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This commit is contained in:
Stefan Kebekus
parent
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@ -1 +1,28 @@
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import Nevanlinna.cauchyRiemann
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import Nevanlinna.analyticAt
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import Nevanlinna.cauchyRiemann
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import Nevanlinna.codiscreteWithin
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import Nevanlinna.divisor
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import Nevanlinna.firstMain
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import Nevanlinna.harmonicAt
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import Nevanlinna.harmonicAt_examples
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import Nevanlinna.harmonicAt_meanValue
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import Nevanlinna.holomorphicAt
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import Nevanlinna.holomorphic_examples
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import Nevanlinna.holomorphic_primitive
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import Nevanlinna.intervalIntegrability
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import Nevanlinna.laplace
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import Nevanlinna.logpos
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import Nevanlinna.mathlibAddOn
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import Nevanlinna.meromorphicAt
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import Nevanlinna.meromorphicOn
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import Nevanlinna.meromorphicOn_divisor
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import Nevanlinna.meromorphicOn_integrability
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import Nevanlinna.partialDeriv
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import Nevanlinna.periodic_integrability
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import Nevanlinna.specialFunctions_CircleIntegral_affine
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import Nevanlinna.specialFunctions_Integral_log_sin
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import Nevanlinna.stronglyMeromorphicAt
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import Nevanlinna.stronglyMeromorphicOn
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import Nevanlinna.stronglyMeromorphicOn_eliminate
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import Nevanlinna.stronglyMeromorphicOn_ratlPolynomial
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import Nevanlinna.stronglyMeromorphic_JensenFormula
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@ -1,59 +0,0 @@
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import Nevanlinna.analyticAt
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import Nevanlinna.divisor
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open scoped Interval Topology
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open Real Filter MeasureTheory intervalIntegral
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noncomputable def AnalyticOnNhd.zeroDivisor
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{f : ℂ → ℂ}
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{U : Set ℂ}
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(hf : AnalyticOnNhd ℂ f U) :
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Divisor U where
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toFun := by
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intro z
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if hz : z ∈ U then
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exact ((hf z hz).order.toNat : ℤ)
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else
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exact 0
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supportInU := by
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intro z hz
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simp at hz
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by_contra h₂z
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simp [h₂z] at hz
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locallyFiniteInU := by
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intro z hz
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apply eventually_nhdsWithin_iff.2
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rw [eventually_nhds_iff]
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rcases AnalyticAt.eventually_eq_zero_or_eventually_ne_zero (hf z hz) with h|h
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· rw [eventually_nhds_iff] at h
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obtain ⟨N, h₁N, h₂N, h₃N⟩ := h
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use N
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constructor
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· intro y h₁y _
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by_cases h₃y : y ∈ U
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· simp [h₃y]
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right
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rw [AnalyticAt.order_eq_top_iff (hf y h₃y)]
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rw [eventually_nhds_iff]
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use N
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· simp [h₃y]
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· tauto
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· rw [eventually_nhdsWithin_iff, eventually_nhds_iff] at h
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obtain ⟨N, h₁N, h₂N, h₃N⟩ := h
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use N
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constructor
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· intro y h₁y h₂y
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by_cases h₃y : y ∈ U
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· simp [h₃y]
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left
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rw [AnalyticAt.order_eq_zero_iff (hf y h₃y)]
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exact h₁N y h₁y h₂y
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· simp [h₃y]
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· tauto
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@ -1,461 +0,0 @@
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import Mathlib.Analysis.Analytic.Constructions
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import Mathlib.Analysis.Analytic.IsolatedZeros
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import Mathlib.Analysis.Complex.Basic
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import Nevanlinna.analyticAt
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noncomputable def AnalyticOnNhd.order
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{f : ℂ → ℂ} {U : Set ℂ} (hf : AnalyticOnNhd ℂ f U) : U → ℕ∞ := fun u ↦ (hf u u.2).order
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theorem AnalyticOnNhd.order_eq_nat_iff
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{f : ℂ → ℂ}
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{U : Set ℂ}
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{z₀ : U}
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(hf : AnalyticOnNhd ℂ f U)
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(n : ℕ) :
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hf.order z₀ = ↑n ↔ ∃ (g : ℂ → ℂ), AnalyticOnNhd ℂ g U ∧ g z₀ ≠ 0 ∧ ∀ z, f z = (z - z₀) ^ n • g z := by
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constructor
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-- Direction →
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intro hn
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obtain ⟨gloc, h₁gloc, h₂gloc, h₃gloc⟩ := (AnalyticAt.order_eq_nat_iff (hf z₀ z₀.2) n).1 hn
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-- Define a candidate function; this is (f z) / (z - z₀) ^ n with the
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-- removable singularity removed
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let g : ℂ → ℂ := fun z ↦ if z = z₀ then gloc z₀ else (f z) / (z - z₀) ^ n
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-- Describe g near z₀
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have g_near_z₀ : ∀ᶠ (z : ℂ) in nhds z₀, g z = gloc z := by
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rw [eventually_nhds_iff]
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obtain ⟨t, h₁t, h₂t, h₃t⟩ := eventually_nhds_iff.1 h₃gloc
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use t
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constructor
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· intro y h₁y
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by_cases h₂y : y = z₀
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· dsimp [g]; simp [h₂y]
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· dsimp [g]; simp [h₂y]
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rw [div_eq_iff_mul_eq, eq_comm, mul_comm]
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exact h₁t y h₁y
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norm_num
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rw [sub_eq_zero]
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tauto
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· constructor
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· assumption
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· assumption
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-- Describe g near points z₁ that are different from z₀
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have g_near_z₁ {z₁ : ℂ} : z₁ ≠ z₀ → ∀ᶠ (z : ℂ) in nhds z₁, g z = f z / (z - z₀) ^ n := by
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intro hz₁
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rw [eventually_nhds_iff]
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use {z₀.1}ᶜ
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constructor
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· intro y hy
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simp at hy
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simp [g, hy]
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· exact ⟨isOpen_compl_singleton, hz₁⟩
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-- Use g and show that it has all required properties
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use g
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constructor
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· -- AnalyticOn ℂ g U
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intro z h₁z
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by_cases h₂z : z = z₀
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· rw [h₂z]
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apply AnalyticAt.congr h₁gloc
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exact Filter.EventuallyEq.symm g_near_z₀
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· simp_rw [eq_comm] at g_near_z₁
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apply AnalyticAt.congr _ (g_near_z₁ h₂z)
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apply AnalyticAt.div
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exact hf z h₁z
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apply AnalyticAt.pow
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apply AnalyticAt.sub
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apply analyticAt_id
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apply analyticAt_const
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simp
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rw [sub_eq_zero]
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tauto
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· constructor
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· simp [g]; tauto
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· intro z
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by_cases h₂z : z = z₀
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· rw [h₂z, g_near_z₀.self_of_nhds]
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exact h₃gloc.self_of_nhds
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· rw [(g_near_z₁ h₂z).self_of_nhds]
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simp [h₂z]
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rw [div_eq_mul_inv, mul_comm, mul_assoc, inv_mul_cancel₀]
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simp; norm_num
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rw [sub_eq_zero]
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tauto
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-- direction ←
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intro h
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obtain ⟨g, h₁g, h₂g, h₃g⟩ := h
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dsimp [AnalyticOnNhd.order]
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rw [AnalyticAt.order_eq_nat_iff]
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use g
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exact ⟨h₁g z₀ z₀.2, ⟨h₂g, Filter.Eventually.of_forall h₃g⟩⟩
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theorem AnalyticOnNhd.support_of_order₁
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{f : ℂ → ℂ}
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{U : Set ℂ}
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(hf : AnalyticOnNhd ℂ f U) :
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Function.support hf.order = U.restrict f⁻¹' {0} := by
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ext u
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simp [AnalyticOnNhd.order]
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rw [not_iff_comm, (hf u u.2).order_eq_zero_iff]
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theorem AnalyticOnNhd.eliminateZeros
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{f : ℂ → ℂ}
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{U : Set ℂ}
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{A : Finset U}
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(hf : AnalyticOnNhd ℂ f U)
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(n : U → ℕ) :
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(∀ a ∈ A, hf.order a = n a) → ∃ (g : ℂ → ℂ), AnalyticOnNhd ℂ g U ∧ (∀ a ∈ A, g a ≠ 0) ∧ ∀ z, f z = (∏ a ∈ A, (z - a) ^ (n a)) • g z := by
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apply Finset.induction (α := U) (p := fun A ↦ (∀ a ∈ A, (hf a.1 a.2).order = n a) → ∃ (g : ℂ → ℂ), AnalyticOnNhd ℂ g U ∧ (∀ a ∈ A, g a ≠ 0) ∧ ∀ z, f z = (∏ a ∈ A, (z - a) ^ (n a)) • g z)
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-- case empty
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simp
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use f
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simp
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exact hf
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-- case insert
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intro b₀ B hb iHyp
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intro hBinsert
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obtain ⟨g₀, h₁g₀, h₂g₀, h₃g₀⟩ := iHyp (fun a ha ↦ hBinsert a (Finset.mem_insert_of_mem ha))
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have : (h₁g₀ b₀ b₀.2).order = n b₀ := by
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rw [← hBinsert b₀ (Finset.mem_insert_self b₀ B)]
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let φ := fun z ↦ (∏ a ∈ B, (z - a.1) ^ n a)
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have : f = fun z ↦ φ z * g₀ z := by
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funext z
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rw [h₃g₀ z]
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rfl
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simp_rw [this]
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have h₁φ : AnalyticAt ℂ φ b₀ := by
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dsimp [φ]
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apply Finset.analyticAt_prod
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intro b _
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apply AnalyticAt.pow
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apply AnalyticAt.sub
