import Mathlib.MeasureTheory.Integral.Periodic import Mathlib.MeasureTheory.Integral.CircleIntegral import Nevanlinna.specialFunctions_Integral_log_sin import Nevanlinna.harmonicAt_examples import Nevanlinna.harmonicAt_meanValue import Nevanlinna.periodic_integrability open scoped Interval Topology open Real Filter MeasureTheory intervalIntegral -- Lemmas for the circleMap lemma l₀ {x₁ x₂ : ℝ} : (circleMap 0 1 x₁) * (circleMap 0 1 x₂) = circleMap 0 1 (x₁+x₂) := by dsimp [circleMap] simp rw [add_mul, Complex.exp_add] lemma l₁ {x : ℝ} : ‖circleMap 0 1 x‖ = 1 := by rw [Complex.norm_eq_abs, abs_circleMap_zero] simp lemma l₂ {x : ℝ} : ‖(circleMap 0 1 x) - a‖ = ‖1 - (circleMap 0 1 (-x)) * a‖ := by calc ‖(circleMap 0 1 x) - a‖ _ = 1 * ‖(circleMap 0 1 x) - a‖ := by exact Eq.symm (one_mul ‖circleMap 0 1 x - a‖) _ = ‖(circleMap 0 1 (-x))‖ * ‖(circleMap 0 1 x) - a‖ := by rw [l₁] _ = ‖(circleMap 0 1 (-x)) * ((circleMap 0 1 x) - a)‖ := by exact Eq.symm (NormedField.norm_mul' (circleMap 0 1 (-x)) (circleMap 0 1 x - a)) _ = ‖(circleMap 0 1 (-x)) * (circleMap 0 1 x) - (circleMap 0 1 (-x)) * a‖ := by rw [mul_sub] _ = ‖(circleMap 0 1 0) - (circleMap 0 1 (-x)) * a‖ := by rw [l₀] simp _ = ‖1 - (circleMap 0 1 (-x)) * a‖ := by congr dsimp [circleMap] simp -- Integral of log ‖circleMap 0 1 x - a‖, for ‖a‖ < 1. lemma int'₀ {a : ℂ} (ha : a ∈ Metric.ball 0 1) : IntervalIntegrable (fun x ↦ log ‖circleMap 0 1 x - a‖) volume 0 (2 * π) := by apply Continuous.intervalIntegrable apply Continuous.log fun_prop simp intro x by_contra h₁a rw [← h₁a] at ha simp at ha lemma int₀ {a : ℂ} (ha : a ∈ Metric.ball 0 1) : ∫ (x : ℝ) in (0)..2 * π, log ‖circleMap 0 1 x - a‖ = 0 := by by_cases h₁a : a = 0 · -- case: a = 0 rw [h₁a] simp -- case: a ≠ 0 simp_rw [l₂] have {x : ℝ} : log ‖1 - circleMap 0 1 (-x) * a‖ = (fun w ↦ log ‖1 - circleMap 0 1 (w) * a‖) (-x) := by rfl conv => left arg 1 intro x rw [this] rw [intervalIntegral.integral_comp_neg ((fun w ↦ log ‖1 - circleMap 0 1 (w) * a‖))] let f₁ := fun w ↦ log ‖1 - circleMap 0 1 w * a‖ have {x : ℝ} : log ‖1 - circleMap 0 1 x * a‖ = f₁ (x + 2 * π) := by dsimp [f₁] congr 4 let A := periodic_circleMap 0 1 x simp at A exact id (Eq.symm A) conv => left arg 1 intro x rw [this] rw [intervalIntegral.integral_comp_add_right f₁ (2 * π)] simp dsimp [f₁] let ρ := ‖a‖⁻¹ have hρ : 1 < ρ := by apply one_lt_inv_iff.mpr constructor · exact norm_pos_iff'.mpr h₁a · exact mem_ball_zero_iff.mp ha let F := fun z ↦ log ‖1 - z * a‖ have hf : ∀ x ∈ Metric.ball 0 ρ, HarmonicAt F x := by intro x hx apply logabs_of_holomorphicAt_is_harmonic apply Differentiable.holomorphicAt fun_prop apply sub_ne_zero_of_ne by_contra h have : ‖x * a‖ < 1 := by calc ‖x * a‖ _ = ‖x‖ * ‖a‖ := by exact NormedField.norm_mul' x a _ < ρ * ‖a‖ := by apply (mul_lt_mul_right _).2 exact mem_ball_zero_iff.mp hx exact norm_pos_iff'.mpr h₁a _ = 1 := by dsimp [ρ] apply inv_mul_cancel₀ exact (AbsoluteValue.ne_zero_iff Complex.abs).mpr h₁a rw [← h] at this simp at this let A := harmonic_meanValue ρ 1 zero_lt_one hρ hf dsimp [F] at A simp at A exact A -- Integral of log ‖circleMap 0 1 x - 1‖ lemma int₁₁ : ∫ (x : ℝ) in (0)..