0
| ↑n / (↑n + 1 + s) * GammaSeq s n | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [GammaSeq, Finset.prod_range_succ', Nat.cast_zero, add_zero, div_mul_div_comm, ← mul_assoc,
← mul_assoc, mul_comm _ (Finset.prod _ _)] | theorem GammaSeq_add_one_left (s : ℂ) {n : ℕ} (hn : n ≠ 0) :
GammaSeq (s + 1) n / s = n / (n + 1 + s) * GammaSeq s n := by
conv_lhs => rw [GammaSeq, Finset.prod_range_succ, div_div]
conv_rhs =>
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.251_0.in2QiCFW52coQT2 | theorem GammaSeq_add_one_left (s : ℂ) {n : ℕ} (hn : n ≠ 0) :
GammaSeq (s + 1) n / s = n / (n + 1 + s) * GammaSeq s n | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
n : ℕ
hn : n ≠ 0
⊢ ↑n ^ (s + 1) * ↑n ! / ((∏ x in Finset.range n, (s + 1 + ↑x)) * (s + 1 + ↑n) * s) =
↑n * ↑n ^ s * ↑n ! / ((∏ k in Finset.range n, (s + ↑(k + 1))) * (↑n + 1 + s) * s) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | congr 3 | theorem GammaSeq_add_one_left (s : ℂ) {n : ℕ} (hn : n ≠ 0) :
GammaSeq (s + 1) n / s = n / (n + 1 + s) * GammaSeq s n := by
conv_lhs => rw [GammaSeq, Finset.prod_range_succ, div_div]
conv_rhs =>
rw [GammaSeq, Finset.prod_range_succ', Nat.cast_zero, add_zero, div_mul_div_comm, ← mul_assoc,
← mul_assoc, ... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.251_0.in2QiCFW52coQT2 | theorem GammaSeq_add_one_left (s : ℂ) {n : ℕ} (hn : n ≠ 0) :
GammaSeq (s + 1) n / s = n / (n + 1 + s) * GammaSeq s n | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case e_a.e_a
s : ℂ
n : ℕ
hn : n ≠ 0
⊢ ↑n ^ (s + 1) = ↑n * ↑n ^ s | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [cpow_add _ _ (Nat.cast_ne_zero.mpr hn), cpow_one, mul_comm] | theorem GammaSeq_add_one_left (s : ℂ) {n : ℕ} (hn : n ≠ 0) :
GammaSeq (s + 1) n / s = n / (n + 1 + s) * GammaSeq s n := by
conv_lhs => rw [GammaSeq, Finset.prod_range_succ, div_div]
conv_rhs =>
rw [GammaSeq, Finset.prod_range_succ', Nat.cast_zero, add_zero, div_mul_div_comm, ← mul_assoc,
← mul_assoc, ... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.251_0.in2QiCFW52coQT2 | theorem GammaSeq_add_one_left (s : ℂ) {n : ℕ} (hn : n ≠ 0) :
GammaSeq (s + 1) n / s = n / (n + 1 + s) * GammaSeq s n | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case e_a.e_a.e_a
s : ℂ
n : ℕ
hn : n ≠ 0
⊢ ∏ x in Finset.range n, (s + 1 + ↑x) = ∏ k in Finset.range n, (s + ↑(k + 1)) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | refine' Finset.prod_congr (by rfl) fun x _ => _ | theorem GammaSeq_add_one_left (s : ℂ) {n : ℕ} (hn : n ≠ 0) :
GammaSeq (s + 1) n / s = n / (n + 1 + s) * GammaSeq s n := by
conv_lhs => rw [GammaSeq, Finset.prod_range_succ, div_div]
conv_rhs =>
rw [GammaSeq, Finset.prod_range_succ', Nat.cast_zero, add_zero, div_mul_div_comm, ← mul_assoc,
← mul_assoc, ... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.251_0.in2QiCFW52coQT2 | theorem GammaSeq_add_one_left (s : ℂ) {n : ℕ} (hn : n ≠ 0) :
GammaSeq (s + 1) n / s = n / (n + 1 + s) * GammaSeq s n | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
n : ℕ
hn : n ≠ 0
⊢ Finset.range n = Finset.range n | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rfl | theorem GammaSeq_add_one_left (s : ℂ) {n : ℕ} (hn : n ≠ 0) :
GammaSeq (s + 1) n / s = n / (n + 1 + s) * GammaSeq s n := by
conv_lhs => rw [GammaSeq, Finset.prod_range_succ, div_div]
conv_rhs =>
rw [GammaSeq, Finset.prod_range_succ', Nat.cast_zero, add_zero, div_mul_div_comm, ← mul_assoc,
← mul_assoc, ... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.251_0.in2QiCFW52coQT2 | theorem GammaSeq_add_one_left (s : ℂ) {n : ℕ} (hn : n ≠ 0) :
GammaSeq (s + 1) n / s = n / (n + 1 + s) * GammaSeq s n | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case e_a.e_a.e_a
s : ℂ
n : ℕ
hn : n ≠ 0
x : ℕ
x✝ : x ∈ Finset.range n
⊢ s + 1 + ↑x = s + ↑(x + 1) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | push_cast | theorem GammaSeq_add_one_left (s : ℂ) {n : ℕ} (hn : n ≠ 0) :
GammaSeq (s + 1) n / s = n / (n + 1 + s) * GammaSeq s n := by
conv_lhs => rw [GammaSeq, Finset.prod_range_succ, div_div]
conv_rhs =>
rw [GammaSeq, Finset.prod_range_succ', Nat.cast_zero, add_zero, div_mul_div_comm, ← mul_assoc,
← mul_assoc, ... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.251_0.in2QiCFW52coQT2 | theorem GammaSeq_add_one_left (s : ℂ) {n : ℕ} (hn : n ≠ 0) :
GammaSeq (s + 1) n / s = n / (n + 1 + s) * GammaSeq s n | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case e_a.e_a.e_a
s : ℂ
n : ℕ
hn : n ≠ 0
x : ℕ
x✝ : x ∈ Finset.range n
⊢ s + 1 + ↑x = s + (↑x + 1) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | ring | theorem GammaSeq_add_one_left (s : ℂ) {n : ℕ} (hn : n ≠ 0) :
GammaSeq (s + 1) n / s = n / (n + 1 + s) * GammaSeq s n := by
conv_lhs => rw [GammaSeq, Finset.prod_range_succ, div_div]
conv_rhs =>
rw [GammaSeq, Finset.prod_range_succ', Nat.cast_zero, add_zero, div_mul_div_comm, ← mul_assoc,
← mul_assoc, ... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.251_0.in2QiCFW52coQT2 | theorem GammaSeq_add_one_left (s : ℂ) {n : ℕ} (hn : n ≠ 0) :
GammaSeq (s + 1) n / s = n / (n + 1 + s) * GammaSeq s n | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case e_a.e_a.e_a
s : ℂ
n : ℕ
hn : n ≠ 0
⊢ s + 1 + ↑n = ↑n + 1 + s | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | abel | theorem GammaSeq_add_one_left (s : ℂ) {n : ℕ} (hn : n ≠ 0) :
GammaSeq (s + 1) n / s = n / (n + 1 + s) * GammaSeq s n := by
conv_lhs => rw [GammaSeq, Finset.prod_range_succ, div_div]
conv_rhs =>
rw [GammaSeq, Finset.prod_range_succ', Nat.cast_zero, add_zero, div_mul_div_comm, ← mul_assoc,