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apply analyticAt_id
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exact analyticAt_const
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have h₂φ : h₁φ.order = (0 : ℕ) := by
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rw [AnalyticAt.order_eq_nat_iff h₁φ 0]
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use φ
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constructor
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· assumption
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· constructor
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· dsimp [φ]
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push_neg
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rw [Finset.prod_ne_zero_iff]
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intro a ha
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simp
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have : ¬ (b₀.1 - a.1 = 0) := by
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by_contra C
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rw [sub_eq_zero] at C
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rw [SetCoe.ext C] at hb
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tauto
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tauto
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· simp
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rw [AnalyticAt.order_mul h₁φ (h₁g₀ b₀ b₀.2)]
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rw [h₂φ]
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simp
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obtain ⟨g₁, h₁g₁, h₂g₁, h₃g₁⟩ := (AnalyticOnNhd.order_eq_nat_iff h₁g₀ (n b₀)).1 this
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use g₁
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constructor
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· exact h₁g₁
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· constructor
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· intro a h₁a
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by_cases h₂a : a = b₀
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· rwa [h₂a]
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· let A' := Finset.mem_of_mem_insert_of_ne h₁a h₂a
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let B' := h₃g₁ a
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let C' := h₂g₀ a A'
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rw [B'] at C'
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exact right_ne_zero_of_smul C'
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· intro z
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let A' := h₃g₀ z
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rw [h₃g₁ z] at A'
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rw [A']
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rw [← smul_assoc]
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congr
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simp
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rw [Finset.prod_insert]
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ring
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exact hb
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theorem XX
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{f : ℂ → ℂ}
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{U : Set ℂ}
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(hU : IsPreconnected U)
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(h₁f : AnalyticOnNhd ℂ f U)
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(h₂f : ∃ u ∈ U, f u ≠ 0) :
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∀ (hu : u ∈ U), (h₁f u hu).order.toNat = (h₁f u hu).order := by
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intro hu
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apply ENat.coe_toNat
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by_contra C
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rw [(h₁f u hu).order_eq_top_iff] at C
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rw [← (h₁f u hu).frequently_zero_iff_eventually_zero] at C
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obtain ⟨u₁, h₁u₁, h₂u₁⟩ := h₂f
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rw [(h₁f.eqOn_zero_of_preconnected_of_frequently_eq_zero hU hu C) h₁u₁] at h₂u₁
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tauto
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theorem discreteZeros
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{f : ℂ → ℂ}
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{U : Set ℂ}
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(hU : IsPreconnected U)
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(h₁f : AnalyticOnNhd ℂ f U)
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(h₂f : ∃ u ∈ U, f u ≠ 0) :
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DiscreteTopology ((U.restrict f)⁻¹' {0}) := by
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simp_rw [← singletons_open_iff_discrete]
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simp_rw [Metric.isOpen_singleton_iff]
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intro z
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let A := XX hU h₁f h₂f z.1.2
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rw [eq_comm] at A
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rw [AnalyticAt.order_eq_nat_iff] at A
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obtain ⟨g, h₁g, h₂g, h₃g⟩ := A
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rw [Metric.eventually_nhds_iff_ball] at h₃g
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have : ∃ ε > 0, ∀ y ∈ Metric.ball (↑z) ε, g y ≠ 0 := by
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have h₄g : ContinuousAt g z := AnalyticAt.continuousAt h₁g
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have : {0}ᶜ ∈ nhds (g z) := by
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exact compl_singleton_mem_nhds_iff.mpr h₂g
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let F := h₄g.preimage_mem_nhds this
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rw [Metric.mem_nhds_iff] at F
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obtain ⟨ε, h₁ε, h₂ε⟩ := F
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use ε
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constructor; exact h₁ε
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intro y hy
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let G := h₂ε hy
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simp at G
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exact G
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obtain ⟨ε₁, h₁ε₁⟩ := this
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obtain ⟨ε₂, h₁ε₂, h₂ε₂⟩ := h₃g
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use min ε₁ ε₂
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constructor
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· have : 0 < min ε₁ ε₂ := by
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rw [lt_min_iff]
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exact And.imp_right (fun _ => h₁ε₂) h₁ε₁
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exact this
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intro y
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intro h₁y
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have h₂y : ↑y ∈ Metric.ball (↑z) ε₂ := by
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simp
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calc dist y z
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_ < min ε₁ ε₂ := by assumption
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_ ≤ ε₂ := by exact min_le_right ε₁ ε₂
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have h₃y : ↑y ∈ Metric.ball (↑z) ε₁ := by
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simp
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calc dist y z
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_ < min ε₁ ε₂ := by assumption
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_ ≤ ε₁ := by exact min_le_left ε₁ ε₂
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have F := h₂ε₂ y.1 h₂y
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have : f y = 0 := by exact y.2
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rw [this] at F
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simp at F
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have : g y.1 ≠ 0 := by
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exact h₁ε₁.2 y h₃y
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simp [this] at F
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ext
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rw [sub_eq_zero] at F
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tauto
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theorem finiteZeros
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{f : ℂ → ℂ}
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{U : Set ℂ}
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(h₁U : IsPreconnected U)
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(h₂U : IsCompact U)
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(h₁f : AnalyticOnNhd ℂ f U)
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(h₂f : ∃ u ∈ U, f u ≠ 0) :
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Set.Finite (U.restrict f⁻¹' {0}) := by
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have closedness : IsClosed (U.restrict f⁻¹' {0}) := by
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apply IsClosed.preimage
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apply continuousOn_iff_continuous_restrict.1
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exact h₁f.continuousOn
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exact isClosed_singleton
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have : CompactSpace U := by
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exact isCompact_iff_compactSpace.mp h₂U
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apply (IsClosed.isCompact closedness).finite
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exact discreteZeros h₁U h₁f h₂f
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theorem AnalyticOnNhdCompact.eliminateZeros
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{f : ℂ → ℂ}
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{U : Set ℂ}
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(h₁U : IsPreconnected U)
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(h₂U : IsCompact U)
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(h₁f : AnalyticOnNhd ℂ f U)
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(h₂f : ∃ u ∈ U, f u ≠ 0) :
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∃ (g : ℂ → ℂ) (A : Finset U), AnalyticOnNhd ℂ g U ∧ (∀ z ∈ U, g z ≠ 0) ∧ ∀ z, f z = (∏ a ∈ A, (z - a) ^ (h₁f.order a).toNat) • g z := by
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let A := (finiteZeros h₁U h₂U h₁f h₂f).toFinset
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let n : U → ℕ := fun z ↦ (h₁f z z.2).order.toNat
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have hn : ∀ a ∈ A, (h₁f a a.2).order = n a := by
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intro a _
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dsimp [n, AnalyticOnNhd.order]
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rw [eq_comm]
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apply XX h₁U
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exact h₂f
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obtain ⟨g, h₁g, h₂g, h₃g⟩ := AnalyticOnNhd.eliminateZeros (A := A) h₁f n hn
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use g
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use A
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|
||||
have inter : ∀ (z : ℂ), f z = (∏ a ∈ A, (z - ↑a) ^ (h₁f (↑a) a.property).order.toNat) • g z := by
|
||||
intro z
|
||||
rw [h₃g z]
|
||||
|
||||
constructor
|
||||
· exact h₁g
|
||||
· constructor
|
||||
· intro z h₁z
|
||||
by_cases h₂z : ⟨z, h₁z⟩ ∈ ↑A.toSet
|
||||
· exact h₂g ⟨z, h₁z⟩ h₂z
|
||||
· have : f z ≠ 0 := by
|
||||
by_contra C
|
||||
have : ⟨z, h₁z⟩ ∈ ↑A.toSet := by
|
||||
dsimp [A]
|
||||
simp
|
||||
exact C
|
||||
tauto
|
||||
rw [inter z] at this
|
||||
exact right_ne_zero_of_smul this
|
||||
· exact inter
|
||||
|
||||
|
||||
theorem AnalyticOnNhdCompact.eliminateZeros₂
|
||||
{f : ℂ → ℂ}
|
||||
{U : Set ℂ}
|
||||
(h₁U : IsPreconnected U)
|
||||
(h₂U : IsCompact U)
|
||||
(h₁f : AnalyticOnNhd ℂ f U)
|
||||
(h₂f : ∃ u ∈ U, f u ≠ 0) :
|
||||
∃ (g : ℂ → ℂ), AnalyticOnNhd ℂ g U ∧ (∀ z ∈ U, g z ≠ 0) ∧ ∀ z, f z = (∏ a ∈ (finiteZeros h₁U h₂U h₁f h₂f).toFinset, (z - a) ^ (h₁f.order a).toNat) • g z := by
|
||||
|
||||
let A := (finiteZeros h₁U h₂U h₁f h₂f).toFinset
|
||||
|
||||
let n : U → ℕ := fun z ↦ (h₁f z z.2).order.toNat
|
||||
|
||||
have hn : ∀ a ∈ A, (h₁f a a.2).order = n a := by
|
||||
intro a _
|
||||
dsimp [n, AnalyticOnNhd.order]
|
||||
rw [eq_comm]
|
||||
apply XX h₁U
|
||||
exact h₂f
|
||||
|
||||
obtain ⟨g, h₁g, h₂g, h₃g⟩ := AnalyticOnNhd.eliminateZeros (A := A) h₁f n hn
|
||||
use g
|
||||
|
||||
have inter : ∀ (z : ℂ), f z = (∏ a ∈ A, (z - ↑a) ^ (h₁f (↑a) a.property).order.toNat) • g z := by
|
||||
intro z
|
||||
rw [h₃g z]
|
||||
|
||||
constructor
|
||||
· exact h₁g
|
||||
· constructor
|
||||
· intro z h₁z
|
||||
by_cases h₂z : ⟨z, h₁z⟩ ∈ ↑A.toSet
|
||||
· exact h₂g ⟨z, h₁z⟩ h₂z
|
||||
· have : f z ≠ 0 := by
|
||||
by_contra C
|
||||
have : ⟨z, h₁z⟩ ∈ ↑A.toSet := by
|
||||
dsimp [A]
|
||||
simp
|
||||
exact C
|
||||
tauto
|
||||
rw [inter z] at this
|
||||
exact right_ne_zero_of_smul this
|
||||
· exact h₃g
|
||||
|
||||
|
||||
theorem AnalyticOnNhdCompact.eliminateZeros₁
|
||||
{f : ℂ → ℂ}
|
||||
{U : Set ℂ}
|
||||
(h₁U : IsPreconnected U)
|
||||
(h₂U : IsCompact U)
|
||||
(h₁f : AnalyticOnNhd ℂ f U)
|
||||
(h₂f : ∃ u ∈ U, f u ≠ 0) :
|
||||
∃ (g : ℂ → ℂ), AnalyticOnNhd ℂ g U ∧ (∀ z ∈ U, g z ≠ 0) ∧ ∀ z, f z = (∏ᶠ a, (z - a) ^ (h₁f.order a).toNat) • g z := by
|
||||
|
||||
let A := (finiteZeros h₁U h₂U h₁f h₂f).toFinset
|
||||
|
||||
let n : U → ℕ := fun z ↦ (h₁f z z.2).order.toNat
|
||||
|
||||
have hn : ∀ a ∈ A, (h₁f a a.2).order = n a := by
|
||||
intro a _
|
||||
dsimp [n, AnalyticOnNhd.order]
|
||||
rw [eq_comm]
|
||||
apply XX h₁U
|
||||
exact h₂f
|
||||
|
||||
obtain ⟨g, h₁g, h₂g, h₃g⟩ := AnalyticOnNhd.eliminateZeros (A := A) h₁f n hn
|
||||
use g
|
||||
|
||||
have inter : ∀ (z : ℂ), f z = (∏ a ∈ A, (z - ↑a) ^ (h₁f (↑a) a.property).order.toNat) • g z := by
|
||||
intro z
|
||||
rw [h₃g z]
|
||||
|
||||
constructor
|
||||
· exact h₁g
|
||||
· constructor
|
||||
· intro z h₁z
|
||||
by_cases h₂z : ⟨z, h₁z⟩ ∈ ↑A.toSet
|
||||
· exact h₂g ⟨z, h₁z⟩ h₂z
|
||||
· have : f z ≠ 0 := by
|
||||
by_contra C
|
||||
have : ⟨z, h₁z⟩ ∈ ↑A.toSet := by
|
||||
dsimp [A]
|
||||
simp
|
||||
exact C
|
||||
tauto
|
||||
rw [inter z] at this
|
||||
exact right_ne_zero_of_smul this
|
||||
· intro z
|
||||
|
||||
let φ : U → ℂ := fun a ↦ (z - ↑a) ^ (h₁f.order a).toNat
|
||||
have hφ : Function.mulSupport φ ⊆ A := by
|
||||
intro x hx
|
||||
simp [φ] at hx
|
||||
have : (h₁f.order x).toNat ≠ 0 := by
|
||||
by_contra C
|
||||
rw [C] at hx
|
||||
simp at hx
|
||||
simp [A]
|
||||
exact AnalyticAt.supp_order_toNat (h₁f x x.2) this
|
||||
rw [finprod_eq_prod_of_mulSupport_subset φ hφ]
|
||||
rw [inter z]
|
||||
rfl
|
||||
@ -1,77 +0,0 @@
|
||||
/-
|
||||
Copyright (c) 2024 Stefan Kebekus. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Stefan Kebekus
|
||||
-/
|
||||
|
||||
import Mathlib.Analysis.InnerProductSpace.PiL2
|
||||
|
||||
/-!
|
||||
|
||||
# Canoncial Elements in Tensor Powers of Real Inner Product Spaces
|
||||
|
||||
Given an `InnerProductSpace ℝ E`, this file defines two canonical tensors, which
|
||||
are relevant when for a coordinate-free definition of differential operators
|
||||
such as the Laplacian.
|
||||
|
||||
* `InnerProductSpace.canonicalContravariantTensor E : E ⊗[ℝ] E →ₗ[ℝ] ℝ`. This is
|
||||
the element corresponding to the inner product.
|
||||
|
||||
* If `E` is finite-dimensional, then `E ⊗[ℝ] E` is canonically isomorphic to its
|
||||
dual. Accordingly, there exists an element
|
||||
`InnerProductSpace.canonicalCovariantTensor E : E ⊗[ℝ] E` that corresponds to
|
||||
`InnerProductSpace.canonicalContravariantTensor E` under this identification.
|
||||
|
||||
The theorem `InnerProductSpace.canonicalCovariantTensorRepresentation` shows
|
||||
that `InnerProductSpace.canonicalCovariantTensor E` can be computed from any
|
||||
orthonormal basis `v` as `∑ i, (v i) ⊗ₜ[ℝ] (v i)`.
