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This commit is contained in:
Stefan Kebekus
parent
db9ce54bcf
commit
371b90c1c6
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@ -3,6 +3,7 @@ import Mathlib.Analysis.Analytic.IsolatedZeros
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import Nevanlinna.analyticOn_zeroSet
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import Nevanlinna.harmonicAt_examples
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import Nevanlinna.harmonicAt_meanValue
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import Nevanlinna.specialFunctions_CircleIntegral_affine
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open Real
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@ -88,6 +89,7 @@ theorem jensen_case_R_eq_one
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tauto
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have : ∫ (x : ℝ) in (0)..2 * π, log ‖f (circleMap 0 1 x)‖ = ∫ (x : ℝ) in (0)..2 * π, G (circleMap 0 1 x) := by
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rw [intervalIntegral.integral_congr_ae]
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rw [MeasureTheory.ae_iff]
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apply Set.Countable.measure_zero
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@ -133,11 +135,16 @@ theorem jensen_case_R_eq_one
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-- ∀ i ∈ (ZeroFinset h₁f).attach, IntervalIntegrable (fun x => ↑(order i) *
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-- log (Complex.abs (circleMap 0 1 x - ↑i))) MeasureTheory.volume 0 (2 * π)
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-- This won't work, because the function **is not** continuous. Need to fix.
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intro i hi
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apply IntervalIntegrable.const_mul
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exact h₁Gi i hi
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have : i.1 ∈ Metric.closedBall (0 : ℂ) 1 := by exact ((ZeroFinset_mem_iff h₁f i).1 i.2).1
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simp at this
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by_cases h₂i : ‖i.1‖ = 1
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-- case pos
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exact int'₂ h₂i
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-- case neg
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have : i.1 ∈ Metric.ball (0 : ℂ) 1 := by sorry
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apply Continuous.intervalIntegrable
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apply continuous_iff_continuousAt.2
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@ -150,14 +157,73 @@ theorem jensen_case_R_eq_one
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simp
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by_contra ha'
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let A := ha i
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rw [← ha'] at A
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simp at A
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conv at h₂i =>
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arg 1
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rw [← ha']
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rw [Complex.norm_eq_abs]
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rw [abs_circleMap_zero 1 x]
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simp
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tauto
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apply ContinuousAt.comp
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apply Complex.continuous_abs.continuousAt
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fun_prop
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-- IntervalIntegrable (fun x => log (Complex.abs (F (circleMap 0 1 x)))) MeasureTheory.volume 0 (2 * π)
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apply Continuous.intervalIntegrable
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apply continuous_iff_continuousAt.2
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intro x
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have : (fun x => log (Complex.abs (F (circleMap 0 1 x)))) = log ∘ Complex.abs ∘ F ∘ (fun x ↦ circleMap 0 1 x) :=
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rfl
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rw [this]
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apply ContinuousAt.comp
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apply Real.continuousAt_log
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simp [h₂F]
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--
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apply ContinuousAt.comp
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apply Complex.continuous_abs.continuousAt
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apply ContinuousAt.comp
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apply DifferentiableAt.continuousAt (𝕜 := ℂ )
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apply HolomorphicAt.differentiableAt
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simp [h₁F]
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--
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apply Continuous.continuousAt
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apply continuous_circleMap
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--
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have : (fun x => ∑ s ∈ (ZeroFinset h₁f).attach, ↑(order s) * log (Complex.abs (circleMap 0 1 x - ↑s)))
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= ∑ s ∈ (ZeroFinset h₁f).attach, (fun x => ↑(order s) * log (Complex.abs (circleMap 0 1 x - ↑s))) := by
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funext x
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simp
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rw [this]
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apply IntervalIntegrable.sum
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intro i h₂i
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apply IntervalIntegrable.const_mul
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have : i.1 ∈ Metric.closedBall (0 : ℂ) 1 := by exact ((ZeroFinset_mem_iff h₁f i).1 i.2).1
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simp at this
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by_cases h₂i : ‖i.1‖ = 1
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-- case pos
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exact int'₂ h₂i
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-- case neg
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have : i.1 ∈ Metric.ball (0 : ℂ) 1 := by sorry
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apply Continuous.intervalIntegrable
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apply continuous_iff_continuousAt.2
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intro x
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have : (fun x => log (Complex.abs (circleMap 0 1 x - ↑i))) = log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 1 x - ↑i) :=
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rfl
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rw [this]
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apply ContinuousAt.comp
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apply Real.continuousAt_log
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simp
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by_contra ha'
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conv at h₂i =>
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arg 1
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rw [← ha']
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rw [Complex.norm_eq_abs]
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rw [abs_circleMap_zero 1 x]
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simp
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tauto