← mul_assoc, ... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.251_0.in2QiCFW52coQT2 | theorem GammaSeq_add_one_left (s : ℂ) {n : ℕ} (hn : n ≠ 0) :
GammaSeq (s + 1) n / s = n / (n + 1 + s) * GammaSeq s n | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case e_a.e_a.e_a
s : ℂ
n : ℕ
hn : n ≠ 0
⊢ s + 1 + ↑n = ↑n + 1 + s | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | abel | theorem GammaSeq_add_one_left (s : ℂ) {n : ℕ} (hn : n ≠ 0) :
GammaSeq (s + 1) n / s = n / (n + 1 + s) * GammaSeq s n := by
conv_lhs => rw [GammaSeq, Finset.prod_range_succ, div_div]
conv_rhs =>
rw [GammaSeq, Finset.prod_range_succ', Nat.cast_zero, add_zero, div_mul_div_comm, ← mul_assoc,
← mul_assoc, ... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.251_0.in2QiCFW52coQT2 | theorem GammaSeq_add_one_left (s : ℂ) {n : ℕ} (hn : n ≠ 0) :
GammaSeq (s + 1) n / s = n / (n + 1 + s) * GammaSeq s n | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
⊢ GammaSeq s n = ∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | have : ∀ x : ℝ, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
⊢ ∀ (x : ℝ), x = x / ↑n * ↑n | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | intro x | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
x : ℝ
⊢ x = x / ↑n * ↑n | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [div_mul_cancel] | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by intro x; | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
x : ℝ
⊢ ↑n ≠ 0 | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | exact Nat.cast_ne_zero.mpr hn | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by intro x; rw [div_mul_cancel]; | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this : ∀ (x : ℝ), x = x / ↑n * ↑n
⊢ GammaSeq s n = ∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | conv_rhs => enter [1, x, 2, 1]; rw [this x] | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this : ∀ (x : ℝ), x = x / ↑n * ↑n
| ∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | enter [1, x, 2, 1]; rw [this x] | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn
conv_rhs => | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this : ∀ (x : ℝ), x = x / ↑n * ↑n
| ∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | enter [1, x, 2, 1]; rw [this x] | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn
conv_rhs => | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this : ∀ (x : ℝ), x = x / ↑n * ↑n
| ∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | enter [1, x, 2, 1] | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn
conv_rhs => | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this : ∀ (x : ℝ), x = x / ↑n * ↑n
x : ℝ
| ↑x | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [this x] | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn
conv_rhs => enter [1, x, 2, 1]; | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this : ∀ (x : ℝ), x = x / ↑n * ↑n
⊢ GammaSeq s n = ∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑(x / ↑n * ↑n) ^ (s - 1) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [GammaSeq_eq_betaIntegral_of_re_pos hs] | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn
conv_rhs => enter [1, x, 2, 1]; rw [this x]
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this : ∀ (x : ℝ), x = x / ↑n * ↑n
⊢ ↑n ^ s * betaIntegral s (↑n + 1) = ∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑(x / ↑n * ↑n) ^ (s - 1) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | have := intervalIntegral.integral_comp_div (a := 0) (b := n)
(fun x => ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1) : ℝ → ℂ) (Nat.cast_ne_zero.mpr hn) | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn
conv_rhs => enter [1, x, 2, 1]; rw [this x]
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this✝ : ∀ (x : ℝ), x = x / ↑n * ↑n
this :
∫ (x : ℝ) in 0 ..↑n, (fun x => ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1)) (x / ↑n) =
↑n • ∫ (x : ℝ) in 0 / ↑n..↑n / ↑n, (fun x => ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1)) x
⊢ ↑n ^ s * betaIntegral s (↑n + 1) = ∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑... | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | dsimp only at this | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn
conv_rhs => enter [1, x, 2, 1]; rw [this x]
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this✝ : ∀ (x : ℝ), x = x / ↑n * ↑n
this :
∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑(x / ↑n * ↑n) ^ (s - 1) =
↑n • ∫ (x : ℝ) in 0 / ↑n..↑n / ↑n, ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1)
⊢ ↑n ^ s * betaIntegral s (↑n + 1) = ∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑(x / ↑n * ↑n... | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [betaIntegral, this, real_smul, zero_div, div_self, add_sub_cancel,
← intervalIntegral.integral_const_mul, ← intervalIntegral.integral_const_mul] | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn
conv_rhs => enter [1, x, 2, 1]; rw [this x]
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this✝ : ∀ (x : ℝ), x = x / ↑n * ↑n
this :
∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑(x / ↑n * ↑n) ^ (s - 1) =
↑n • ∫ (x : ℝ) in 0 / ↑n..↑n / ↑n, ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1)
⊢ ∫ (x : ℝ) in 0 ..1, ↑n ^ s * (↑x ^ (s - 1) * (1 - ↑x) ^ ↑n) =
∫ (x : ℝ) in 0 ..1, ↑↑n *... | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | swap | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn
conv_rhs => enter [1, x, 2, 1]; rw [this x]
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this✝ : ∀ (x : ℝ), x = x / ↑n * ↑n
this :
∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑(x / ↑n * ↑n) ^ (s - 1) =