|
||||
|
||||
-/
|
||||
|
||||
open InnerProductSpace
|
||||
open TensorProduct
|
||||
|
||||
|
||||
noncomputable def InnerProductSpace.canonicalContravariantTensor
|
||||
{E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
|
||||
: E ⊗[ℝ] E →ₗ[ℝ] ℝ := TensorProduct.lift bilinFormOfRealInner
|
||||
|
||||
|
||||
noncomputable def InnerProductSpace.canonicalCovariantTensor
|
||||
(E : Type*) [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
|
||||
: E ⊗[ℝ] E := ∑ i, ((stdOrthonormalBasis ℝ E) i) ⊗ₜ[ℝ] ((stdOrthonormalBasis ℝ E) i)
|
||||
|
||||
theorem InnerProductSpace.canonicalCovariantTensorRepresentation
|
||||
(E : Type*) [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
|
||||
[Fintype ι]
|
||||
(v : OrthonormalBasis ι ℝ E) :
|
||||
InnerProductSpace.canonicalCovariantTensor E = ∑ i, (v i) ⊗ₜ[ℝ] (v i) := by
|
||||
|
||||
let w := stdOrthonormalBasis ℝ E
|
||||
conv =>
|
||||
right
|
||||
arg 2
|
||||
intro i
|
||||
rw [← w.sum_repr' (v i)]
|
||||
simp_rw [TensorProduct.sum_tmul, TensorProduct.tmul_sum, TensorProduct.smul_tmul_smul]
|
||||
|
||||
conv =>
|
||||
right
|
||||
rw [Finset.sum_comm]
|
||||
arg 2
|
||||
intro y
|
||||
rw [Finset.sum_comm]
|
||||
arg 2
|
||||
intro x
|
||||
rw [← Finset.sum_smul]
|
||||
arg 1
|
||||
arg 2
|
||||
intro i
|
||||
rw [← real_inner_comm (w x)]
|
||||
simp_rw [OrthonormalBasis.sum_inner_mul_inner v]
|
||||
|
||||
have {i} : ∑ j, ⟪w i, w j⟫_ℝ • w i ⊗ₜ[ℝ] w j = w i ⊗ₜ[ℝ] w i := by
|
||||
rw [Fintype.sum_eq_single i, orthonormal_iff_ite.1 w.orthonormal]; simp
|
||||
intro _ _; rw [orthonormal_iff_ite.1 w.orthonormal]; simp; tauto
|
||||
simp_rw [this]
|
||||
rfl
|
||||
@ -1,104 +0,0 @@
|
||||
import Mathlib.Analysis.Calculus.ContDiff.Basic
|
||||
import Mathlib.Analysis.InnerProductSpace.PiL2
|
||||
|
||||
/-
|
||||
|
||||
Let E, F, G be vector spaces over nontrivally normed field 𝕜, a homogeneus
|
||||
linear differential operator of order n is a map that attaches to every point e
|
||||
of E a linear evaluation
|
||||
|
||||
{Continuous 𝕜-multilinear maps E → F in n variables} → G
|
||||
|
||||
In other words, homogeneus linear differential operator of order n is an
|
||||
instance of the type:
|
||||
|
||||
D : E → (ContinuousMultilinearMap 𝕜 (fun _ : Fin n ↦ E) F) →ₗ[𝕜] G
|
||||
|
||||
Given any map f : E → F, one obtains a map D f : E → G by sending a point e to
|
||||
the evaluation (D e), applied to the n.th derivative of f at e
|
||||
|
||||
fun e ↦ D e (iteratedFDeriv 𝕜 n f e)
|
||||
|
||||
-/
|
||||
|
||||
@[ext]
|
||||
class HomLinDiffOp
|
||||
(𝕜 : Type*) [NontriviallyNormedField 𝕜]
|
||||
(n : ℕ)
|
||||
(E : Type*) [NormedAddCommGroup E] [NormedSpace 𝕜 E]
|
||||
(F : Type*) [NormedAddCommGroup F] [NormedSpace 𝕜 F]
|
||||
(G : Type*) [NormedAddCommGroup G] [NormedSpace 𝕜 G]
|
||||
where
|
||||
tensorfield : E → ( E [×n]→L[𝕜] F) →L[𝕜] G
|
||||
-- tensorfield : E → (ContinuousMultilinearMap 𝕜 (fun _ : Fin n ↦ E) F) →ₗ[𝕜] G
|
||||
|
||||
|
||||
namespace HomLinDiffOp
|
||||
|
||||
noncomputable def toFun
|
||||
{𝕜 : Type*} [NontriviallyNormedField 𝕜]
|
||||
{n : ℕ}
|
||||
{E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
|
||||
{F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
|
||||
{G : Type*} [NormedAddCommGroup G] [NormedSpace 𝕜 G]
|
||||
(o : HomLinDiffOp 𝕜 n E F G)
|
||||
: (E → F) → (E → G) :=
|
||||
fun f z ↦ o.tensorfield z (iteratedFDeriv 𝕜 n f z)
|
||||
|
||||
|
||||
noncomputable def Laplace
|
||||
{𝕜 : Type*} [RCLike 𝕜]
|
||||
{n : ℕ}
|
||||
: HomLinDiffOp 𝕜 2 (EuclideanSpace 𝕜 (Fin n)) 𝕜 𝕜
|
||||
where
|
||||
tensorfield := by
|
||||
intro _
|
||||
|
||||
let v := stdOrthonormalBasis 𝕜 (EuclideanSpace 𝕜 (Fin n))
|
||||
rw [finrank_euclideanSpace_fin] at v
|
||||
|
||||
exact {
|
||||
toFun := fun f' ↦ ∑ i, f' ![v i, v i]
|
||||
map_add' := by
|
||||
intro f₁ f₂
|
||||
exact Finset.sum_add_distrib
|
||||
map_smul' := by
|
||||
intro m f
|
||||
exact Eq.symm (Finset.mul_sum Finset.univ (fun i ↦ f ![v i, v i]) m)
|
||||
cont := by
|
||||
simp
|
||||
apply continuous_finset_sum
|
||||
intro i _
|
||||
exact ContinuousMultilinearMap.continuous_eval_const ![v i, v i]
|
||||
}
|
||||
|
||||
|
||||
noncomputable def Gradient
|
||||
{𝕜 : Type*} [RCLike 𝕜]
|
||||
{n : ℕ}
|
||||
: HomLinDiffOp 𝕜 1 (EuclideanSpace 𝕜 (Fin n)) 𝕜 (EuclideanSpace 𝕜 (Fin n))
|
||||
where
|
||||
tensorfield := by
|
||||
intro _
|
||||
|
||||
let v := stdOrthonormalBasis 𝕜 (EuclideanSpace 𝕜 (Fin n))
|
||||
rw [finrank_euclideanSpace_fin] at v
|
||||
|
||||
exact {
|
||||
toFun := fun f' ↦ ∑ i, (f' ![v i]) • (v i)
|
||||
map_add' := by
|
||||
intro f₁ f₂
|
||||
simp; simp_rw [add_smul, Finset.sum_add_distrib]
|
||||
map_smul' := by
|
||||
intro m f
|
||||
simp; simp_rw [Finset.smul_sum, ←smul_assoc,smul_eq_mul]
|
||||
cont := by
|
||||
simp
|
||||
apply continuous_finset_sum
|
||||
intro i _
|
||||
apply Continuous.smul
|
||||
exact ContinuousMultilinearMap.continuous_eval_const ![v i]
|
||||
exact continuous_const
|
||||
}
|
||||
|
||||
end HomLinDiffOp
|
||||
@ -1,176 +0,0 @@
|
||||
import Mathlib.Analysis.Complex.TaylorSeries
|
||||
import Nevanlinna.cauchyRiemann
|
||||
|
||||
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
|
||||
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F]
|
||||
variable {G : Type*} [NormedAddCommGroup G] [NormedSpace ℂ G]
|
||||
|
||||
def HolomorphicAt (f : E → F) (x : E) : Prop :=
|
||||
∃ s ∈ nhds x, ∀ z ∈ s, DifferentiableAt ℂ f z
|
||||
|
||||
|
||||
theorem HolomorphicAt_iff
|
||||
{f : E → F}
|
||||
{x : E} :
|
||||
HolomorphicAt f x ↔ ∃ s :
|
||||
Set E, IsOpen s ∧ x ∈ s ∧ (∀ z ∈ s, DifferentiableAt ℂ f z) := by
|
||||
constructor
|
||||
· intro hf
|
||||
obtain ⟨t, h₁t, h₂t⟩ := hf
|
||||
obtain ⟨s, h₁s, h₂s, h₃s⟩ := mem_nhds_iff.1 h₁t
|
||||
use s
|
||||
constructor
|
||||
· assumption
|
||||
· constructor
|
||||
· assumption
|
||||
· intro z hz
|
||||
exact h₂t z (h₁s hz)
|
||||
· intro hyp
|
||||
obtain ⟨s, h₁s, h₂s, hf⟩ := hyp
|
||||
use s
|
||||
constructor
|
||||
· apply (IsOpen.mem_nhds_iff h₁s).2 h₂s
|
||||
· assumption
|
||||
|
||||
|
||||
theorem HolomorphicAt_analyticAt
|
||||
[CompleteSpace F]
|
||||
{f : ℂ → F}
|
||||
{x : ℂ} :
|
||||
HolomorphicAt f x → AnalyticAt ℂ f x := by
|
||||
intro hf
|
||||
obtain ⟨s, h₁s, h₂s, h₃s⟩ := HolomorphicAt_iff.1 hf
|
||||
apply DifferentiableOn.analyticAt (s := s)
|
||||
intro z hz
|
||||
apply DifferentiableAt.differentiableWithinAt
|
||||
apply h₃s
|
||||
exact hz
|
||||
exact IsOpen.mem_nhds h₁s h₂s
|
||||
|
||||
|
||||
theorem HolomorphicAt_differentiableAt
|
||||
{f : E → F}
|
||||
{x : E} :
|
||||
HolomorphicAt f x → DifferentiableAt ℂ f x := by
|
||||
intro hf
|
||||
obtain ⟨s, _, h₂s, h₃s⟩ := HolomorphicAt_iff.1 hf
|
||||
exact h₃s x h₂s
|
||||
|
||||
|
||||
theorem HolomorphicAt_isOpen
|
||||
(f : E → F) :
|
||||
IsOpen { x : E | HolomorphicAt f x } := by
|
||||
|
||||
rw [← subset_interior_iff_isOpen]
|
||||
intro x hx
|
||||
simp at hx
|
||||
obtain ⟨s, h₁s, h₂s, h₃s⟩ := HolomorphicAt_iff.1 hx
|
||||
use s
|
||||
constructor
|
||||
· simp
|
||||
constructor
|
||||
· exact h₁s
|
||||
· intro x hx
|
||||
simp
|
||||
use s
|
||||
constructor
|
||||
· exact IsOpen.mem_nhds h₁s hx
|
||||
· exact h₃s
|
||||
· exact h₂s
|
||||
|
||||
|
||||
theorem HolomorphicAt_comp
|
||||
{g : E → F}
|
||||
{f : F → G}
|
||||
{z : E}
|
||||
(hf : HolomorphicAt f (g z))
|
||||
(hg : HolomorphicAt g z) :
|
||||
HolomorphicAt (f ∘ g) z := by
|
||||
obtain ⟨UE, h₁UE, h₂UE⟩ := hg
|
||||
obtain ⟨UF, h₁UF, h₂UF⟩ := hf
|
||||
use UE ∩ g⁻¹' UF
|
||||
constructor
|
||||
· simp
|
||||
constructor
|
||||
· assumption
|
||||
· apply ContinuousAt.preimage_mem_nhds
|
||||
apply (h₂UE z (mem_of_mem_nhds h₁UE)).continuousAt
|
||||
assumption
|
||||
· intro x hx
|
||||
apply DifferentiableAt.comp
|
||||
apply h₂UF
|
||||
exact hx.2
|
||||
apply h₂UE
|
||||
exact hx.1
|
||||
|
||||
|
||||
theorem HolomorphicAt_neg
|
||||
{f : E → F}
|
||||
{z : E}
|
||||
(hf : HolomorphicAt f z) :
|
||||
HolomorphicAt (-f) z := by
|
||||
obtain ⟨UF, h₁UF, h₂UF⟩ := hf
|
||||
use UF
|
||||
constructor
|
||||
· assumption
|
||||
· intro z hz
|
||||
apply differentiableAt_neg_iff.mp
|
||||
simp
|
||||
exact h₂UF z hz
|
||||
|
||||
|
||||
theorem HolomorphicAt_contDiffAt
|
||||
[CompleteSpace F]
|
||||
{f : ℂ → F}
|
||||
{z : ℂ}
|
||||
(hf : HolomorphicAt f z) :
|
||||
ContDiffAt ℝ 2 f z := by
|
||||
|
||||
let t := {x | HolomorphicAt f x}
|
||||
have ht : IsOpen t := HolomorphicAt_isOpen f
|
||||
have hz : z ∈ t := by exact hf
|
||||
|
||||
-- ContDiffAt ℝ 2 f z
|
||||
apply ContDiffOn.contDiffAt _ ((IsOpen.mem_nhds_iff ht).2 hz)
|
||||
suffices h : ContDiffOn ℂ 2 f t from by
|
||||
apply ContDiffOn.restrict_scalars ℝ h
|
||||
apply DifferentiableOn.contDiffOn _ ht
