128 lines
3.2 KiB
Plaintext
128 lines
3.2 KiB
Plaintext
import Mathlib.MeasureTheory.Integral.CircleIntegral
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import Nevanlinna.divisor
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import Nevanlinna.stronglyMeromorphicOn
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import Nevanlinna.meromorphicOn_divisor
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open Real
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-- Lang p. 164
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theorem MeromorphicOn.restrict
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{f : ℂ → ℂ}
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(h₁f : MeromorphicOn f ⊤)
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(r : ℝ) :
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MeromorphicOn f (Metric.closedBall 0 r) := by
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exact fun x a => h₁f x trivial
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noncomputable def MeromorphicOn.N_zero
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{f : ℂ → ℂ}
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(hf : MeromorphicOn f ⊤) :
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ℝ → ℝ :=
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fun r ↦ ∑ᶠ z, (max 0 ((hf.restrict r).divisor z)) * log (r * ‖z‖⁻¹)
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noncomputable def MeromorphicOn.N_infty
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{f : ℂ → ℂ}
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(hf : MeromorphicOn f ⊤) :
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ℝ → ℝ :=
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fun r ↦ ∑ᶠ z, (max 0 (-((hf.restrict r).divisor z))) * log (r * ‖z‖⁻¹)
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theorem Nevanlinna_counting
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{f : ℂ → ℂ}
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(hf : MeromorphicOn f ⊤) :
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hf.N_zero - hf.N_infty = fun r ↦ ∑ᶠ z, ((hf.restrict r).divisor z) * log (r * ‖z‖⁻¹) := by
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funext r
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simp only [Pi.sub_apply]
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unfold MeromorphicOn.N_zero MeromorphicOn.N_infty
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let A := (hf.restrict r).divisor.finiteSupport (isCompact_closedBall 0 r)
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repeat
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rw [finsum_eq_sum_of_support_subset (s := A.toFinset)]
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rw [← Finset.sum_sub_distrib]
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simp_rw [← sub_mul]
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congr
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funext x
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congr
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by_cases h : 0 ≤ (hf.restrict r).divisor x
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· simp [h]
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· have h' : 0 ≤ -((hf.restrict r).divisor x) := by
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simp at h
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apply Int.le_neg_of_le_neg
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simp
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exact Int.le_of_lt h
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simp at h
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simp [h']
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linarith
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--
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repeat
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intro x
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contrapose
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simp
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intro hx
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rw [hx]
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tauto
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noncomputable def logpos : ℝ → ℝ := fun r ↦ max 0 (log r)
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theorem loglogpos {r : ℝ} : log r = logpos r - logpos r⁻¹ := by
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unfold logpos
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rw [log_inv]
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by_cases h : 0 ≤ log r
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· simp [h]
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· simp at h
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have : 0 ≤ -log r := Left.nonneg_neg_iff.2 (le_of_lt h)
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simp [h, this]
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exact neg_nonneg.mp this
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--
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noncomputable def MeromorphicOn.m_infty
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{f : ℂ → ℂ}
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(_ : MeromorphicOn f ⊤) :
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ℝ → ℝ :=
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fun r ↦ (2 * π)⁻¹ * ∫ x in (0)..(2 * π), logpos ‖f (circleMap 0 r x)‖
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theorem Nevanlinna_proximity
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{f : ℂ → ℂ}
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{r : ℝ}
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(h₁f : MeromorphicOn f ⊤) :
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(2 * π)⁻¹ * ∫ x in (0)..(2 * π), log ‖f (circleMap 0 r x)‖ = (h₁f.m_infty r) - (h₁f.inv.m_infty r) := by
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unfold MeromorphicOn.m_infty
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rw [← mul_sub]; congr
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rw [← intervalIntegral.integral_sub]; congr
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funext x
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simp_rw [loglogpos]; congr
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exact Eq.symm (IsAbsoluteValue.abv_inv Norm.norm (f (circleMap 0 r x)))
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--
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sorry
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noncomputable def MeromorphicOn.T_infty
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{f : ℂ → ℂ}
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(hf : MeromorphicOn f ⊤) :
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ℝ → ℝ :=
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hf.m_infty + hf.N_infty
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theorem Nevanlinna_firstMain₁
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{f : ℂ → ℂ}
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(h₁f : MeromorphicOn f ⊤)
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(h₂f : StronglyMeromorphicAt f 0)
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(h₃f : f 0 ≠ 0) :
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(fun r ↦ log ‖f 0‖) + h₁f.inv.T_infty = h₁f.T_infty := by
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funext r
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simp
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unfold MeromorphicOn.T_infty
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unfold MeromorphicOn.N_infty
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unfold MeromorphicOn.m_infty
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simp
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sorry
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theorem Nevanlinna_firstMain₂
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{f : ℂ → ℂ}
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{a : ℂ}
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{r : ℝ}
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(h₁f : MeromorphicOn f ⊤) :
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|(h₁f.T_infty r) - ((h₁f.sub (MeromorphicOn.const a)).T_infty r)| ≤ logpos ‖a‖ + log 2 := by
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sorry
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