↑n • ∫ (x : ℝ) in 0 / ↑n..↑n / ↑n, ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1)
⊢ ↑n ≠ 0 | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | exact Nat.cast_ne_zero.mpr hn | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn
conv_rhs => enter [1, x, 2, 1]; rw [this x]
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this✝ : ∀ (x : ℝ), x = x / ↑n * ↑n
this :
∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑(x / ↑n * ↑n) ^ (s - 1) =
↑n • ∫ (x : ℝ) in 0 / ↑n..↑n / ↑n, ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1)
⊢ ∫ (x : ℝ) in 0 ..1, ↑n ^ s * (↑x ^ (s - 1) * (1 - ↑x) ^ ↑n) =
∫ (x : ℝ) in 0 ..1, ↑↑n *... | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | simp_rw [intervalIntegral.integral_of_le zero_le_one] | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn
conv_rhs => enter [1, x, 2, 1]; rw [this x]
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this✝ : ∀ (x : ℝ), x = x / ↑n * ↑n
this :
∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑(x / ↑n * ↑n) ^ (s - 1) =
↑n • ∫ (x : ℝ) in 0 / ↑n..↑n / ↑n, ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1)
⊢ ∫ (x : ℝ) in Ioc 0 1, ↑n ^ s * (↑x ^ (s - 1) * (1 - ↑x) ^ ↑n) ∂volume =
∫ (x : ℝ) in Io... | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | refine' set_integral_congr measurableSet_Ioc fun x hx => _ | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn
conv_rhs => enter [1, x, 2, 1]; rw [this x]
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this✝ : ∀ (x : ℝ), x = x / ↑n * ↑n
this :
∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑(x / ↑n * ↑n) ^ (s - 1) =
↑n • ∫ (x : ℝ) in 0 / ↑n..↑n / ↑n, ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1)
x : ℝ
hx : x ∈ Ioc 0 1
⊢ ↑n ^ s * (↑x ^ (s - 1) * (1 - ↑x) ^ ↑n) = ↑↑n * (↑((1 - x) ^ n) * ↑(... | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | push_cast | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn
conv_rhs => enter [1, x, 2, 1]; rw [this x]
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this✝ : ∀ (x : ℝ), x = x / ↑n * ↑n
this :
∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑(x / ↑n * ↑n) ^ (s - 1) =
↑n • ∫ (x : ℝ) in 0 / ↑n..↑n / ↑n, ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1)
x : ℝ
hx : x ∈ Ioc 0 1
⊢ ↑n ^ s * (↑x ^ (s - 1) * (1 - ↑x) ^ ↑n) = ↑n * ((1 - ↑x) ^ n * (↑x *... | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | have hn' : (n : ℂ) ≠ 0 := Nat.cast_ne_zero.mpr hn | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn
conv_rhs => enter [1, x, 2, 1]; rw [this x]
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this✝ : ∀ (x : ℝ), x = x / ↑n * ↑n
this :
∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑(x / ↑n * ↑n) ^ (s - 1) =
↑n • ∫ (x : ℝ) in 0 / ↑n..↑n / ↑n, ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1)
x : ℝ
hx : x ∈ Ioc 0 1
hn' : ↑n ≠ 0
⊢ ↑n ^ s * (↑x ^ (s - 1) * (1 - ↑x) ^ ↑n) = ↑n * ((1 - ↑x... | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | have A : (n : ℂ) ^ s = (n : ℂ) ^ (s - 1) * n := by
conv_lhs => rw [(by ring : s = s - 1 + 1), cpow_add _ _ hn']
simp | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn
conv_rhs => enter [1, x, 2, 1]; rw [this x]
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this✝ : ∀ (x : ℝ), x = x / ↑n * ↑n
this :
∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑(x / ↑n * ↑n) ^ (s - 1) =
↑n • ∫ (x : ℝ) in 0 / ↑n..↑n / ↑n, ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1)
x : ℝ
hx : x ∈ Ioc 0 1
hn' : ↑n ≠ 0
⊢ ↑n ^ s = ↑n ^ (s - 1) * ↑n | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | conv_lhs => rw [(by ring : s = s - 1 + 1), cpow_add _ _ hn'] | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn
conv_rhs => enter [1, x, 2, 1]; rw [this x]
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this✝ : ∀ (x : ℝ), x = x / ↑n * ↑n
this :
∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑(x / ↑n * ↑n) ^ (s - 1) =
↑n • ∫ (x : ℝ) in 0 / ↑n..↑n / ↑n, ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1)
x : ℝ
hx : x ∈ Ioc 0 1
hn' : ↑n ≠ 0
| ↑n ^ s | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [(by ring : s = s - 1 + 1), cpow_add _ _ hn'] | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn
conv_rhs => enter [1, x, 2, 1]; rw [this x]
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this✝ : ∀ (x : ℝ), x = x / ↑n * ↑n
this :
∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑(x / ↑n * ↑n) ^ (s - 1) =
↑n • ∫ (x : ℝ) in 0 / ↑n..↑n / ↑n, ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1)
x : ℝ
hx : x ∈ Ioc 0 1
hn' : ↑n ≠ 0
| ↑n ^ s | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [(by ring : s = s - 1 + 1), cpow_add _ _ hn'] | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn
conv_rhs => enter [1, x, 2, 1]; rw [this x]
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this✝ : ∀ (x : ℝ), x = x / ↑n * ↑n
this :
∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑(x / ↑n * ↑n) ^ (s - 1) =
↑n • ∫ (x : ℝ) in 0 / ↑n..↑n / ↑n, ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1)
x : ℝ
hx : x ∈ Ioc 0 1
hn' : ↑n ≠ 0
| ↑n ^ s | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [(by ring : s = s - 1 + 1), cpow_add _ _ hn'] | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn
conv_rhs => enter [1, x, 2, 1]; rw [this x]
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this✝ : ∀ (x : ℝ), x = x / ↑n * ↑n
this :
∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑(x / ↑n * ↑n) ^ (s - 1) =
↑n • ∫ (x : ℝ) in 0 / ↑n..↑n / ↑n, ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1)
x : ℝ