|
||||
intro w hw
|
||||
apply DifferentiableAt.differentiableWithinAt
|
||||
exact HolomorphicAt_differentiableAt hw
|
||||
|
||||
|
||||
theorem CauchyRiemann'₅
|
||||
{f : ℂ → F}
|
||||
{z : ℂ}
|
||||
(h : DifferentiableAt ℂ f z) :
|
||||
partialDeriv ℝ Complex.I f z = Complex.I • partialDeriv ℝ 1 f z := by
|
||||
unfold partialDeriv
|
||||
|
||||
conv =>
|
||||
left
|
||||
rw [DifferentiableAt.fderiv_restrictScalars ℝ h]
|
||||
simp
|
||||
rw [← mul_one Complex.I]
|
||||
rw [← smul_eq_mul]
|
||||
conv =>
|
||||
right
|
||||
right
|
||||
rw [DifferentiableAt.fderiv_restrictScalars ℝ h]
|
||||
simp
|
||||
|
||||
theorem CauchyRiemann'₆
|
||||
{f : ℂ → F}
|
||||
{z : ℂ}
|
||||
(h : HolomorphicAt f z) :
|
||||
partialDeriv ℝ Complex.I f =ᶠ[nhds z] Complex.I • partialDeriv ℝ 1 f := by
|
||||
|
||||
obtain ⟨s, h₁s, hz, h₂f⟩ := HolomorphicAt_iff.1 h
|
||||
|
||||
apply Filter.eventuallyEq_iff_exists_mem.2
|
||||
use s
|
||||
constructor
|
||||
· exact IsOpen.mem_nhds h₁s hz
|
||||
· intro w hw
|
||||
apply CauchyRiemann'₅
|
||||
exact h₂f w hw
|
||||
@ -1,358 +0,0 @@
|
||||
import Init.Classical
|
||||
import Mathlib.Analysis.Analytic.Meromorphic
|
||||
import Mathlib.Topology.ContinuousOn
|
||||
import Mathlib.Analysis.Analytic.IsolatedZeros
|
||||
import Nevanlinna.holomorphic
|
||||
import Nevanlinna.analyticOnNhd_zeroSet
|
||||
|
||||
|
||||
noncomputable def zeroDivisor
|
||||
(f : ℂ → ℂ) :
|
||||
ℂ → ℕ := by
|
||||
intro z
|
||||
by_cases hf : AnalyticAt ℂ f z
|
||||
· exact hf.order.toNat
|
||||
· exact 0
|
||||
|
||||
|
||||
theorem analyticAtZeroDivisorSupport
|
||||
{f : ℂ → ℂ}
|
||||
{z : ℂ}
|
||||
(h : z ∈ Function.support (zeroDivisor f)) :
|
||||
AnalyticAt ℂ f z := by
|
||||
|
||||
by_contra h₁f
|
||||
simp at h
|
||||
dsimp [zeroDivisor] at h
|
||||
simp [h₁f] at h
|
||||
|
||||
|
||||
theorem zeroDivisor_eq_ord_AtZeroDivisorSupport
|
||||
{f : ℂ → ℂ}
|
||||
{z : ℂ}
|
||||
(h : z ∈ Function.support (zeroDivisor f)) :
|
||||
zeroDivisor f z = (analyticAtZeroDivisorSupport h).order.toNat := by
|
||||
|
||||
unfold zeroDivisor
|
||||
simp [analyticAtZeroDivisorSupport h]
|
||||
|
||||
|
||||
theorem zeroDivisor_eq_ord_AtZeroDivisorSupport'
|
||||
{f : ℂ → ℂ}
|
||||
{z : ℂ}
|
||||
(h : z ∈ Function.support (zeroDivisor f)) :
|
||||
zeroDivisor f z = (analyticAtZeroDivisorSupport h).order := by
|
||||
|
||||
unfold zeroDivisor
|
||||
simp [analyticAtZeroDivisorSupport h]
|
||||
sorry
|
||||
|
||||
|
||||
lemma toNatEqSelf_iff {n : ℕ∞} : n.toNat = n ↔ ∃ m : ℕ, m = n := by
|
||||
constructor
|
||||
· intro H₁
|
||||
rw [← ENat.some_eq_coe, ← WithTop.ne_top_iff_exists]
|
||||
by_contra H₂
|
||||
rw [H₂] at H₁
|
||||
simp at H₁
|
||||
· intro H
|
||||
obtain ⟨m, hm⟩ := H
|
||||
rw [← hm]
|
||||
simp
|
||||
|
||||
|
||||
lemma natural_if_toNatNeZero {n : ℕ∞} : n.toNat ≠ 0 → ∃ m : ℕ, m = n := by
|
||||
rw [← ENat.some_eq_coe, ← WithTop.ne_top_iff_exists]
|
||||
contrapose; simp; tauto
|
||||
|
||||
|
||||
theorem zeroDivisor_localDescription
|
||||
{f : ℂ → ℂ}
|
||||
{z₀ : ℂ}
|
||||
(h : z₀ ∈ Function.support (zeroDivisor f)) :
|
||||
∃ (g : ℂ → ℂ), AnalyticAt ℂ g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : ℂ) in nhds z₀, f z = (z - z₀) ^ (zeroDivisor f z₀) • g z := by
|
||||
|
||||
have A : zeroDivisor f ↑z₀ ≠ 0 := by exact h
|
||||
let B := zeroDivisor_eq_ord_AtZeroDivisorSupport h
|
||||
rw [B] at A
|
||||
have C := natural_if_toNatNeZero A
|
||||
obtain ⟨m, hm⟩ := C
|
||||
have h₂m : m ≠ 0 := by
|
||||
rw [← hm] at A
|
||||
simp at A
|
||||
assumption
|
||||
rw [eq_comm] at hm
|
||||
let E := AnalyticAt.order_eq_nat_iff (analyticAtZeroDivisorSupport h) m
|
||||
let F := hm
|
||||
rw [E] at F
|
||||
|
||||
have : m = zeroDivisor f z₀ := by
|
||||
rw [B, hm]
|
||||
simp
|
||||
rwa [this] at F
|
||||
|
||||
|
||||
theorem zeroDivisor_zeroSet
|
||||
{f : ℂ → ℂ}
|
||||
{z₀ : ℂ}
|
||||
(h : z₀ ∈ Function.support (zeroDivisor f)) :
|
||||
f z₀ = 0 := by
|
||||
obtain ⟨g, _, _, h₃⟩ := zeroDivisor_localDescription h
|
||||
rw [Filter.Eventually.self_of_nhds h₃]
|
||||
simp
|
||||
left
|
||||
exact h
|
||||
|
||||
|
||||
theorem zeroDivisor_support_iff
|
||||
{f : ℂ → ℂ}
|
||||
{z₀ : ℂ} :
|
||||
z₀ ∈ Function.support (zeroDivisor f) ↔
|
||||
f z₀ = 0 ∧
|
||||
AnalyticAt ℂ f z₀ ∧
|
||||
∃ (g : ℂ → ℂ), AnalyticAt ℂ g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : ℂ) in nhds z₀, f z = (z - z₀) ^ (zeroDivisor f z₀) • g z := by
|
||||
constructor
|
||||
· intro hz
|
||||
constructor
|
||||
· exact zeroDivisor_zeroSet hz
|
||||
· constructor
|
||||
· exact analyticAtZeroDivisorSupport hz
|
||||
· exact zeroDivisor_localDescription hz
|
||||
· intro ⟨h₁, h₂, h₃⟩
|
||||
have : zeroDivisor f z₀ = (h₂.order).toNat := by
|
||||
unfold zeroDivisor
|
||||
simp [h₂]
|
||||
simp [this]
|
||||
simp [(h₂.order_eq_nat_iff (zeroDivisor f z₀)).2 h₃]
|
||||
obtain ⟨g, h₁g, h₂g, h₃g⟩ := h₃
|
||||
rw [Filter.Eventually.self_of_nhds h₃g] at h₁
|
||||
simp [h₂g] at h₁
|
||||
assumption
|
||||
|
||||
|
||||
theorem topOnPreconnected
|
||||
{f : ℂ → ℂ}
|
||||
{U : Set ℂ}
|
||||
(hU : IsPreconnected U)
|
||||
(h₁f : AnalyticOnNhd ℂ f U)
|
||||
(h₂f : ∃ z ∈ U, f z ≠ 0) :
|
||||
∀ (hz : z ∈ U), (h₁f z hz).order ≠ ⊤ := by
|
||||
|
||||
by_contra H
|
||||
push_neg at H
|
||||
obtain ⟨z', hz'⟩ := H
|
||||
rw [AnalyticAt.order_eq_top_iff] at hz'
|
||||
rw [← AnalyticAt.frequently_zero_iff_eventually_zero (h₁f z z')] at hz'
|
||||
have A := AnalyticOnNhd.eqOn_zero_of_preconnected_of_frequently_eq_zero h₁f hU z' hz'
|
||||
tauto
|
||||
|
||||
|
||||
theorem supportZeroSet
|
||||
{f : ℂ → ℂ}
|
||||
{U : Set ℂ}
|
||||
(hU : IsPreconnected U)
|
||||
(h₁f : AnalyticOnNhd ℂ f U)
|
||||
(h₂f : ∃ z ∈ U, f z ≠ 0) :
|
||||
U ∩ Function.support (zeroDivisor f) = U ∩ f⁻¹' {0} := by
|
||||
|
||||
ext x
|
||||
constructor
|
||||
· intro hx
|
||||
constructor
|
||||
· exact hx.1
|
||||
exact zeroDivisor_zeroSet hx.2
|
||||
· simp
|
||||
intro h₁x h₂x
|
||||
constructor
|
||||
· exact h₁x
|
||||
· let A := zeroDivisor_support_iff (f := f) (z₀ := x)
|
||||
simp at A
|
||||
rw [A]
|
||||
constructor
|
||||
· exact h₂x
|
||||
· constructor
|
||||
· exact h₁f x h₁x
|
||||
· have B := AnalyticAt.order_eq_nat_iff (h₁f x h₁x) (zeroDivisor f x)
|
||||
simp at B
|
||||
rw [← B]
|
||||
dsimp [zeroDivisor]
|
||||
simp [h₁f x h₁x]
|
||||
refine Eq.symm (ENat.coe_toNat ?h.mpr.right.right.right.a)
|
||||
exact topOnPreconnected hU h₁f h₂f h₁x
|
||||
|
||||
/-
|
||||
theorem discreteZeros
|
||||
{f : ℂ → ℂ} :
|
||||
DiscreteTopology (Function.support (zeroDivisor f)) := by
|
||||
|
||||
simp_rw [← singletons_open_iff_discrete, Metric.isOpen_singleton_iff]
|
||||
intro z
|
||||
|
||||
have A : zeroDivisor f ↑z ≠ 0 := by exact z.2
|
||||
let B := zeroDivisor_eq_ord_AtZeroDivisorSupport z.2
|
||||
rw [B] at A
|
||||
have C := natural_if_toNatNeZero A
|
||||
obtain ⟨m, hm⟩ := C
|
||||
have h₂m : m ≠ 0 := by
|
||||
rw [← hm] at A
|
||||
simp at A
|
||||
assumption
|
||||
rw [eq_comm] at hm
|
||||
let E := AnalyticAt.order_eq_nat_iff (analyticAtZeroDivisorSupport z.2) m
|
||||
rw [E] at hm
|
||||
|
||||
obtain ⟨g, h₁g, h₂g, h₃g⟩ := hm
|
||||
|
||||
rw [Metric.eventually_nhds_iff_ball] at h₃g
|
||||
have : ∃ ε > 0, ∀ y ∈ Metric.ball (↑z) ε, g y ≠ 0 := by
|
||||
have h₄g : ContinuousAt g z := AnalyticAt.continuousAt h₁g
|
||||
have : {0}ᶜ ∈ nhds (g z) := by
|
||||
exact compl_singleton_mem_nhds_iff.mpr h₂g
|
||||
|
||||
let F := h₄g.preimage_mem_nhds this
|
||||
rw [Metric.mem_nhds_iff] at F
|
||||
obtain ⟨ε, h₁ε, h₂ε⟩ := F
|
||||
use ε
|
||||
constructor; exact h₁ε
|
||||
intro y hy
|
||||
let G := h₂ε hy
|
||||
simp at G
|
||||
exact G
|
||||
obtain ⟨ε₁, h₁ε₁⟩ := this
|
||||
|
||||
obtain ⟨ε₂, h₁ε₂, h₂ε₂⟩ := h₃g
|
||||
use min ε₁ ε₂
|
||||
constructor
|
||||
· have : 0 < min ε₁ ε₂ := by
|
||||
rw [lt_min_iff]
|
||||
exact And.imp_right (fun _ => h₁ε₂) h₁ε₁
|
||||
exact this
|
||||
|
||||
intro y
|
||||
intro h₁y
|
||||
|
||||
have h₂y : ↑y ∈ Metric.ball (↑z) ε₂ := by
|
||||
simp
|
||||
calc dist y z
|
||||
_ < min ε₁ ε₂ := by assumption
|
||||
_ ≤ ε₂ := by exact min_le_right ε₁ ε₂
|
||||
|
||||
have h₃y : ↑y ∈ Metric.ball (↑z) ε₁ := by
|
||||
simp
|
||||
calc dist y z
|
||||
_ < min ε₁ ε₂ := by assumption
|
||||
_ ≤ ε₁ := by exact min_le_left ε₁ ε₂
|
||||
|
||||
let F := h₂ε₂ y.1 h₂y
|
||||
rw [zeroDivisor_zeroSet y.2] at F
|
||||
simp at F
|
||||
|
||||
simp [h₂m] at F
|
||||
|
||||
have : g y.1 ≠ 0 := by
|
||||
exact h₁ε₁.2 y h₃y
|
||||
simp [this] at F
|
||||
ext
|
||||
rwa [sub_eq_zero] at F
|
||||
-/
|
||||
|
||||
theorem zeroDivisor_finiteOnCompact
|
||||
{f : ℂ → ℂ}
|
||||
{U : Set ℂ}
|
||||
(hU : IsPreconnected U)
|
||||
(h₁f : AnalyticOnNhd ℂ f U)
|
||||
(h₂f : ∃ z ∈ U, f z ≠ 0) -- not needed!