π, log (4 * sin x ^ 2) = 0 := by have t₁ {x : ℝ} : x ∈ Set.Ioo 0 π → log (4 * sin x ^ 2) = log 4 + 2 * log (sin x) := by intro hx rw [log_mul, log_pow] rfl exact Ne.symm (NeZero.ne' 4) apply pow_ne_zero 2 apply (fun a => Ne.symm (ne_of_lt a)) exact sin_pos_of_mem_Ioo hx have t₂ : Set.EqOn (fun y ↦ log (4 * sin y ^ 2)) (fun y ↦ log 4 + 2 * log (sin y)) (Set.Ioo 0 π) := by intro x hx simp rw [t₁ hx] rw [intervalIntegral.integral_congr_volume pi_pos t₂] rw [intervalIntegral.integral_add] rw [intervalIntegral.integral_const_mul] simp rw [integral_log_sin₂] have : (4 : ℝ) = 2 * 2 := by norm_num rw [this, log_mul] ring norm_num norm_num -- IntervalIntegrable (fun x => log 4) volume 0 π simp -- IntervalIntegrable (fun x => 2 * log (sin x)) volume 0 π apply IntervalIntegrable.const_mul exact intervalIntegrable_log_sin lemma logAffineHelper {x : ℝ} : log ‖circleMap 0 1 x - 1‖ = log (4 * sin (x / 2) ^ 2) / 2 := by dsimp [Complex.abs] rw [log_sqrt (Complex.normSq_nonneg (circleMap 0 1 x - 1))] congr calc Complex.normSq (circleMap 0 1 x - 1) _ = (cos x - 1) * (cos x - 1) + sin x * sin x := by dsimp [circleMap] rw [Complex.normSq_apply] simp _ = sin x ^ 2 + cos x ^ 2 + 1 - 2 * cos x := by ring _ = 2 - 2 * cos x := by rw [sin_sq_add_cos_sq] norm_num _ = 2 - 2 * cos (2 * (x / 2)) := by rw [← mul_div_assoc] congr; norm_num _ = 4 - 4 * cos (x / 2) ^ 2 := by rw [cos_two_mul] ring _ = 4 * sin (x / 2) ^ 2 := by nth_rw 1 [← mul_one 4, ← sin_sq_add_cos_sq (x / 2)] ring 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 simp_rw [logAffineHelper] apply IntervalIntegrable.div_const rw [← IntervalIntegrable.comp_mul_left_iff (c := 2) (Ne.symm (NeZero.ne' 2))] simp 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 int''₁ : -- Integrability of log ‖circleMap 0 1 x - 1‖ for arbitrary intervals ∀ (t₁ t₂ : ℝ), IntervalIntegrable (fun x ↦ log ‖circleMap 0 1 x - 1‖) volume t₁ t₂ := by intro t₁ t₂ apply periodic_integrability4 (T := 2 * π) (t := 0) -- have : (fun x => log ‖circleMap 0 1 x - 1‖) = (fun x => log ‖x - 1‖) ∘ (circleMap 0 1) := rfl rw [this] apply Function.Periodic.comp exact periodic_circleMap 0 1 -- exact two_pi_pos -- rw [zero_add] exact int'₁ lemma int₁ : ∫ x in (0)..(2 * π), log ‖circleMap 0 1 x - 1‖ = 0 := by simp_rw [logAffineHelper] simp have : ∫ (x : ℝ) in (0)..2 * π, log (4 * sin (x / 2) ^ 2) = 2 * ∫ (x : ℝ) in (0)..π, log (4 * sin x ^ 2) := by have : 1 = 2 * (2 : ℝ)⁻¹ := by exact Eq.symm (mul_inv_cancel_of_invertible 2) nth_rw 1 [← one_mul (∫ (x : ℝ) in (0)..2 * π, log (4 * sin (x / 2) ^ 2))] rw [← mul_inv_cancel_of_invertible 2, mul_assoc] let f := fun y ↦ log (4 * sin y ^ 2) have {x : ℝ} : log (4 * sin (x / 2) ^ 2) = f (x / 2) := by simp conv => left right right arg 1 intro x rw [this] rw [intervalIntegral.inv_mul_integral_comp_div 2] simp rw [this] simp exact int₁₁ -- Integral of log ‖circleMap 0 1 x - a‖, for ‖a‖ = 1 lemma int'₂ {a : ℂ} (ha : ‖a‖ = 1) : IntervalIntegrable (fun x ↦ log ‖circleMap 0 1 x - a‖) volume 0 (2 * π) := by simp_rw [l₂] have {x : ℝ} : log ‖1 - circleMap 0 1 (-x) * a‖ = (fun w ↦ log ‖1 - circleMap 0 