hx : x ∈ Ioc 0 1
hn' : ↑n ≠ 0
⊢ s = s - 1 + 1 | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | ring | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn
conv_rhs => enter [1, x, 2, 1]; rw [this x]
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this✝ : ∀ (x : ℝ), x = x / ↑n * ↑n
this :
∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑(x / ↑n * ↑n) ^ (s - 1) =
↑n • ∫ (x : ℝ) in 0 / ↑n..↑n / ↑n, ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1)
x : ℝ
hx : x ∈ Ioc 0 1
hn' : ↑n ≠ 0
⊢ ↑n ^ (s - 1) * ↑n ^ 1 = ↑n ^ (s - 1) * ↑n | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | simp | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn
conv_rhs => enter [1, x, 2, 1]; rw [this x]
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this✝ : ∀ (x : ℝ), x = x / ↑n * ↑n
this :
∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑(x / ↑n * ↑n) ^ (s - 1) =
↑n • ∫ (x : ℝ) in 0 / ↑n..↑n / ↑n, ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1)
x : ℝ
hx : x ∈ Ioc 0 1
hn' : ↑n ≠ 0
A : ↑n ^ s = ↑n ^ (s - 1) * ↑n
⊢ ↑n ^ s * (↑x ^ (s - 1) *... | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | have B : ((x : ℂ) * ↑n) ^ (s - 1) = (x : ℂ) ^ (s - 1) * (n : ℂ) ^ (s - 1) := by
rw [← ofReal_nat_cast,
mul_cpow_ofReal_nonneg hx.1.le (Nat.cast_pos.mpr (Nat.pos_of_ne_zero hn)).le] | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn
conv_rhs => enter [1, x, 2, 1]; rw [this x]
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this✝ : ∀ (x : ℝ), x = x / ↑n * ↑n
this :
∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑(x / ↑n * ↑n) ^ (s - 1) =
↑n • ∫ (x : ℝ) in 0 / ↑n..↑n / ↑n, ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1)
x : ℝ
hx : x ∈ Ioc 0 1
hn' : ↑n ≠ 0
A : ↑n ^ s = ↑n ^ (s - 1) * ↑n
⊢ (↑x * ↑n) ^ (s - 1) = ↑x... | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [← ofReal_nat_cast,
mul_cpow_ofReal_nonneg hx.1.le (Nat.cast_pos.mpr (Nat.pos_of_ne_zero hn)).le] | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn
conv_rhs => enter [1, x, 2, 1]; rw [this x]
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this✝ : ∀ (x : ℝ), x = x / ↑n * ↑n
this :
∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑(x / ↑n * ↑n) ^ (s - 1) =
↑n • ∫ (x : ℝ) in 0 / ↑n..↑n / ↑n, ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1)
x : ℝ
hx : x ∈ Ioc 0 1
hn' : ↑n ≠ 0
A : ↑n ^ s = ↑n ^ (s - 1) * ↑n
B : (↑x * ↑n) ^ (s - 1) = ... | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [A, B, cpow_nat_cast] | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn
conv_rhs => enter [1, x, 2, 1]; rw [this x]
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this✝ : ∀ (x : ℝ), x = x / ↑n * ↑n
this :
∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑(x / ↑n * ↑n) ^ (s - 1) =
↑n • ∫ (x : ℝ) in 0 / ↑n..↑n / ↑n, ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1)
x : ℝ
hx : x ∈ Ioc 0 1
hn' : ↑n ≠ 0
A : ↑n ^ s = ↑n ^ (s - 1) * ↑n
B : (↑x * ↑n) ^ (s - 1) = ... | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | ring | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) := by
have : ∀ x : ℝ, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn
conv_rhs => enter [1, x, 2, 1]; rw [this x]
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.264_0.in2QiCFW52coQT2 | theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
GammaSeq s n = ∫ x : ℝ in (0)..n, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
⊢ Tendsto (fun n => ∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)) atTop (𝓝 (Gamma s)) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [Gamma_eq_integral hs] | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
⊢ Tendsto (fun n => ∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)) atTop (𝓝 (GammaIntegral s)) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | let f : ℕ → ℝ → ℂ := fun n =>
indicator (Ioc 0 (n : ℝ)) fun x : ℝ => ((1 - x / n) ^ n : ℝ) * (x : ℂ) ^ (s - 1) | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
⊢ Tendsto (fun n => ∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)) atTop (𝓝 (GammaIntegral s)) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | have f_ible : ∀ n : ℕ, Integrable (f n) (volume.restrict (Ioi 0)) := by
intro n
rw [integrable_indicator_iff (measurableSet_Ioc : MeasurableSet (Ioc (_ : ℝ) _)), IntegrableOn,
Measure.restrict_restrict_of_subset Ioc_subset_Ioi_self, ← IntegrableOn, ←
intervalIntegrable_iff_integrableOn_Ioc_of_le (by... | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
⊢ ∀ (n : ℕ), Integrable (f n) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | intro n | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
n : ℕ
⊢ Integrable (f n) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [integrable_indicator_iff (measurableSet_Ioc : MeasurableSet (Ioc (_ : ℝ) _)), IntegrableOn,
Measure.restrict_restrict_of_subset Ioc_subset_Ioi_self, ← IntegrableOn, ←
intervalIntegrable_iff_integrableOn_Ioc_of_le (by positivity : (0 : ℝ) ≤ n)] | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
n : ℕ
⊢ 0 ≤ ↑n | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | positivity | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
n : ℕ
⊢ IntervalIntegrable (fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)) volume 0 ↑n | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | apply IntervalIntegrable.continuousOn_mul | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hf
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
n : ℕ
⊢ IntervalIntegrable (fun x => ↑x ^ (s - 1)) volume 0 ↑n | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | refine' intervalIntegral.intervalIntegrable_cpow' _ | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hf