|
||||
(h₂U : IsCompact U) :
|
||||
Set.Finite (U ∩ Function.support (zeroDivisor f)) := by
|
||||
|
||||
have hinter : IsCompact (U ∩ Function.support (zeroDivisor f)) := by
|
||||
apply IsCompact.of_isClosed_subset h₂U
|
||||
rw [supportZeroSet]
|
||||
apply h₁f.continuousOn.preimage_isClosed_of_isClosed
|
||||
exact IsCompact.isClosed h₂U
|
||||
exact isClosed_singleton
|
||||
assumption
|
||||
assumption
|
||||
assumption
|
||||
exact Set.inter_subset_left
|
||||
|
||||
apply hinter.finite
|
||||
apply DiscreteTopology.of_subset (s := Function.support (zeroDivisor f))
|
||||
exact discreteZeros (f := f)
|
||||
exact Set.inter_subset_right
|
||||
|
||||
|
||||
|
||||
|
||||
noncomputable def zeroDivisorDegree
|
||||
{f : ℂ → ℂ}
|
||||
{U : Set ℂ}
|
||||
(h₁U : IsPreconnected U) -- not needed!
|
||||
(h₂U : IsCompact U)
|
||||
(h₁f : AnalyticOnNhd ℂ f U)
|
||||
(h₂f : ∃ z ∈ U, f z ≠ 0) : -- not needed!
|
||||
ℕ := (zeroDivisor_finiteOnCompact h₁U h₁f h₂f h₂U).toFinset.card
|
||||
|
||||
|
||||
lemma zeroDivisorDegreeZero
|
||||
{f : ℂ → ℂ}
|
||||
{U : Set ℂ}
|
||||
(h₁U : IsPreconnected U) -- not needed!
|
||||
(h₂U : IsCompact U)
|
||||
(h₁f : AnalyticOnNhd ℂ f U)
|
||||
(h₂f : ∃ z ∈ U, f z ≠ 0) : -- not needed!
|
||||
0 = zeroDivisorDegree h₁U h₂U h₁f h₂f ↔ U ∩ (zeroDivisor f).support = ∅ := by
|
||||
sorry
|
||||
|
||||
|
||||
lemma eliminatingZeros₀
|
||||
{U : Set ℂ}
|
||||
(h₁U : IsPreconnected U)
|
||||
(h₂U : IsCompact U) :
|
||||
∀ n : ℕ, ∀ f : ℂ → ℂ, (h₁f : AnalyticOnNhd ℂ f U) → (h₂f : ∃ z ∈ U, f z ≠ 0) →
|
||||
(n = zeroDivisorDegree h₁U h₂U h₁f h₂f) →
|
||||
∃ F : ℂ → ℂ, (AnalyticOnNhd ℂ F U) ∧ (f = F * ∏ᶠ a ∈ (U ∩ (zeroDivisor f).support), fun z ↦ (z - a) ^ (zeroDivisor f a)) := by
|
||||
|
||||
intro n
|
||||
induction' n with n ih
|
||||
-- case zero
|
||||
intro f h₁f h₂f h₃f
|
||||
use f
|
||||
rw [zeroDivisorDegreeZero] at h₃f
|
||||
rw [h₃f]
|
||||
simpa
|
||||
|
||||
-- case succ
|
||||
intro f h₁f h₂f h₃f
|
||||
|
||||
let Supp := (zeroDivisor_finiteOnCompact h₁U h₁f h₂f h₂U).toFinset
|
||||
|
||||
have : Supp.Nonempty := by
|
||||
rw [← Finset.one_le_card]
|
||||
calc 1
|
||||
_ ≤ n + 1 := by exact Nat.le_add_left 1 n
|
||||
_ = zeroDivisorDegree h₁U h₂U h₁f h₂f := by exact h₃f
|
||||
_ = Supp.card := by rfl
|
||||
obtain ⟨z₀, hz₀⟩ := this
|
||||
dsimp [Supp] at hz₀
|
||||
simp only [Set.Finite.mem_toFinset, Set.mem_inter_iff] at hz₀
|
||||
|
||||
|
||||
|
||||
|
||||
let A := AnalyticOnNhd.order_eq_nat_iff h₁f hz₀.1 (zeroDivisor f z₀)
|
||||
let B := zeroDivisor_eq_ord_AtZeroDivisorSupport hz₀.2
|
||||
let B := zeroDivisor_eq_ord_AtZeroDivisorSupport' hz₀.2
|
||||
rw [eq_comm] at B
|
||||
let C := A B
|
||||
obtain ⟨g₀, h₁g₀, h₂g₀, h₃g₀⟩ := C
|
||||
|
||||
have h₄g₀ : ∃ z ∈ U, g₀ z ≠ 0 := by sorry
|
||||
have h₅g₀ : n = zeroDivisorDegree h₁U h₂U h₁g₀ h₄g₀ := by sorry
|
||||
|
||||
obtain ⟨F, h₁F, h₂F⟩ := ih g₀ h₁g₀ h₄g₀ h₅g₀
|
||||
use F
|
||||
constructor
|
||||
· assumption
|
||||
·
|
||||
sorry
|
||||
@ -1,449 +0,0 @@
|
||||
import Mathlib.Analysis.Complex.CauchyIntegral
|
||||
import Mathlib.Analysis.Analytic.IsolatedZeros
|
||||
import Nevanlinna.analyticOnNhd_zeroSet
|
||||
import Nevanlinna.harmonicAt_examples
|
||||
import Nevanlinna.harmonicAt_meanValue
|
||||
import Nevanlinna.specialFunctions_CircleIntegral_affine
|
||||
|
||||
open Real
|
||||
|
||||
|
||||
|
||||
theorem jensen_case_R_eq_one
|
||||
(f : ℂ → ℂ)
|
||||
(h₁f : AnalyticOnNhd ℂ f (Metric.closedBall 0 1))
|
||||
(h₂f : f 0 ≠ 0) :
|
||||
log ‖f 0‖ = -∑ᶠ s, (h₁f.order s).toNat * log (‖s.1‖⁻¹) + (2 * π)⁻¹ * ∫ (x : ℝ) in (0)..(2 * π), log ‖f (circleMap 0 1 x)‖ := by
|
||||
|
||||
have h₁U : IsPreconnected (Metric.closedBall (0 : ℂ) 1) :=
|
||||
(convex_closedBall (0 : ℂ) 1).isPreconnected
|
||||
|
||||
have h₂U : IsCompact (Metric.closedBall (0 : ℂ) 1) :=
|
||||
isCompact_closedBall 0 1
|
||||
|
||||
have h'₂f : ∃ u ∈ (Metric.closedBall (0 : ℂ) 1), f u ≠ 0 := by
|
||||
use 0; simp; exact h₂f
|
||||
|
||||
obtain ⟨F, h₁F, h₂F, h₃F⟩ := AnalyticOnNhdCompact.eliminateZeros₂ h₁U h₂U h₁f h'₂f
|
||||
|
||||
have h'₁F : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, HolomorphicAt F z := by
|
||||
intro z h₁z
|
||||
apply AnalyticAt.holomorphicAt
|
||||
exact h₁F z h₁z
|
||||
|
||||
let G := fun z ↦ log ‖F z‖ + ∑ s ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, (h₁f.order s).toNat * log ‖z - s‖
|
||||
|
||||
have decompose_f : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, f z ≠ 0 → log ‖f z‖ = G z := by
|
||||
intro z h₁z h₂z
|
||||
|
||||
conv =>
|
||||
left
|
||||
arg 1
|
||||
rw [h₃F]
|
||||
rw [smul_eq_mul]
|
||||
rw [norm_mul]
|
||||
rw [norm_prod]
|
||||
left
|
||||
arg 2
|
||||
intro b
|
||||
rw [norm_pow]
|
||||
simp only [Complex.norm_eq_abs, Finset.univ_eq_attach]
|
||||
rw [Real.log_mul]
|
||||
rw [Real.log_prod]
|
||||
conv =>
|
||||
left
|
||||
left
|
||||
arg 2
|
||||
intro s
|
||||
rw [Real.log_pow]
|
||||
dsimp [G]
|
||||
abel
|
||||
|
||||
-- ∀ x ∈ ⋯.toFinset, Complex.abs (z - ↑x) ^ (h'₁f.order x).toNat ≠ 0
|
||||
have : ∀ x ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, Complex.abs (z - ↑x) ^ (h₁f.order x).toNat ≠ 0 := by
|
||||
intro s hs
|
||||
simp at hs
|
||||
simp
|
||||
intro h₂s
|
||||
rw [h₂s] at h₂z
|
||||
tauto
|
||||
exact this
|
||||
|
||||
-- ∏ x ∈ ⋯.toFinset, Complex.abs (z - ↑x) ^ (h'₁f.order x).toNat ≠ 0
|
||||
have : ∀ x ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, Complex.abs (z - ↑x) ^ (h₁f.order x).toNat ≠ 0 := by
|
||||
intro s hs
|
||||
simp at hs
|
||||
simp
|
||||
intro h₂s
|
||||
rw [h₂s] at h₂z
|
||||
tauto
|
||||
rw [Finset.prod_ne_zero_iff]
|
||||
exact this
|
||||
|
||||
-- Complex.abs (F z) ≠ 0
|
||||
simp
|
||||
exact h₂F z h₁z
|
||||
|
||||
|
||||
have int_logAbs_f_eq_int_G : ∫ (x : ℝ) in (0)..2 * π, log ‖f (circleMap 0 1 x)‖ = ∫ (x : ℝ) in (0)..2 * π, G (circleMap 0 1 x) := by
|
||||
|
||||
rw [intervalIntegral.integral_congr_ae]
|
||||
rw [MeasureTheory.ae_iff]
|
||||
apply Set.Countable.measure_zero
|
||||
simp
|
||||
|
||||
have t₀ : {a | a ∈ Ι 0 (2 * π) ∧ ¬log ‖f (circleMap 0 1 a)‖ = G (circleMap 0 1 a)}
|
||||
⊆ (circleMap 0 1)⁻¹' (Metric.closedBall 0 1 ∩ f⁻¹' {0}) := by
|
||||
intro a ha
|
||||
simp at ha
|
||||
simp
|
||||
by_contra C
|
||||
have : (circleMap 0 1 a) ∈ Metric.closedBall 0 1 :=
|
||||
circleMap_mem_closedBall 0 (zero_le_one' ℝ) a
|
||||
exact ha.2 (decompose_f (circleMap 0 1 a) this C)
|
||||
|
||||
apply Set.Countable.mono t₀
|
||||
apply Set.Countable.preimage_circleMap
|
||||
apply Set.Finite.countable
|
||||
let A := finiteZeros h₁U h₂U h₁f h'₂f
|
||||
|
||||
have : (Metric.closedBall 0 1 ∩ f ⁻¹' {0}) = (Metric.closedBall 0 1).restrict f ⁻¹' {0} := by
|
||||
ext z
|
||||
simp
|
||||
tauto
|
||||
rw [this]
|
||||
exact Set.Finite.image Subtype.val A
|
||||
exact Ne.symm (zero_ne_one' ℝ)
|
||||
|
||||
|
||||
have decompose_int_G : ∫ (x : ℝ) in (0)..2 * π, G (circleMap 0 1 x)
|
||||
= (∫ (x : ℝ) in (0)..2 * π, log (Complex.abs (F (circleMap 0 1 x))))
|
||||
+ ∑ x ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, (h₁f.order x).toNat * ∫ (x_1 : ℝ) in (0)..2 * π, log (Complex.abs (circleMap 0 1 x_1 - ↑x)) := by
|
||||
dsimp [G]
|
||||
rw [intervalIntegral.integral_add]
|
||||
rw [intervalIntegral.integral_finset_sum]
|
||||
simp_rw [intervalIntegral.integral_const_mul]
|
||||
|
||||
-- ∀ i ∈ (finiteZeros h₁U h₂U h'₁f h'₂f).toFinset,
|
||||
-- IntervalIntegrable (fun x => (h'₁f.order i).toNat *
|
||||
-- log (Complex.abs (circleMap 0 1 x - ↑i))) MeasureTheory.volume 0 (2 * π)
|
||||
intro i _
|
||||
apply IntervalIntegrable.const_mul
|
||||
--simp at this
|
||||
by_cases h₂i : ‖i.1‖ = 1
|
||||
-- case pos
|
||||
exact int'₂ h₂i
|
||||
-- case neg
|
||||
apply Continuous.intervalIntegrable
|
||||
apply continuous_iff_continuousAt.2
|
||||
intro x
|
||||
have : (fun x => log (Complex.abs (circleMap 0 1 x - ↑i))) = log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 1 x - ↑i) :=
|
||||
rfl
|
||||
rw [this]
|
||||
apply ContinuousAt.comp
|
||||
apply Real.continuousAt_log
|
||||
simp
|
||||
|
||||
by_contra ha'
|
||||
conv at h₂i =>
|
||||
arg 1
|
||||
rw [← ha']
|
||||
rw [Complex.norm_eq_abs]
|
||||
rw [abs_circleMap_zero 1 x]
|
||||
simp
|
||||
tauto
|
||||
apply ContinuousAt.comp
|
||||
apply Complex.continuous_abs.continuousAt
