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Update specialFunctions_CircleIntegral_affine.lean

This commit is contained in:
Stefan Kebekus 2024-08-22 08:47:20 +02:00
parent 77dea4115e
commit db9ce54bcf
1 changed files with 24 additions and 16 deletions
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40
Nevanlinna/specialFunctions_CircleIntegral_affine.lean
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@ -51,16 +51,16 @@ lemma int₀
-- case: a ≠ 0 -- case: a ≠ 0
simp_rw [l₂] simp_rw [l₂]
have {x : ℝ} : Real.log ‖1 - circleMap 0 1 (-x) * a‖ = (fun w ↦ Real.log ‖1 - circleMap 0 1 (w) * a‖) (-x) := by rfl have {x : ℝ} : log ‖1 - circleMap 0 1 (-x) * a‖ = (fun w ↦ log ‖1 - circleMap 0 1 (w) * a‖) (-x) := by rfl
conv => conv =>
left left
arg 1 arg 1
intro x intro x
rw [this] rw [this]
rw [intervalIntegral.integral_comp_neg ((fun w ↦ Real.log ‖1 - circleMap 0 1 (w) * a‖))] rw [intervalIntegral.integral_comp_neg ((fun w ↦ log ‖1 - circleMap 0 1 (w) * a‖))]
let f₁ := fun w ↦ Real.log ‖1 - circleMap 0 1 w * a‖ let f₁ := fun w ↦ log ‖1 - circleMap 0 1 w * a‖
have {x : ℝ} : Real.log ‖1 - circleMap 0 1 x * a‖ = f₁ (x + 2 * Real.pi) := by have {x : ℝ} : log ‖1 - circleMap 0 1 x * a‖ = f₁ (x + 2 * π) := by
dsimp [f₁] dsimp [f₁]
congr 4 congr 4
let A := periodic_circleMap 0 1 x let A := periodic_circleMap 0 1 x
@ -71,7 +71,7 @@ lemma int₀
arg 1 arg 1
intro x intro x
rw [this] rw [this]
rw [intervalIntegral.integral_comp_add_right f₁ (2 * Real.pi)] rw [intervalIntegral.integral_comp_add_right f₁ (2 * π)]
simp simp
dsimp [f₁] dsimp [f₁]
@ -82,7 +82,7 @@ lemma int₀
· exact norm_pos_iff'.mpr h₁a · exact norm_pos_iff'.mpr h₁a
· exact mem_ball_zero_iff.mp ha · exact mem_ball_zero_iff.mp ha
let F := fun z ↦ Real.log ‖1 - z * a‖ let F := fun z ↦ log ‖1 - z * a‖
have hf : ∀ x ∈ Metric.ball 0 ρ, HarmonicAt F x := by have hf : ∀ x ∈ Metric.ball 0 ρ, HarmonicAt F x := by
intro x hx intro x hx
@ -105,7 +105,7 @@ lemma int₀
rw [← h] at this rw [← h] at this
simp at this simp at this
let A := harmonic_meanValue ρ 1 Real.zero_lt_one hρ hf let A := harmonic_meanValue ρ 1 zero_lt_one hρ hf
dsimp [F] at A dsimp [F] at A
simp at A simp at A
exact A exact A
@ -163,7 +163,7 @@ lemma logAffineHelper {x : ℝ} : log ‖circleMap 0 1 x - 1‖ = log (4 * sin (
_ = 2 - 2 * cos (2 * (x / 2)) := by _ = 2 - 2 * cos (2 * (x / 2)) := by
rw [← mul_div_assoc] rw [← mul_div_assoc]
congr; norm_num congr; norm_num
_ = 4 - 4 * Real.cos (x / 2) ^ 2 := by _ = 4 - 4 * cos (x / 2) ^ 2 := by
rw [cos_two_mul] rw [cos_two_mul]