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
n : ℕ
⊢ -1 < (s - 1).re | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rwa [sub_re, one_re, ← zero_sub, sub_lt_sub_iff_right] | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hg
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
n : ℕ
⊢ ContinuousOn (fun x => ↑((1 - x / ↑n) ^ n)) (uIcc 0 ↑n) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | apply Continuous.continuousOn | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hg.h
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
n : ℕ
⊢ Continuous fun x => ↑((1 - x / ↑n) ^ n) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | exact IsROrC.continuous_ofReal.comp -- Porting note: was `continuity`
((continuous_const.sub (continuous_id'.div_const ↑n)).pow n) | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n)
⊢ Tendsto (fun n => ∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)) atTop (𝓝 (GammaIntegral s)) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | have f_tends : ∀ x : ℝ, x ∈ Ioi (0 : ℝ) →
Tendsto (fun n : ℕ => f n x) atTop (𝓝 <| ↑(Real.exp (-x)) * (x : ℂ) ^ (s - 1)) := by
intro x hx
apply Tendsto.congr'
show ∀ᶠ n : ℕ in atTop, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) = f n x
· refine' Eventually.mp (eventually_ge_atTop ⌈x⌉₊) (eventually_of_f... | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n)
⊢ ∀ x ∈ Ioi 0, Tendsto (fun n => f n x) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1))) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | intro x hx | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n)
x : ℝ
hx : x ∈ Ioi 0
⊢ Tendsto (fun n => f n x) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1))) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | apply Tendsto.congr' | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hl
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n)
x : ℝ
hx : x ∈ Ioi 0
⊢ ?f₁ =ᶠ[atTop] fun n => f n x
case h
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s... | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | show ∀ᶠ n : ℕ in atTop, ↑((1 - x / n) ^ n) * (x : ℂ) ^ (s - 1) = f n x | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hl
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n)
x : ℝ
hx : x ∈ Ioi 0
⊢ ∀ᶠ (n : ℕ) in atTop, ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1) = f n x | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | refine' Eventually.mp (eventually_ge_atTop ⌈x⌉₊) (eventually_of_forall fun n hn => _) | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hl
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n)
x : ℝ
hx : x ∈ Ioi 0
n : ℕ
hn : ⌈x⌉₊ ≤ n
⊢ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1) = f n x | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [Nat.ceil_le] at hn | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hl
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n)
x : ℝ
hx : x ∈ Ioi 0
n : ℕ
hn : x ≤ ↑n
⊢ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1) = f n x | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | dsimp only | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hl
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n)
x : ℝ
hx : x ∈ Ioi 0
n : ℕ
hn : x ≤ ↑n
⊢ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1) = indicator (Ioc 0 ↑n) (fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)) x | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [indicator_of_mem] | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hl.h
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n)
x : ℝ
hx : x ∈ Ioi 0
n : ℕ
hn : x ≤ ↑n
⊢ x ∈ Ioc 0 ↑n | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | exact ⟨hx, hn⟩ | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n)
x : ℝ
hx : x ∈ Ioi 0
⊢ Tendsto (fun x_1 => ↑((1 - x / ↑x_1) ^ x_1) * ↑x ^ (s - 1)) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1))) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | simp_rw [mul_comm] | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n)
x : ℝ
hx : x ∈ Ioi 0
⊢ Tendsto (fun x_1 => ↑x ^ (s - 1) * ↑((1 - x / ↑x_1) ^ x_1)) atTop (𝓝 (↑x ^ (s - 1) * ↑(rexp (-x)))) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | refine' (Tendsto.comp (continuous_ofReal.tendsto _) _).const_mul _ | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n)
x : ℝ
hx : x ∈ Ioi 0
⊢ Tendsto (fun x_1 => (1 - x / ↑x_1) ^ x_1) atTop (𝓝 (rexp (-x))) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | convert tendsto_one_plus_div_pow_exp (-x) using 1 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h.e'_3
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n)
x : ℝ
hx : x ∈ Ioi 0
⊢ (fun x_1 => (1 - x / ↑x_1) ^ x_1) = fun x_1 => (1 + -x / ↑x_1) ^ x_1 | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | ext1 n | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h.e'_3.h
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n)
x : ℝ
hx : x ∈ Ioi 0
n : ℕ
⊢ (1 - x / ↑n) ^ n = (1 + -x / ↑n) ^ n | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [neg_div, ← sub_eq_add_neg] | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n)
f_tends : ∀ x ∈ Ioi 0, Tendsto (fun n => f n x) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1)))
⊢ Tendsto (fun n => ∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)) atTop ... | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | convert tendsto_integral_of_dominated_convergence _ (fun n => (f_ible n).1)
(Real.GammaIntegral_convergent hs) _