|
||||
fun_prop
|
||||
-- IntervalIntegrable (fun x => log (Complex.abs (F (circleMap 0 1 x)))) MeasureTheory.volume 0 (2 * π)
|
||||
apply Continuous.intervalIntegrable
|
||||
apply continuous_iff_continuousAt.2
|
||||
intro x
|
||||
have : (fun x => log (Complex.abs (F (circleMap 0 1 x)))) = log ∘ Complex.abs ∘ F ∘ (fun x ↦ circleMap 0 1 x) :=
|
||||
rfl
|
||||
rw [this]
|
||||
apply ContinuousAt.comp
|
||||
apply Real.continuousAt_log
|
||||
simp [h₂F]
|
||||
-- ContinuousAt (⇑Complex.abs ∘ F ∘ fun x => circleMap 0 1 x) x
|
||||
apply ContinuousAt.comp
|
||||
apply Complex.continuous_abs.continuousAt
|
||||
apply ContinuousAt.comp
|
||||
apply DifferentiableAt.continuousAt (𝕜 := ℂ )
|
||||
apply HolomorphicAt.differentiableAt
|
||||
simp [h'₁F]
|
||||
-- ContinuousAt (fun x => circleMap 0 1 x) x
|
||||
apply Continuous.continuousAt
|
||||
apply continuous_circleMap
|
||||
|
||||
have : (fun x => ∑ s ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, (h₁f.order s).toNat * log (Complex.abs (circleMap 0 1 x - ↑s)))
|
||||
= ∑ s ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, (fun x => (h₁f.order s).toNat * log (Complex.abs (circleMap 0 1 x - ↑s))) := by
|
||||
funext x
|
||||
simp
|
||||
rw [this]
|
||||
apply IntervalIntegrable.sum
|
||||
intro i _
|
||||
apply IntervalIntegrable.const_mul
|
||||
--have : i.1 ∈ Metric.closedBall (0 : ℂ) 1 := i.2
|
||||
--simp at this
|
||||
by_cases h₂i : ‖i.1‖ = 1
|
||||
-- case pos
|
||||
exact int'₂ h₂i
|
||||
-- case neg
|
||||
--have : i.1 ∈ Metric.ball (0 : ℂ) 1 := by sorry
|
||||
apply Continuous.intervalIntegrable
|
||||
apply continuous_iff_continuousAt.2
|
||||
intro x
|
||||
have : (fun x => log (Complex.abs (circleMap 0 1 x - ↑i))) = log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 1 x - ↑i) :=
|
||||
rfl
|
||||
rw [this]
|
||||
apply ContinuousAt.comp
|
||||
apply Real.continuousAt_log
|
||||
simp
|
||||
|
||||
by_contra ha'
|
||||
conv at h₂i =>
|
||||
arg 1
|
||||
rw [← ha']
|
||||
rw [Complex.norm_eq_abs]
|
||||
rw [abs_circleMap_zero 1 x]
|
||||
simp
|
||||
tauto
|
||||
apply ContinuousAt.comp
|
||||
apply Complex.continuous_abs.continuousAt
|
||||
fun_prop
|
||||
|
||||
have t₁ : (∫ (x : ℝ) in (0)..2 * Real.pi, log ‖F (circleMap 0 1 x)‖) = 2 * Real.pi * log ‖F 0‖ := by
|
||||
let logAbsF := fun w ↦ Real.log ‖F w‖
|
||||
have t₀ : ∀ z ∈ Metric.closedBall 0 1, HarmonicAt logAbsF z := by
|
||||
intro z hz
|
||||
apply logabs_of_holomorphicAt_is_harmonic
|
||||
apply h'₁F z hz
|
||||
exact h₂F z hz
|
||||
|
||||
apply harmonic_meanValue₁ 1 Real.zero_lt_one t₀
|
||||
|
||||
simp_rw [← Complex.norm_eq_abs] at decompose_int_G
|
||||
rw [t₁] at decompose_int_G
|
||||
|
||||
conv at decompose_int_G =>
|
||||
right
|
||||
right
|
||||
arg 2
|
||||
intro x
|
||||
right
|
||||
rw [int₃ x.2]
|
||||
simp at decompose_int_G
|
||||
|
||||
rw [int_logAbs_f_eq_int_G]
|
||||
rw [decompose_int_G]
|
||||
rw [h₃F]
|
||||
simp
|
||||
have {l : ℝ} : π⁻¹ * 2⁻¹ * (2 * π * l) = l := by
|
||||
calc π⁻¹ * 2⁻¹ * (2 * π * l)
|
||||
_ = π⁻¹ * (2⁻¹ * 2) * π * l := by ring
|
||||
_ = π⁻¹ * π * l := by ring
|
||||
_ = (π⁻¹ * π) * l := by ring
|
||||
_ = 1 * l := by
|
||||
rw [inv_mul_cancel₀]
|
||||
exact pi_ne_zero
|
||||
_ = l := by simp
|
||||
rw [this]
|
||||
rw [log_mul]
|
||||
rw [log_prod]
|
||||
simp
|
||||
|
||||
rw [finsum_eq_sum_of_support_subset _ (s := (finiteZeros h₁U h₂U h₁f h'₂f).toFinset)]
|
||||
simp
|
||||
simp
|
||||
intro x ⟨h₁x, _⟩
|
||||
simp
|
||||
|
||||
dsimp [AnalyticOnNhd.order] at h₁x
|
||||
simp only [Function.mem_support, ne_eq, Nat.cast_eq_zero, not_or] at h₁x
|
||||
exact AnalyticAt.supp_order_toNat (AnalyticOnNhd.order.proof_1 h₁f x) h₁x
|
||||
|
||||
--
|
||||
intro x hx
|
||||
simp at hx
|
||||
simp
|
||||
intro h₁x
|
||||
nth_rw 1 [← h₁x] at h₂f
|
||||
tauto
|
||||
|
||||
--
|
||||
rw [Finset.prod_ne_zero_iff]
|
||||
intro x hx
|
||||
simp at hx
|
||||
simp
|
||||
intro h₁x
|
||||
nth_rw 1 [← h₁x] at h₂f
|
||||
tauto
|
||||
|
||||
--
|
||||
simp
|
||||
apply h₂F
|
||||
simp
|
||||
|
||||
|
||||
lemma const_mul_circleMap_zero
|
||||
{R θ : ℝ} :
|
||||
circleMap 0 R θ = R * circleMap 0 1 θ := by
|
||||
rw [circleMap_zero, circleMap_zero]
|
||||
simp
|
||||
|
||||
|
||||
theorem jensen
|
||||
{R : ℝ}
|
||||
(hR : 0 < R)
|
||||
(f : ℂ → ℂ)
|
||||
(h₁f : AnalyticOnNhd ℂ f (Metric.closedBall 0 R))
|
||||
(h₂f : f 0 ≠ 0) :
|
||||
log ‖f 0‖ = -∑ᶠ s, (h₁f.order s).toNat * log (R * ‖s.1‖⁻¹) + (2 * π)⁻¹ * ∫ (x : ℝ) in (0)..(2 * π), log ‖f (circleMap 0 R x)‖ := by
|
||||
|
||||
|
||||
let ℓ : ℂ ≃L[ℂ] ℂ :=
|
||||
{
|
||||
toFun := fun x ↦ R * x
|
||||
map_add' := fun x y => DistribSMul.smul_add R x y
|
||||
map_smul' := fun m x => mul_smul_comm m (↑R) x
|
||||
invFun := fun x ↦ R⁻¹ * x
|
||||
left_inv := by
|
||||
intro x
|
||||
simp
|
||||
rw [← mul_assoc, mul_comm, inv_mul_cancel₀, mul_one]
|
||||
simp
|
||||
exact ne_of_gt hR
|
||||
right_inv := by
|
||||
intro x
|
||||
simp
|
||||
rw [← mul_assoc, mul_inv_cancel₀, one_mul]
|
||||
simp
|
||||
exact ne_of_gt hR
|
||||
continuous_toFun := continuous_const_smul R
|
||||
continuous_invFun := continuous_const_smul R⁻¹
|
||||
}
|
||||
|
||||
|
||||
let F := f ∘ ℓ
|
||||
|
||||
have h₁F : AnalyticOnNhd ℂ F (Metric.closedBall 0 1) := by
|
||||
apply AnalyticOnNhd.comp (t := Metric.closedBall 0 R)
|
||||
exact h₁f
|
||||
intro x _
|
||||
apply ℓ.toContinuousLinearMap.analyticAt x
|
||||
|
||||
intro x hx
|
||||
have : ℓ x = R * x := by rfl
|
||||
rw [this]
|
||||
simp
|
||||
simp at hx
|
||||
rw [abs_of_pos hR]
|
||||
calc R * Complex.abs x
|
||||
_ ≤ R * 1 := by exact (mul_le_mul_iff_of_pos_left hR).mpr hx
|
||||
_ = R := by simp
|
||||
|
||||
have h₂F : F 0 ≠ 0 := by
|
||||
dsimp [F]
|
||||
have : ℓ 0 = R * 0 := by rfl
|
||||
rw [this]
|
||||
simpa
|
||||
|
||||
let A := jensen_case_R_eq_one F h₁F h₂F
|
||||
|
||||
dsimp [F] at A
|
||||
have {x : ℂ} : ℓ x = R * x := by rfl
|
||||
repeat
|
||||
simp_rw [this] at A
|
||||
simp at A
|
||||
simp
|
||||
rw [A]
|
||||
simp_rw [← const_mul_circleMap_zero]
|
||||
simp
|
||||
|
||||
let e : (Metric.closedBall (0 : ℂ) 1) → (Metric.closedBall (0 : ℂ) R) := by
|
||||
intro ⟨x, hx⟩
|
||||
have hy : R • x ∈ Metric.closedBall (0 : ℂ) R := by
|
||||
simp
|
||||
simp at hx
|
||||
have : R = |R| := by exact Eq.symm (abs_of_pos hR)
|
||||
rw [← this]
|
||||
norm_num
|
||||
calc R * Complex.abs x
|
||||
_ ≤ R * 1 := by exact (mul_le_mul_iff_of_pos_left hR).mpr hx
|
||||
_ = R := by simp
|
||||
exact ⟨R • x, hy⟩
|
||||
|
||||
let e' : (Metric.closedBall (0 : ℂ) R) → (Metric.closedBall (0 : ℂ) 1) := by
|
||||
intro ⟨x, hx⟩
|
||||
have hy : R⁻¹ • x ∈ Metric.closedBall (0 : ℂ) 1 := by
|
||||
simp
|
||||
simp at hx
|
||||
have : R = |R| := by exact Eq.symm (abs_of_pos hR)
|
||||
rw [← this]
|
||||
norm_num
|
||||
calc R⁻¹ * Complex.abs x
|
||||
_ ≤ R⁻¹ * R := by
|
||||
apply mul_le_mul_of_nonneg_left hx
|
||||
apply inv_nonneg.mpr
|
||||
exact abs_eq_self.mp (id (Eq.symm this))
|
||||
_ = 1 := by
|
||||
apply inv_mul_cancel₀
|
||||
exact Ne.symm (ne_of_lt hR)
|
||||
exact ⟨R⁻¹ • x, hy⟩
|
||||
|
||||
apply finsum_eq_of_bijective e
|
||||
|
||||
|
||||
apply Function.bijective_iff_has_inverse.mpr
|
||||
use e'
|
||||
constructor
|
||||
· apply Function.leftInverse_iff_comp.mpr
|
||||
funext x
|
||||
dsimp only [e, e', id_eq, eq_mp_eq_cast, Function.comp_apply]
|
||||
conv =>
|
||||
left
|
||||
arg 1
|
||||
rw [← smul_assoc, smul_eq_mul]
|
||||
rw [inv_mul_cancel₀ (Ne.symm (ne_of_lt hR))]
|
||||
rw [one_smul]
|
||||
· apply Function.rightInverse_iff_comp.mpr
|
||||
funext x
|
||||
dsimp only [e, e', id_eq, eq_mp_eq_cast, Function.comp_apply]
|
||||
conv =>
|
||||
left
|
||||
arg 1
|
||||
rw [← smul_assoc, smul_eq_mul]
|
||||
rw [mul_inv_cancel₀ (Ne.symm (ne_of_lt hR))]
|
||||
rw [one_smul]
|
||||
|
||||
intro x
|
||||
simp
|
||||
by_cases hx : x = (0 : ℂ)
|
||||
rw [hx]
|
||||
simp
|
||||
|
||||
rw [log_mul, log_mul, log_inv, log_inv]
|
||||
have : R = |R| := by exact Eq.symm (abs_of_pos hR)
|
||||
rw [← this]
|
||||
simp
|
||||
left
|
||||
congr 1
|
||||
|
||||
dsimp [AnalyticOnNhd.order]
|
||||
rw [← AnalyticAt.order_comp_CLE ℓ]
|
||||
|
||||
--
|
||||
simpa
|
||||
--
|
||||
have : R = |R| := by exact Eq.symm (abs_of_pos hR)
|
||||
rw [← this]
|
||||
apply inv_ne_zero
|
||||