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apply ContinuousAt.comp
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apply Complex.continuous_abs.continuousAt
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fun_prop
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sorry
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sorry
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@ -0,0 +1,109 @@
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import Mathlib.MeasureTheory.Integral.Periodic
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import Mathlib.MeasureTheory.Integral.CircleIntegral
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import Nevanlinna.specialFunctions_Integral_log_sin
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import Nevanlinna.harmonicAt_examples
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import Nevanlinna.harmonicAt_meanValue
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import Mathlib.Algebra.Periodic
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open scoped Interval Topology
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open Real Filter MeasureTheory intervalIntegral
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theorem periodic_integrability₁
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{E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E]
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{f : ℝ → E}
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{t T : ℝ}
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{n : ℕ}
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(h₁f : Function.Periodic f T)
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(h₂f : IntervalIntegrable f MeasureTheory.volume t (t + T)) :
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IntervalIntegrable f MeasureTheory.volume t (t + n * T) := by
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induction' n with n hn
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simp
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apply IntervalIntegrable.trans (b := t + n * T)
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exact hn
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let A := IntervalIntegrable.comp_sub_right h₂f (n * T)
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have : f = fun x ↦ f (x - n * T) := by simp [Function.Periodic.sub_nat_mul_eq h₁f n]
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simp_rw [← this] at A
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have : (t + T + ↑n * T) = (t + ↑(n + 1) * T) := by simp; ring
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simp_rw [this] at A
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exact A
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theorem periodic_integrability₂
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{E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E]
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{f : ℝ → E}
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{t T : ℝ}
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{n : ℕ}
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(h₁f : Function.Periodic f T)
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(h₂f : IntervalIntegrable f MeasureTheory.volume t (t + T)) :
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IntervalIntegrable f MeasureTheory.volume (t - n * T) t := by
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induction' n with n hn
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simp
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--
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apply IntervalIntegrable.trans (b := t - n * T)
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--
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have A := IntervalIntegrable.comp_add_right h₂f (((n + 1): ℕ) * T)
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have : f = fun x ↦ f (x + ((n + 1) : ℕ) * T) := by
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funext x
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have : x = x + ↑(n + 1) * T - ↑(n + 1) * T := by ring
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nth_rw 1 [this]
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rw [Function.Periodic.sub_nat_mul_eq h₁f]
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simp_rw [← this] at A
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have : (t + T - ↑(n + 1) * T) = (t - ↑n * T) := by simp; ring
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simp_rw [this] at A
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exact A
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--
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exact hn
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theorem periodic_integrability₃
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{E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E]
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{f : ℝ → E}
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{t T : ℝ}
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{n₁ n₂ : ℕ}
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(h₁f : Function.Periodic f T)
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(h₂f : IntervalIntegrable f MeasureTheory.volume t (t + T)) :
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IntervalIntegrable f MeasureTheory.volume (t - n₁ * T) (t + n₂ * T) := by
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apply IntervalIntegrable.trans (b := t)
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exact periodic_integrability₂ h₁f h₂f
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exact periodic_integrability₁ h₁f h₂f
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theorem periodic_integrability4
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{E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E]
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{f : ℝ → E}
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{t T : ℝ}
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{a₁ a₂ : ℝ}
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(h₁f : Function.Periodic f T)
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(hT : 0 < T)
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(h₂f : IntervalIntegrable f MeasureTheory.volume t (t + T)) :
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IntervalIntegrable f MeasureTheory.volume a₁ a₂ := by
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obtain ⟨n₁, hn₁⟩ : ∃ n₁ : ℕ, t - n₁ * T ≤ min a₁ a₂ := by
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obtain ⟨n₁, hn₁⟩ := exists_nat_ge ((t -min a₁ a₂) / T)
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use n₁
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rw [sub_le_iff_le_add]
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rw [div_le_iff hT] at hn₁
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rw [sub_le_iff_le_add] at hn₁
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rw [add_comm]
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exact hn₁
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obtain ⟨n₂, hn₂⟩ : ∃ n₂ : ℕ, max a₁ a₂ ≤ t + n₂ * T := by
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obtain ⟨n₂, hn₂⟩ := exists_nat_ge ((max a₁ a₂ - t) / T)
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use n₂
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rw [← sub_le_iff_le_add]
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rw [div_le_iff hT] at hn₂
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linarith
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have : Set.uIcc a₁ a₂ ⊆ Set.uIcc (t - n₁ * T) (t + n₂ * T) := by
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apply Set.uIcc_subset_uIcc_iff_le.mpr
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constructor
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· calc min (t - ↑n₁ * T) (t + ↑n₂ * T)
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_ ≤ (t - ↑n₁ * T) := by exact min_le_left (t - ↑n₁ * T) (t + ↑n₂ * T)
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_ ≤ min a₁ a₂ := by exact hn₁