((ae_restrict_iff' measurableSet_Ioi).mpr (ae_of_all _ f_tends)) using 1 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h.e'_3
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n)
f_tends : ∀ x ∈ Ioi 0, Tendsto (fun n => f n x) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1)))
⊢ (fun n => ∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)) = ... | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | ext1 n | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h.e'_3.h
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n)
f_tends : ∀ x ∈ Ioi 0, Tendsto (fun n => f n x) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1)))
n : ℕ
⊢ ∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1) = ∫ (... | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [integral_indicator (measurableSet_Ioc : MeasurableSet (Ioc (_ : ℝ) _)),
intervalIntegral.integral_of_le (by positivity : 0 ≤ (n : ℝ)),
Measure.restrict_restrict_of_subset Ioc_subset_Ioi_self] | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n)
f_tends : ∀ x ∈ Ioi 0, Tendsto (fun n => f n x) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1)))
n : ℕ
⊢ 0 ≤ ↑n | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | positivity | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n)
f_tends : ∀ x ∈ Ioi 0, Tendsto (fun n => f n x) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1)))
⊢ ∀ (n : ℕ), ∀ᵐ (a : ℝ) ∂Measure.restrict volume (Ioi 0), ‖f n a‖ ≤ rexp (-a) * a ^... | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | intro n | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n)
f_tends : ∀ x ∈ Ioi 0, Tendsto (fun n => f n x) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1)))
n : ℕ
⊢ ∀ᵐ (a : ℝ) ∂Measure.restrict volume (Ioi 0), ‖f n a‖ ≤ rexp (-a) * a ^ (s.r... | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | refine' (ae_restrict_iff' measurableSet_Ioi).mpr (ae_of_all _ fun x hx => _) | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n)
f_tends : ∀ x ∈ Ioi 0, Tendsto (fun n => f n x) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1)))
n : ℕ
x : ℝ
hx : x ∈ Ioi 0
⊢ ‖f n x‖ ≤ rexp (-x) * x ^ (s.re - 1) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | dsimp only | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n)
f_tends : ∀ x ∈ Ioi 0, Tendsto (fun n => f n x) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1)))
n : ℕ
x : ℝ
hx : x ∈ Ioi 0
⊢ ‖indicator (Ioc 0 ↑n) (fun x => ↑((1 - x / ↑n) ^ n) * ... | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rcases lt_or_le (n : ℝ) x with (hxn | hxn) | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case inl
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n)
f_tends : ∀ x ∈ Ioi 0, Tendsto (fun n => f n x) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1)))
n : ℕ
x : ℝ
hx : x ∈ Ioi 0
hxn : ↑n < x
⊢ ‖indicator (Ioc 0 ↑n) (fun x => ... | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [indicator_of_not_mem (not_mem_Ioc_of_gt hxn), norm_zero,
mul_nonneg_iff_right_nonneg_of_pos (exp_pos _)] | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case inl
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n)
f_tends : ∀ x ∈ Ioi 0, Tendsto (fun n => f n x) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1)))
n : ℕ
x : ℝ
hx : x ∈ Ioi 0
hxn : ↑n < x
⊢ 0 ≤ x ^ (s.re - 1) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | exact rpow_nonneg_of_nonneg (le_of_lt hx) _ | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case inr
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n)
f_tends : ∀ x ∈ Ioi 0, Tendsto (fun n => f n x) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1)))
n : ℕ
x : ℝ
hx : x ∈ Ioi 0
hxn : x ≤ ↑n
⊢ ‖indicator (Ioc 0 ↑n) (fun x => ... | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [indicator_of_mem (mem_Ioc.mpr ⟨mem_Ioi.mp hx, hxn⟩), norm_mul, Complex.norm_eq_abs,
Complex.abs_of_nonneg
(pow_nonneg (sub_nonneg.mpr <| div_le_one_of_le hxn <| by positivity) _),
Complex.norm_eq_abs, abs_cpow_eq_rpow_re_of_pos hx, sub_re, one_re,
mul_le_mul_right (rpow_pos_of_pos ... | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n)
f_tends : ∀ x ∈ Ioi 0, Tendsto (fun n => f n x) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1)))
n : ℕ
x : ℝ
hx : x ∈ Ioi 0
hxn : x ≤ ↑n
⊢ 0 ≤ ↑n | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | positivity | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case inr
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => indicator (Ioc 0 ↑n) fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n)
f_tends : ∀ x ∈ Ioi 0, Tendsto (fun n => f n x) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1)))
n : ℕ
x : ℝ
hx : x ∈ Ioi 0
hxn : x ≤ ↑n
⊢ (1 - x / ↑n) ^ n ≤ rexp (-x) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | exact one_sub_div_pow_le_exp_neg hxn | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.288_0.in2QiCFW52coQT2 | /-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 -... | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
⊢ Tendsto (GammaSeq s) atTop (𝓝 (Gamma s)) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | suffices ∀ m : ℕ, -↑m < re s → Tendsto (GammaSeq s) atTop (𝓝 <| GammaAux m s) by
rw [Gamma]
apply this
rw [neg_lt]
rcases lt_or_le 0 (re s) with (hs | hs)
· exact (neg_neg_of_pos hs).trans_le (Nat.cast_nonneg _)
· refine' (Nat.lt_floor_add_one _).trans_le _
rw [sub_eq_neg_add, Nat.floor_a... | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) := by
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.351_0.in2QiCFW52coQT2 | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