exact Ne.symm (ne_of_lt hR)
|
||||
--
|
||||
exact Ne.symm (ne_of_lt hR)
|
||||
--
|
||||
simp
|
||||
constructor
|
||||
· assumption
|
||||
· exact Ne.symm (ne_of_lt hR)
|
||||
@ -1,666 +0,0 @@
|
||||
import Mathlib.Analysis.Complex.TaylorSeries
|
||||
import Mathlib.Analysis.SpecialFunctions.Integrals
|
||||
import Mathlib.MeasureTheory.Integral.DivergenceTheorem
|
||||
import Mathlib.MeasureTheory.Integral.IntervalIntegral
|
||||
import Mathlib.MeasureTheory.Function.LocallyIntegrable
|
||||
import Nevanlinna.partialDeriv
|
||||
import Nevanlinna.cauchyRiemann
|
||||
|
||||
/-
|
||||
noncomputable def partialDeriv
|
||||
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
|
||||
{F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] : E → (E → F) → (E → F) :=
|
||||
fun v ↦ (fun f ↦ (fun w ↦ fderiv ℝ f w v))
|
||||
|
||||
|
||||
theorem partialDeriv_compContLinAt
|
||||
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
|
||||
{F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F]
|
||||
{G : Type*} [NormedAddCommGroup G] [NormedSpace ℂ G]
|
||||
{f : E → F}
|
||||
{l : F →L[ℝ] G}
|
||||
{v : E}
|
||||
{x : E}
|
||||
(h : DifferentiableAt ℝ f x) :
|
||||
(partialDeriv v (l ∘ f)) x = (l ∘ partialDeriv v f) x:= by
|
||||
unfold partialDeriv
|
||||
rw [fderiv.comp x (ContinuousLinearMap.differentiableAt l) h]
|
||||
simp
|
||||
|
||||
|
||||
theorem partialDeriv_compCLE
|
||||
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
|
||||
{F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F]
|
||||
{G : Type*} [NormedAddCommGroup G] [NormedSpace ℂ G]
|
||||
{f : E → F}
|
||||
{l : F ≃L[ℝ] G} {v : E} : partialDeriv v (l ∘ f) = l ∘ partialDeriv v f := by
|
||||
funext x
|
||||
by_cases hyp : DifferentiableAt ℝ f x
|
||||
· let lCLM : F →L[ℝ] G := l
|
||||
suffices shyp : partialDeriv v (lCLM ∘ f) x = (lCLM ∘ partialDeriv v f) x from by tauto
|
||||
apply partialDeriv_compContLinAt
|
||||
exact hyp
|
||||
· unfold partialDeriv
|
||||
rw [fderiv_zero_of_not_differentiableAt]
|
||||
simp
|
||||
rw [fderiv_zero_of_not_differentiableAt]
|
||||
simp
|
||||
exact hyp
|
||||
rw [ContinuousLinearEquiv.comp_differentiableAt_iff]
|
||||
exact hyp
|
||||
|
||||
theorem partialDeriv_smul'₂
|
||||
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
|
||||
{F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F]
|
||||
{f : E → F} {a : ℂ} {v : E} :
|
||||
partialDeriv v (a • f) = a • partialDeriv v f := by
|
||||
|
||||
funext w
|
||||
by_cases ha : a = 0
|
||||
· unfold partialDeriv
|
||||
have : a • f = fun y ↦ a • f y := by rfl
|
||||
rw [this, ha]
|
||||
simp
|
||||
· -- Now a is not zero. We present scalar multiplication with a as a continuous linear equivalence.
|
||||
let smulCLM : F ≃L[ℝ] F :=
|
||||
{
|
||||
toFun := fun x ↦ a • x
|
||||
map_add' := fun x y => DistribSMul.smul_add a x y
|
||||
map_smul' := fun m x => (smul_comm ((RingHom.id ℝ) m) a x).symm
|
||||
invFun := fun x ↦ a⁻¹ • x
|
||||
left_inv := by
|
||||
intro x
|
||||
simp
|
||||
rw [← smul_assoc, smul_eq_mul, mul_comm, mul_inv_cancel ha, one_smul]
|
||||
right_inv := by
|
||||
intro x
|
||||
simp
|
||||
rw [← smul_assoc, smul_eq_mul, mul_inv_cancel ha, one_smul]
|
||||
continuous_toFun := continuous_const_smul a
|
||||
continuous_invFun := continuous_const_smul a⁻¹
|
||||
}
|
||||
|
||||
have : a • f = smulCLM ∘ f := by tauto
|
||||
rw [this]
|
||||
rw [partialDeriv_compCLE]
|
||||
tauto
|
||||
|
||||
theorem CauchyRiemann₄
|
||||
{F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F]
|
||||
{f : ℂ → F} :
|
||||
(Differentiable ℂ f) → partialDeriv ℝ Complex.I f = Complex.I • partialDeriv ℝ 1 f := by
|
||||
intro h
|
||||
unfold partialDeriv
|
||||
|
||||
conv =>
|
||||
left
|
||||
intro w
|
||||
rw [DifferentiableAt.fderiv_restrictScalars ℝ (h w)]
|
||||
simp
|
||||
rw [← mul_one Complex.I]
|
||||
rw [← smul_eq_mul]
|
||||
conv =>
|
||||
right
|
||||
right
|
||||
intro w
|
||||
rw [DifferentiableAt.fderiv_restrictScalars ℝ (h w)]
|
||||
funext w
|
||||
simp
|
||||
-/
|
||||
|
||||
|
||||
theorem MeasureTheory.integral2_divergence₃
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
|
||||
(f g : ℝ × ℝ → E)
|
||||
(h₁f : ContDiff ℝ 1 f)
|
||||
(h₁g : ContDiff ℝ 1 g)
|
||||
(a₁ : ℝ)
|
||||
(a₂ : ℝ)
|
||||
(b₁ : ℝ)
|
||||
(b₂ : ℝ) :
|
||||
∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ((fderiv ℝ f) (x, y)) (1, 0) + ((fderiv ℝ g) (x, y)) (0, 1) = (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) - ∫ (y : ℝ) in a₂..b₂, f (a₁, y) := by
|
||||
|
||||
apply integral2_divergence_prod_of_hasFDerivWithinAt_off_countable f g (fderiv ℝ f) (fderiv ℝ g) a₁ a₂ b₁ b₂ ∅
|
||||
exact Set.countable_empty
|
||||
-- ContinuousOn f (Set.uIcc a₁ b₁ ×ˢ Set.uIcc a₂ b₂)
|
||||
exact h₁f.continuous.continuousOn
|
||||
--
|
||||
exact h₁g.continuous.continuousOn
|
||||
--
|
||||
rw [Set.diff_empty]
|
||||
intro x _
|
||||
exact DifferentiableAt.hasFDerivAt ((h₁f.differentiable le_rfl) x)
|
||||
--
|
||||
rw [Set.diff_empty]
|
||||
intro y _
|
||||
exact DifferentiableAt.hasFDerivAt ((h₁g.differentiable le_rfl) y)
|
||||
--
|
||||
apply ContinuousOn.integrableOn_compact
|
||||
apply IsCompact.prod
|
||||
exact isCompact_uIcc
|
||||
exact isCompact_uIcc
|
||||
apply ContinuousOn.add
|
||||
apply Continuous.continuousOn
|
||||
exact Continuous.clm_apply (ContDiff.continuous_fderiv h₁f le_rfl) continuous_const
|
||||
apply Continuous.continuousOn
|
||||
exact Continuous.clm_apply (ContDiff.continuous_fderiv h₁g le_rfl) continuous_const
|
||||
|
||||
|
||||
theorem integral_divergence₄
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
|
||||
(f g : ℂ → E)
|
||||
(h₁f : ContDiff ℝ 1 f)
|
||||
(h₁g : ContDiff ℝ 1 g)
|
||||
(a₁ : ℝ)
|
||||
(a₂ : ℝ)
|
||||
(b₁ : ℝ)
|
||||
(b₂ : ℝ) :
|
||||
∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ((fderiv ℝ f) ⟨x, y⟩ ) 1 + ((fderiv ℝ g) ⟨x, y⟩) Complex.I = (((∫ (x : ℝ) in a₁..b₁, g ⟨x, b₂⟩) - ∫ (x : ℝ) in a₁..b₁, g ⟨x, a₂⟩) + ∫ (y : ℝ) in a₂..b₂, f ⟨b₁, y⟩) - ∫ (y : ℝ) in a₂..b₂, f ⟨a₁, y⟩ := by
|
||||
|
||||
let fr : ℝ × ℝ → E := f ∘ Complex.equivRealProdCLM.symm
|
||||
let gr : ℝ × ℝ → E := g ∘ Complex.equivRealProdCLM.symm
|
||||
|
||||
have sfr {x y : ℝ} : f { re := x, im := y } = fr (x, y) := by exact rfl
|
||||
have sgr {x y : ℝ} : g { re := x, im := y } = gr (x, y) := by exact rfl
|
||||
repeat (conv in f { re := _, im := _ } => rw [sfr])
|
||||
repeat (conv in g { re := _, im := _ } => rw [sgr])
|
||||
|
||||
have sfr' {x y : ℝ} {z : ℂ} : (fderiv ℝ f { re := x, im := y }) z = fderiv ℝ fr (x, y) (Complex.equivRealProdCLM z) := by
|
||||
rw [fderiv.comp]
|
||||
rw [Complex.equivRealProdCLM.symm.fderiv]
|
||||
tauto
|
||||
apply Differentiable.differentiableAt
|
||||
exact h₁f.differentiable le_rfl
|
||||
exact Complex.equivRealProdCLM.symm.differentiableAt
|
||||
conv in ⇑(fderiv ℝ f { re := _, im := _ }) _ => rw [sfr']
|
||||
|
||||
have sgr' {x y : ℝ} {z : ℂ} : (fderiv ℝ g { re := x, im := y }) z = fderiv ℝ gr (x, y) (Complex.equivRealProdCLM z) := by
|
||||
rw [fderiv.comp]
|
||||
rw [Complex.equivRealProdCLM.symm.fderiv]
|
||||
tauto
|
||||
apply Differentiable.differentiableAt
|
||||
exact h₁g.differentiable le_rfl
|
||||
exact Complex.equivRealProdCLM.symm.differentiableAt
|
||||
conv in ⇑(fderiv ℝ g { re := _, im := _ }) _ => rw [sgr']
|
||||
|
||||
apply MeasureTheory.integral2_divergence₃ fr gr _ _ a₁ a₂ b₁ b₂
|
||||
-- ContDiff ℝ 1 fr
|
||||
exact (ContinuousLinearEquiv.contDiff_comp_iff (ContinuousLinearEquiv.symm Complex.equivRealProdCLM)).mpr h₁f
|
||||
-- ContDiff ℝ 1 gr
|
||||
exact (ContinuousLinearEquiv.contDiff_comp_iff (ContinuousLinearEquiv.symm Complex.equivRealProdCLM)).mpr h₁g
|
||||
|
||||
|
||||
theorem integral_divergence₅
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
(F : ℂ → E)
|
||||
(hF : Differentiable ℂ F)
|
||||
(lowerLeft upperRight : ℂ) :
|
||||
(∫ (x : ℝ) in lowerLeft.re..upperRight.re, F ⟨x, lowerLeft.im⟩) + Complex.I • ∫ (x : ℝ) in lowerLeft.im..upperRight.im, F ⟨upperRight.re, x⟩ =
|
||||
(∫ (x : ℝ) in lowerLeft.re..upperRight.re, F ⟨x, upperRight.im⟩) + Complex.I • ∫ (x : ℝ) in lowerLeft.im..upperRight.im, F ⟨lowerLeft.re, x⟩ := by
|
||||
|
||||
let h₁f : ContDiff ℝ 1 F := (hF.contDiff : ContDiff ℂ 1 F).restrict_scalars ℝ
|
||||
|
||||
let h₁g : ContDiff ℝ 1 (-Complex.I • F) := by