1 (w) * a‖) (-x) := by rfl conv => arg 1 intro x rw [this] rw [IntervalIntegrable.iff_comp_neg] let f₁ := fun w ↦ log ‖1 - circleMap 0 1 w * a‖ have {x : ℝ} : log ‖1 - circleMap 0 1 x * a‖ = f₁ (x + 2 * π) := by dsimp [f₁] congr 4 let A := periodic_circleMap 0 1 x simp at A exact id (Eq.symm A) conv => arg 1 intro x arg 0 intro w rw [this] simp have : 0 = 2 * π - 2 * π := by ring rw [this] have : -(2 * π ) = 0 - 2 * π := by ring rw [this] apply IntervalIntegrable.comp_add_right _ (2 * π) --f₁ (2 * π) dsimp [f₁] have : ∃ ζ, a = circleMap 0 1 ζ := by apply Set.exists_range_iff.mp use a simp exact ha obtain ⟨ζ, hζ⟩ := this rw [hζ] simp_rw [l₀] have : 2 * π = (2 * π + ζ) - ζ := by ring rw [this] have : 0 = ζ - ζ := by ring rw [this] have : (fun w => log (Complex.abs (1 - circleMap 0 1 (w + ζ)))) = fun x ↦ (fun w ↦ log (Complex.abs (1 - circleMap 0 1 (w)))) (x + ζ) := rfl rw [this] apply IntervalIntegrable.comp_add_right (f := (fun w ↦ log (Complex.abs (1 - circleMap 0 1 (w))))) _ ζ have : Function.Periodic (fun x ↦ log (Complex.abs (1 - circleMap 0 1 x))) (2 * π) := by have : (fun x ↦ log (Complex.abs (1 - circleMap 0 1 x))) = ( (fun x ↦ log (Complex.abs (1 - x))) ∘ (circleMap 0 1) ) := by rfl rw [this] apply Function.Periodic.comp exact periodic_circleMap 0 1 let A := int''₁ (2 * π + ζ) ζ have {x : ℝ} : ‖circleMap 0 1 x - 1‖ = Complex.abs (1 - circleMap 0 1 x) := AbsoluteValue.map_sub Complex.abs (circleMap 0 1 x) 1 simp_rw [this] at A exact A lemma int₂ {a : ℂ} (ha : ‖a‖ = 1) : ∫ x in (0)..(2 * π), log ‖circleMap 0 1 x - a‖ = 0 := by simp_rw [l₂] have {x : ℝ} : log ‖1 - circleMap 0 1 (-x) * a‖ = (fun w ↦ log ‖1 - circleMap 0 1 (w) * a‖) (-x) := by rfl conv => left arg 1 intro x rw [this] rw [intervalIntegral.integral_comp_neg ((fun w ↦ log ‖1 - circleMap 0 1 (w) * a‖))] let f₁ := fun w ↦ log ‖1 - circleMap 0 1 w * a‖ have {x : ℝ} : log ‖1 - circleMap 0 1 x * a‖ = f₁ (x + 2 * π) := by dsimp [f₁] congr 4 let A := periodic_circleMap 0 1 x simp at A exact id (Eq.symm A) conv => left arg 1 intro x rw [this] rw [intervalIntegral.integral_comp_add_right f₁ (2 * π)] simp dsimp [f₁] have : ∃ ζ, a = circleMap 0 1 ζ := by apply Set.exists_range_iff.mp use a simp exact ha obtain ⟨ζ, hζ⟩ := this rw [hζ] simp_rw [l₀] rw [intervalIntegral.integral_comp_add_right (f := fun x ↦ log (Complex.abs (1 - circleMap 0 1 (x))))] have : Function.Periodic (fun x ↦ log (Complex.abs (1 - circleMap 0 1 x))) (2 * π) := by have : (fun x ↦ log (Complex.abs (1 - circleMap 0 1 x))) = ( (fun x ↦ log (Complex.abs (1 - x))) ∘ (circleMap 0 1) ) := by rfl rw [this] apply Function.Periodic.comp exact periodic_circleMap 0 1 let A := Function.Periodic.intervalIntegral_add_eq this ζ 0 simp at A simp rw [add_comm] rw [A] have {x : ℝ} : log (Complex.abs (1 - circleMap 0 1 x)) = log ‖circleMap 0 1 x - 1‖ := by rw [← AbsoluteValue.map_neg Complex.abs] simp simp_rw [this] exact int₁ lemma int₃ {a : ℂ} (ha : a ∈ Metric.closedBall 0 1) : ∫ x in (0)..(2 * π), log ‖circleMap 0 1 x - a‖ = 0 := by by_cases h₁a : a ∈ Metric.ball 0 1 · exact int₀ h₁a · apply int₂ simp at ha simp at h₁a simp linarith