this : ∀ (m : ℕ), -↑m < s.re → Tendsto (GammaSeq s) atTop (𝓝 (GammaAux m s))
⊢ Tendsto (GammaSeq s) atTop (𝓝 (Gamma s)) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [Gamma] | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) := by
suffices ∀ m : ℕ, -↑m < re s → Tendsto (GammaSeq s) atTop (𝓝 <| GammaAux m s) by
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.351_0.in2QiCFW52coQT2 | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
this : ∀ (m : ℕ), -↑m < s.re → Tendsto (GammaSeq s) atTop (𝓝 (GammaAux m s))
⊢ Tendsto (GammaSeq s) atTop (𝓝 (GammaAux ⌊1 - s.re⌋₊ s)) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | apply this | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) := by
suffices ∀ m : ℕ, -↑m < re s → Tendsto (GammaSeq s) atTop (𝓝 <| GammaAux m s) by
rw [Gamma]
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.351_0.in2QiCFW52coQT2 | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case a
s : ℂ
this : ∀ (m : ℕ), -↑m < s.re → Tendsto (GammaSeq s) atTop (𝓝 (GammaAux m s))
⊢ -↑⌊1 - s.re⌋₊ < s.re | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [neg_lt] | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) := by
suffices ∀ m : ℕ, -↑m < re s → Tendsto (GammaSeq s) atTop (𝓝 <| GammaAux m s) by
rw [Gamma]
apply this
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.351_0.in2QiCFW52coQT2 | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case a
s : ℂ
this : ∀ (m : ℕ), -↑m < s.re → Tendsto (GammaSeq s) atTop (𝓝 (GammaAux m s))
⊢ -s.re < ↑⌊1 - s.re⌋₊ | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rcases lt_or_le 0 (re s) with (hs | hs) | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) := by
suffices ∀ m : ℕ, -↑m < re s → Tendsto (GammaSeq s) atTop (𝓝 <| GammaAux m s) by
rw [Gamma]
apply this
rw [neg_lt]
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.351_0.in2QiCFW52coQT2 | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case a.inl
s : ℂ
this : ∀ (m : ℕ), -↑m < s.re → Tendsto (GammaSeq s) atTop (𝓝 (GammaAux m s))
hs : 0 < s.re
⊢ -s.re < ↑⌊1 - s.re⌋₊ | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | exact (neg_neg_of_pos hs).trans_le (Nat.cast_nonneg _) | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) := by
suffices ∀ m : ℕ, -↑m < re s → Tendsto (GammaSeq s) atTop (𝓝 <| GammaAux m s) by
rw [Gamma]
apply this
rw [neg_lt]
rcases lt_or_le 0 (re s) with (hs |... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.351_0.in2QiCFW52coQT2 | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case a.inr
s : ℂ
this : ∀ (m : ℕ), -↑m < s.re → Tendsto (GammaSeq s) atTop (𝓝 (GammaAux m s))
hs : s.re ≤ 0
⊢ -s.re < ↑⌊1 - s.re⌋₊ | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | refine' (Nat.lt_floor_add_one _).trans_le _ | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) := by
suffices ∀ m : ℕ, -↑m < re s → Tendsto (GammaSeq s) atTop (𝓝 <| GammaAux m s) by
rw [Gamma]
apply this
rw [neg_lt]
rcases lt_or_le 0 (re s) with (hs |... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.351_0.in2QiCFW52coQT2 | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case a.inr
s : ℂ
this : ∀ (m : ℕ), -↑m < s.re → Tendsto (GammaSeq s) atTop (𝓝 (GammaAux m s))
hs : s.re ≤ 0
⊢ ↑⌊-s.re⌋₊ + 1 ≤ ↑⌊1 - s.re⌋₊ | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [sub_eq_neg_add, Nat.floor_add_one (neg_nonneg.mpr hs), Nat.cast_add_one] | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) := by
suffices ∀ m : ℕ, -↑m < re s → Tendsto (GammaSeq s) atTop (𝓝 <| GammaAux m s) by
rw [Gamma]
apply this
rw [neg_lt]
rcases lt_or_le 0 (re s) with (hs |... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.351_0.in2QiCFW52coQT2 | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
⊢ ∀ (m : ℕ), -↑m < s.re → Tendsto (GammaSeq s) atTop (𝓝 (GammaAux m s)) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | intro m | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) := by
suffices ∀ m : ℕ, -↑m < re s → Tendsto (GammaSeq s) atTop (𝓝 <| GammaAux m s) by
rw [Gamma]
apply this
rw [neg_lt]
rcases lt_or_le 0 (re s) with (hs |... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.351_0.in2QiCFW52coQT2 | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s : ℂ
m : ℕ
⊢ -↑m < s.re → Tendsto (GammaSeq s) atTop (𝓝 (GammaAux m s)) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | induction' m with m IH generalizing s | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) := by
suffices ∀ m : ℕ, -↑m < re s → Tendsto (GammaSeq s) atTop (𝓝 <| GammaAux m s) by
rw [Gamma]
apply this
rw [neg_lt]
rcases lt_or_le 0 (re s) with (hs |... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.351_0.in2QiCFW52coQT2 | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case zero
s : ℂ
⊢ -↑Nat.zero < s.re → Tendsto (GammaSeq s) atTop (𝓝 (GammaAux Nat.zero s)) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | intro hs | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) := by
suffices ∀ m : ℕ, -↑m < re s → Tendsto (GammaSeq s) atTop (𝓝 <| GammaAux m s) by
rw [Gamma]
apply this
rw [neg_lt]
rcases lt_or_le 0 (re s) with (hs |... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.351_0.in2QiCFW52coQT2 | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case zero
s : ℂ
hs : -↑Nat.zero < s.re
⊢ Tendsto (GammaSeq s) atTop (𝓝 (GammaAux Nat.zero s)) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [Nat.cast_zero, neg_zero] at hs | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) := by
suffices ∀ m : ℕ, -↑m < re s → Tendsto (GammaSeq s) atTop (𝓝 <| GammaAux m s) by
rw [Gamma]
apply this
rw [neg_lt]