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· calc max a₁ a₂
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_ ≤ t + n₂ * T := by exact hn₂
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_ ≤ max (t - ↑n₁ * T) (t + ↑n₂ * T) := by exact le_max_right (t - ↑n₁ * T) (t + ↑n₂ * T)
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apply IntervalIntegrable.mono_set _ this
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exact periodic_integrability₃ h₁f h₂f
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@ -3,12 +3,19 @@ import Mathlib.MeasureTheory.Integral.CircleIntegral
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import Nevanlinna.specialFunctions_Integral_log_sin
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import Nevanlinna.harmonicAt_examples
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import Nevanlinna.harmonicAt_meanValue
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import Nevanlinna.periodic_integrability
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open scoped Interval Topology
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open Real Filter MeasureTheory intervalIntegral
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-- Integrability of periodic functions
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-- Lemmas for the circleMap
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lemma l₀ {x₁ x₂ : ℝ} : (circleMap 0 1 x₁) * (circleMap 0 1 x₂) = circleMap 0 1 (x₁+x₂) := by
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dsimp [circleMap]
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simp
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@ -187,6 +194,23 @@ lemma int'₁ : -- Integrability of log ‖circleMap 0 1 x - 1‖
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apply IntervalIntegrable.const_mul
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exact intervalIntegrable_log_sin
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lemma int''₁ : -- Integrability of log ‖circleMap 0 1 x - 1‖ for arbitrary intervals
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∀ (t₁ t₂ : ℝ), IntervalIntegrable (fun x ↦ log ‖circleMap 0 1 x - 1‖) volume t₁ t₂ := by
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intro t₁ t₂
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apply periodic_integrability4 (T := 2 * π) (t := 0)
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--
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have : (fun x => log ‖circleMap 0 1 x - 1‖) = (fun x => log ‖x - 1‖) ∘ (circleMap 0 1) := rfl
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rw [this]
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apply Function.Periodic.comp
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exact periodic_circleMap 0 1
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--
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exact two_pi_pos
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--
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rw [zero_add]
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exact int'₁
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lemma int₁ :
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∫ x in (0)..(2 * π), log ‖circleMap 0 1 x - 1‖ = 0 := by
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@ -213,6 +237,71 @@ lemma int₁ :
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exact int₁₁
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-- Integral of log ‖circleMap 0 1 x - a‖, for ‖a‖ = 1
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lemma int'₂
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{a : ℂ}
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(ha : ‖a‖ = 1) :
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IntervalIntegrable (fun x ↦ log ‖circleMap 0 1 x - a‖) volume 0 (2 * π) := by
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simp_rw [l₂]
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have {x : ℝ} : log ‖1 - circleMap 0 1 (-x) * a‖ = (fun w ↦ log ‖1 - circleMap 0 1 (w) * a‖) (-x) := by rfl
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conv =>
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arg 1
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intro x
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rw [this]
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rw [IntervalIntegrable.iff_comp_neg]
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let f₁ := fun w ↦ log ‖1 - circleMap 0 1 w * a‖
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have {x : ℝ} : log ‖1 - circleMap 0 1 x * a‖ = f₁ (x + 2 * π) := by
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dsimp [f₁]
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congr 4
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let A := periodic_circleMap 0 1 x
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simp at A
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exact id (Eq.symm A)
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conv =>
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arg 1
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intro x
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arg 0
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intro w
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rw [this]
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simp
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have : 0 = 2 * π - 2 * π := by ring
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rw [this]
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have : -(2 * π ) = 0 - 2 * π := by ring
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rw [this]
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apply IntervalIntegrable.comp_add_right _ (2 * π) --f₁ (2 * π)
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dsimp [f₁]
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have : ∃ ζ, a = circleMap 0 1 ζ := by
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apply Set.exists_range_iff.mp
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use a
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simp
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exact ha
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obtain ⟨ζ, hζ⟩ := this
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rw [hζ]
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simp_rw [l₀]
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have : 2 * π = (2 * π + ζ) - ζ := by ring
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rw [this]
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have : 0 = ζ - ζ := by ring
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rw [this]
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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
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rw [this]
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apply IntervalIntegrable.comp_add_right (f := (fun w ↦ log (Complex.abs (1 - circleMap 0 1 (w))))) _ ζ
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have : Function.Periodic (fun x ↦ log (Complex.abs (1 - circleMap 0 1 x))) (2 * π) := by
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have : (fun x ↦ log (Complex.abs (1 - circleMap 0 1 x))) = ( (fun x ↦ log (Complex.abs (1 - x))) ∘ (circleMap 0 1) ) := by rfl
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rw [this]
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apply Function.Periodic.comp
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exact periodic_circleMap 0 1
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let A := int''₁ (2 * π + ζ) ζ
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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
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simp_rw [this] at A
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exact A
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lemma int₂
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{a : ℂ}
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(ha : ‖a‖ = 1) :
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