116 lines
3.1 KiB
Plaintext
116 lines
3.1 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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noncomputable def MeromorphicOn.N_zero
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{f : ℂ → ℂ}
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(h₁f : MeromorphicOn f ⊤) :
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ℝ → ℝ :=
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fun r ↦ ∑ᶠ z ∈ Metric.closedBall (0 : ℂ) r, (max 0 (h₁f.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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(h₁f : MeromorphicOn f ⊤) :
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ℝ → ℝ :=
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fun r ↦ ∑ᶠ z ∈ Metric.closedBall (0 : ℂ) r, (max 0 (-(h₁f.divisor z))) * log (r * ‖z‖⁻¹)
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theorem Nevanlinna_counting
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{f : ℂ → ℂ}
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(h₁f : MeromorphicOn f ⊤) :
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h₁f.N_zero - h₁f.N_infty = fun r ↦ ∑ᶠ z ∈ Metric.closedBall (0 : ℂ) r, (h₁f.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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rw [finsum_eq_sum]
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sorry
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have h₁fr : MeromorphicOn f (Metric.ball (0 : ℂ) r) := by
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sorry
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let Sr :=
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rw [finsum_eq_sum_of_support_subset _ h₄f]
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have h₂U : IsCompact (Metric.closedBall (0 : ℂ) R) :=
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isCompact_closedBall 0 R
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have h'₂f : ∃ u : (Metric.closedBall (0 : ℂ) R), f u ≠ 0 := by
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use ⟨0, Metric.mem_closedBall_self (le_of_lt hR)⟩
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have h₃f : Set.Finite (Function.support h₁f.divisor) := by
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exact Divisor.finiteSupport h₂U (StronglyMeromorphicOn.meromorphicOn h₁f).divisor
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sorry
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--
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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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(h₁f : 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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