ring ring
_ = 4 * sin (x / 2) ^ 2 := by _ = 4 * sin (x / 2) ^ 2 := by
@ -172,12 +172,20 @@ lemma logAffineHelper {x : ℝ} : log ‖circleMap 0 1 x - 1‖ = log (4 * sin (
lemma int'₁ : -- Integrability of log ‖circleMap 0 1 x - 1‖ lemma int'₁ : -- Integrability of log ‖circleMap 0 1 x - 1‖
IntervalIntegrable (fun x ↦ log ‖circleMap 0 1 x - 1‖) volume 0 (2 * π) := by IntervalIntegrable (fun x ↦ log ‖circleMap 0 1 x - 1‖) volume 0 (2 * π) := by
simp_rw [logAffineHelper] simp_rw [logAffineHelper]
apply IntervalIntegrable.div_const
rw [← IntervalIntegrable.comp_mul_left_iff (c := 2) (Ne.symm (NeZero.ne' 2))] rw [← IntervalIntegrable.comp_mul_left_iff (c := 2) (Ne.symm (NeZero.ne' 2))]
simp simp
sorry have h₁ : Set.EqOn (fun x => log (4 * sin x ^ 2)) (fun x => log 4 + 2 * log (sin x)) (Set.Ioo 0 π) := by
intro x hx
simp [log_mul (Ne.symm (NeZero.ne' 4)), log_pow, ne_of_gt (sin_pos_of_mem_Ioo hx)]
rw [IntervalIntegrable.integral_congr_Ioo pi_nonneg h₁]
apply IntervalIntegrable.add
simp
apply IntervalIntegrable.const_mul
exact intervalIntegrable_log_sin
lemma int₁ : lemma int₁ :
∫ x in (0)..(2 * π), log ‖circleMap 0 1 x - 1‖ = 0 := by ∫ x in (0)..(2 * π), log ‖circleMap 0 1 x - 1‖ = 0 := by
@ -208,19 +216,19 @@ lemma int₁ :
lemma int₂ lemma int₂
{a : ℂ} {a : ℂ}
(ha : ‖a‖ = 1) : (ha : ‖a‖ = 1) :
∫ x in (0)..(2 * Real.pi), log ‖circleMap 0 1 x - a‖ = 0 := by ∫ x in (0)..(2 * π), log ‖circleMap 0 1 x - a‖ = 0 := by
simp_rw [l₂] simp_rw [l₂]
have {x : ℝ} : Real.log ‖1 - circleMap 0 1 (-x) * a‖ = (fun w ↦ Real.log ‖1 - circleMap 0 1 (w) * a‖) (-x) := by rfl have {x : ℝ} : log ‖1 - circleMap 0 1 (-x) * a‖ = (fun w ↦ log ‖1 - circleMap 0 1 (w) * a‖) (-x) := by rfl
conv => conv =>
left left
arg 1 arg 1
intro x intro x
rw [this] rw [this]
rw [intervalIntegral.integral_comp_neg ((fun w ↦ Real.log ‖1 - circleMap 0 1 (w) * a‖))] rw [intervalIntegral.integral_comp_neg ((fun w ↦ log ‖1 - circleMap 0 1 (w) * a‖))]
let f₁ := fun w ↦ Real.log ‖1 - circleMap 0 1 w * a‖ let f₁ := fun w ↦ log ‖1 - circleMap 0 1 w * a‖
have {x : ℝ} : Real.log ‖1 - circleMap 0 1 x * a‖ = f₁ (x + 2 * Real.pi) := by have {x : ℝ} : log ‖1 - circleMap 0 1 x * a‖ = f₁ (x + 2 * π) := by
dsimp [f₁] dsimp [f₁]
congr 4 congr 4
let A := periodic_circleMap 0 1 x let A := periodic_circleMap 0 1 x
@ -231,7 +239,7 @@ lemma int₂
arg 1 arg 1
intro x intro x
rw [this] rw [this]
rw [intervalIntegral.integral_comp_add_right f₁ (2 * Real.pi)] rw [intervalIntegral.integral_comp_add_right f₁ (2 * π)]
simp simp
dsimp [f₁] dsimp [f₁]