rcases lt_or_le 0 (re s) with (hs |... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.351_0.in2QiCFW52coQT2 | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case zero
s : ℂ
hs : 0 < s.re
⊢ Tendsto (GammaSeq s) atTop (𝓝 (GammaAux Nat.zero s)) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [← Gamma_eq_GammaAux] | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) := by
suffices ∀ m : ℕ, -↑m < re s → Tendsto (GammaSeq s) atTop (𝓝 <| GammaAux m s) by
rw [Gamma]
apply this
rw [neg_lt]
rcases lt_or_le 0 (re s) with (hs |... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.351_0.in2QiCFW52coQT2 | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case zero
s : ℂ
hs : 0 < s.re
⊢ Tendsto (GammaSeq s) atTop (𝓝 (Gamma s)) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | refine' Tendsto.congr' _ (approx_Gamma_integral_tendsto_Gamma_integral hs) | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) := by
suffices ∀ m : ℕ, -↑m < re s → Tendsto (GammaSeq s) atTop (𝓝 <| GammaAux m s) by
rw [Gamma]
apply this
rw [neg_lt]
rcases lt_or_le 0 (re s) with (hs |... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.351_0.in2QiCFW52coQT2 | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case zero
s : ℂ
hs : 0 < s.re
⊢ (fun n => ∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)) =ᶠ[atTop] GammaSeq s | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | refine' (eventually_ne_atTop 0).mp (eventually_of_forall fun n hn => _) | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) := by
suffices ∀ m : ℕ, -↑m < re s → Tendsto (GammaSeq s) atTop (𝓝 <| GammaAux m s) by
rw [Gamma]
apply this
rw [neg_lt]
rcases lt_or_le 0 (re s) with (hs |... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.351_0.in2QiCFW52coQT2 | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case zero
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
⊢ (fun n => ∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)) n = GammaSeq s n | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | exact (GammaSeq_eq_approx_Gamma_integral hs hn).symm | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) := by
suffices ∀ m : ℕ, -↑m < re s → Tendsto (GammaSeq s) atTop (𝓝 <| GammaAux m s) by
rw [Gamma]
apply this
rw [neg_lt]
rcases lt_or_le 0 (re s) with (hs |... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.351_0.in2QiCFW52coQT2 | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case zero.h1
s : ℂ
hs : 0 < s.re
⊢ -s.re < ↑Nat.zero | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rwa [Nat.cast_zero, neg_lt_zero] | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) := by
suffices ∀ m : ℕ, -↑m < re s → Tendsto (GammaSeq s) atTop (𝓝 <| GammaAux m s) by
rw [Gamma]
apply this
rw [neg_lt]
rcases lt_or_le 0 (re s) with (hs |... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.351_0.in2QiCFW52coQT2 | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case succ
m : ℕ
IH : ∀ (s : ℂ), -↑m < s.re → Tendsto (GammaSeq s) atTop (𝓝 (GammaAux m s))
s : ℂ
⊢ -↑(Nat.succ m) < s.re → Tendsto (GammaSeq s) atTop (𝓝 (GammaAux (Nat.succ m) s)) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | intro hs | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) := by
suffices ∀ m : ℕ, -↑m < re s → Tendsto (GammaSeq s) atTop (𝓝 <| GammaAux m s) by
rw [Gamma]
apply this
rw [neg_lt]
rcases lt_or_le 0 (re s) with (hs |... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.351_0.in2QiCFW52coQT2 | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case succ
m : ℕ
IH : ∀ (s : ℂ), -↑m < s.re → Tendsto (GammaSeq s) atTop (𝓝 (GammaAux m s))
s : ℂ
hs : -↑(Nat.succ m) < s.re
⊢ Tendsto (GammaSeq s) atTop (𝓝 (GammaAux (Nat.succ m) s)) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [Nat.cast_succ, neg_add, ← sub_eq_add_neg, sub_lt_iff_lt_add, ← one_re, ← add_re] at hs | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) := by
suffices ∀ m : ℕ, -↑m < re s → Tendsto (GammaSeq s) atTop (𝓝 <| GammaAux m s) by
rw [Gamma]
apply this
rw [neg_lt]
rcases lt_or_le 0 (re s) with (hs |... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.351_0.in2QiCFW52coQT2 | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case succ
m : ℕ
IH : ∀ (s : ℂ), -↑m < s.re → Tendsto (GammaSeq s) atTop (𝓝 (GammaAux m s))
s : ℂ
hs : -↑m < (s + 1).re
⊢ Tendsto (GammaSeq s) atTop (𝓝 (GammaAux (Nat.succ m) s)) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [GammaAux] | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) := by
suffices ∀ m : ℕ, -↑m < re s → Tendsto (GammaSeq s) atTop (𝓝 <| GammaAux m s) by
rw [Gamma]
apply this
rw [neg_lt]
rcases lt_or_le 0 (re s) with (hs |... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.351_0.in2QiCFW52coQT2 | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case succ
m : ℕ
IH : ∀ (s : ℂ), -↑m < s.re → Tendsto (GammaSeq s) atTop (𝓝 (GammaAux m s))
s : ℂ
hs : -↑m < (s + 1).re
⊢ Tendsto (GammaSeq s) atTop (𝓝 ((fun s => GammaAux m (s + 1) / s) s)) | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | have := @Tendsto.congr' _ _ _ ?_ _ _
((eventually_ne_atTop 0).mp (eventually_of_forall fun n hn => ?_)) ((IH _ hs).div_const s) | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) := by
suffices ∀ m : ℕ, -↑m < re s → Tendsto (GammaSeq s) atTop (𝓝 <| GammaAux m s) by
rw [Gamma]
apply this
rw [neg_lt]
rcases lt_or_le 0 (re s) with (hs |... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.351_0.in2QiCFW52coQT2 | /-- Euler's limit formula for the complex Gamma function. -/
theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
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