|
||||
have : -Complex.I • F = fun x ↦ -Complex.I • F x := by rfl
|
||||
rw [this]
|
||||
apply ContDiff.comp
|
||||
exact contDiff_const_smul _
|
||||
exact h₁f
|
||||
|
||||
let A := integral_divergence₄ (-Complex.I • F) F h₁g h₁f lowerLeft.re upperRight.im upperRight.re lowerLeft.im
|
||||
have {z : ℂ} : fderiv ℝ F z Complex.I = partialDeriv ℝ Complex.I F z := by rfl
|
||||
conv at A in (fderiv ℝ F _) _ => rw [this]
|
||||
have {z : ℂ} : fderiv ℝ (-Complex.I • F) z 1 = partialDeriv ℝ 1 (-Complex.I • F) z := by rfl
|
||||
conv at A in (fderiv ℝ (-Complex.I • F) _) _ => rw [this]
|
||||
conv at A =>
|
||||
left
|
||||
arg 1
|
||||
intro x
|
||||
arg 1
|
||||
intro y
|
||||
rw [CauchyRiemann₄ hF]
|
||||
rw [partialDeriv_smul'₂]
|
||||
simp
|
||||
simp at A
|
||||
|
||||
have {t₁ t₂ t₃ t₄ : E} : 0 = (t₁ - t₂) + t₃ + t₄ → t₁ + t₃ = t₂ - t₄ := by
|
||||
intro hyp
|
||||
calc
|
||||
t₁ + t₃ = t₁ + t₃ - 0 := by rw [sub_zero (t₁ + t₃)]
|
||||
_ = t₁ + t₃ - (t₁ - t₂ + t₃ + t₄) := by rw [hyp]
|
||||
_ = t₂ - t₄ := by abel
|
||||
let B := this A
|
||||
repeat
|
||||
rw [intervalIntegral.integral_symm lowerLeft.im upperRight.im] at B
|
||||
simp at B
|
||||
exact B
|
||||
|
||||
|
||||
noncomputable def primitive
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] :
|
||||
ℂ → (ℂ → E) → (ℂ → E) := by
|
||||
intro z₀
|
||||
intro f
|
||||
exact fun z ↦ (∫ (x : ℝ) in z₀.re..z.re, f ⟨x, z₀.im⟩) + Complex.I • ∫ (x : ℝ) in z₀.im..z.im, f ⟨z.re, x⟩
|
||||
|
||||
|
||||
theorem primitive_zeroAtBasepoint
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
(f : ℂ → E)
|
||||
(z₀ : ℂ) :
|
||||
(primitive z₀ f) z₀ = 0 := by
|
||||
unfold primitive
|
||||
simp
|
||||
|
||||
|
||||
theorem primitive_fderivAtBasepointZero
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
(f : ℂ → E)
|
||||
(hf : Continuous f) :
|
||||
HasDerivAt (primitive 0 f) (f 0) 0 := by
|
||||
unfold primitive
|
||||
simp
|
||||
apply hasDerivAt_iff_isLittleO.2
|
||||
simp
|
||||
rw [Asymptotics.isLittleO_iff]
|
||||
intro c hc
|
||||
|
||||
have {z : ℂ} {e : E} : z • e = (∫ (_ : ℝ) in (0)..(z.re), e) + Complex.I • ∫ (_ : ℝ) in (0)..(z.im), e:= by
|
||||
simp
|
||||
rw [smul_comm]
|
||||
rw [← smul_assoc]
|
||||
simp
|
||||
have : z.re • e = (z.re : ℂ) • e := by exact rfl
|
||||
rw [this, ← add_smul]
|
||||
simp
|
||||
conv =>
|
||||
left
|
||||
intro x
|
||||
left
|
||||
arg 1
|
||||
arg 2
|
||||
rw [this]
|
||||
have {A B C D :E} : (A + B) - (C + D) = (A - C) + (B - D) := by
|
||||
abel
|
||||
have t₀ {r : ℝ} : IntervalIntegrable (fun x => f { re := x, im := 0 }) MeasureTheory.volume 0 r := by
|
||||
apply Continuous.intervalIntegrable
|
||||
apply Continuous.comp
|
||||
exact hf
|
||||
have : (fun x => ({ re := x, im := 0 } : ℂ)) = Complex.ofRealLI := by rfl
|
||||
rw [this]
|
||||
continuity
|
||||
have t₁ {r : ℝ} : IntervalIntegrable (fun _ => f 0) MeasureTheory.volume 0 r := by
|
||||
apply Continuous.intervalIntegrable
|
||||
apply Continuous.comp
|
||||
exact hf
|
||||
fun_prop
|
||||
have t₂ {a b : ℝ} : IntervalIntegrable (fun x_1 => f { re := a, im := x_1 }) MeasureTheory.volume 0 b := by
|
||||
apply Continuous.intervalIntegrable
|
||||
apply Continuous.comp hf
|
||||
have : (Complex.mk a) = (fun x => Complex.I • Complex.ofRealCLM x + { re := a, im := 0 }) := by
|
||||
funext x
|
||||
apply Complex.ext
|
||||
rw [Complex.add_re]
|
||||
simp
|
||||
simp
|
||||
rw [this]
|
||||
apply Continuous.add
|
||||
continuity
|
||||
fun_prop
|
||||
|
||||
have t₃ {a : ℝ} : IntervalIntegrable (fun _ => f 0) MeasureTheory.volume 0 a := by
|
||||
apply Continuous.intervalIntegrable
|
||||
apply Continuous.comp
|
||||
exact hf
|
||||
fun_prop
|
||||
conv =>
|
||||
left
|
||||
intro x
|
||||
left
|
||||
arg 1
|
||||
rw [this]
|
||||
rw [← smul_sub]
|
||||
rw [← intervalIntegral.integral_sub t₀ t₁]
|
||||
rw [← intervalIntegral.integral_sub t₂ t₃]
|
||||
rw [Filter.eventually_iff_exists_mem]
|
||||
|
||||
let s := f⁻¹' Metric.ball (f 0) (c / (4 : ℝ))
|
||||
have h₁s : IsOpen s := IsOpen.preimage hf Metric.isOpen_ball
|
||||
have h₂s : 0 ∈ s := by
|
||||
apply Set.mem_preimage.mpr
|
||||
apply Metric.mem_ball_self
|
||||
linarith
|
||||
|
||||
obtain ⟨ε, h₁ε, h₂ε⟩ := Metric.isOpen_iff.1 h₁s 0 h₂s
|
||||
|
||||
have h₃ε : ∀ y ∈ Metric.ball 0 ε, ‖(f y) - (f 0)‖ < (c / (4 : ℝ)) := by
|
||||
intro y hy
|
||||
apply mem_ball_iff_norm.mp (h₂ε hy)
|
||||
|
||||
use Metric.ball 0 (ε / (4 : ℝ))
|
||||
constructor
|
||||
· apply Metric.ball_mem_nhds 0
|
||||
linarith
|
||||
· intro y hy
|
||||
have h₁y : |y.re| < ε / 4 := by
|
||||
calc |y.re|
|
||||
_ ≤ Complex.abs y := by apply Complex.abs_re_le_abs
|
||||
_ < ε / 4 := by
|
||||
let A := mem_ball_iff_norm.1 hy
|
||||
simp at A
|
||||
linarith
|
||||
have h₂y : |y.im| < ε / 4 := by
|
||||
calc |y.im|
|
||||
_ ≤ Complex.abs y := by apply Complex.abs_im_le_abs
|
||||
_ < ε / 4 := by
|
||||
let A := mem_ball_iff_norm.1 hy
|
||||
simp at A
|
||||
linarith
|
||||
|
||||
have intervalComputation {x' y' : ℝ} (h : x' ∈ Ι 0 y') : |x'| ≤ |y'| := by
|
||||
let A := h.1
|
||||
let B := h.2
|
||||
rcases le_total 0 y' with hy | hy
|
||||
· simp [hy] at A
|
||||
simp [hy] at B
|
||||
rw [abs_of_nonneg hy]
|
||||
rw [abs_of_nonneg (le_of_lt A)]
|
||||
exact B
|
||||
· simp [hy] at A
|
||||
simp [hy] at B
|
||||
rw [abs_of_nonpos hy]
|
||||
rw [abs_of_nonpos]
|
||||
linarith [h.1]
|
||||
exact B
|
||||
|
||||
have t₁ : ‖(∫ (x : ℝ) in (0)..(y.re), f { re := x, im := 0 } - f 0)‖ ≤ (c / (4 : ℝ)) * |y.re - 0| := by
|
||||
apply intervalIntegral.norm_integral_le_of_norm_le_const
|
||||
intro x hx
|
||||
|
||||
have h₁x : |x| < ε / 4 := by
|
||||
calc |x|
|
||||
_ ≤ |y.re| := intervalComputation hx
|
||||
_ < ε / 4 := h₁y
|
||||
apply le_of_lt
|
||||
apply h₃ε { re := x, im := 0 }
|
||||
rw [mem_ball_iff_norm]
|
||||
simp
|
||||
have : { re := x, im := 0 } = (x : ℂ) := by rfl
|
||||
rw [this]
|
||||
rw [Complex.abs_ofReal]
|
||||
linarith
|
||||
|
||||
have t₂ : ‖∫ (x : ℝ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖ ≤ (c / (4 : ℝ)) * |y.im - 0| := by
|
||||
apply intervalIntegral.norm_integral_le_of_norm_le_const
|
||||
intro x hx
|
||||
|
||||
have h₁x : |x| < ε / 4 := by
|
||||
calc |x|
|
||||
_ ≤ |y.im| := intervalComputation hx
|
||||
_ < ε / 4 := h₂y
|
||||
|
||||
apply le_of_lt
|
||||
apply h₃ε { re := y.re, im := x }
|
||||
simp
|
||||
|
||||
calc Complex.abs { re := y.re, im := x }
|
||||
_ ≤ |y.re| + |x| := by
|
||||
apply Complex.abs_le_abs_re_add_abs_im { re := y.re, im := x }
|
||||
_ < ε := by
|
||||
linarith
|
||||
|
||||
calc ‖(∫ (x : ℝ) in (0)..(y.re), f { re := x, im := 0 } - f 0) + Complex.I • ∫ (x : ℝ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖
|
||||
_ ≤ ‖(∫ (x : ℝ) in (0)..(y.re), f { re := x, im := 0 } - f 0)‖ + ‖Complex.I • ∫ (x : ℝ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖ := by
|
||||
apply norm_add_le
|
||||
_ ≤ ‖(∫ (x : ℝ) in (0)..(y.re), f { re := x, im := 0 } - f 0)‖ + ‖∫ (x : ℝ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖ := by
|
||||
simp
|
||||
rw [norm_smul]
|
||||
simp
|
||||
_ ≤ (c / (4 : ℝ)) * |y.re - 0| + (c / (4 : ℝ)) * |y.im - 0| := by
|
||||
apply add_le_add
|
||||
exact t₁
|
||||
exact t₂
|
||||
_ ≤ (c / (4 : ℝ)) * (|y.re| + |y.im|) := by
|
||||
simp
|
||||
rw [mul_add]
|
||||
_ ≤ (c / (4 : ℝ)) * (4 * ‖y‖) := by
|
||||
have : |y.re| + |y.im| ≤ 4 * ‖y‖ := by
|
||||
calc |y.re| + |y.im|
|
||||
_ ≤ ‖y‖ + ‖y‖ := by
|
||||
apply add_le_add
|
||||
apply Complex.abs_re_le_abs
|
||||
apply Complex.abs_im_le_abs
|
||||
_ ≤ 4 * ‖y‖ := by
|
||||
rw [← two_mul]
|
||||
apply mul_le_mul
|
||||
linarith
|
||||
rfl
|
||||
exact norm_nonneg y
|
||||
linarith
|
||||
|
||||
apply mul_le_mul
|
||||
rfl
|
||||
exact this
|
||||
apply add_nonneg
|
||||
apply abs_nonneg
|
||||
apply abs_nonneg
|
||||
linarith
|
||||
_ ≤ c * ‖y‖ := by
|
||||
linarith
|
||||
|
||||
|
||||
theorem primitive_translation
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
(f : ℂ → E)
|
||||
(z₀ t : ℂ) :
|
||||
primitive z₀ (f ∘ fun z ↦ (z - t)) = ((primitive (z₀ - t) f) ∘ fun z ↦ (z - t)) := by
|
||||
funext z
|
||||
unfold primitive
|
||||
simp
|
||||
|
||||
let g : ℝ → E := fun x ↦ f ( {re := x, im := z₀.im - t.im} )
|
||||
have {x : ℝ} : f ({ re := x, im := z₀.im } - t) = g (1*x - t.re) := by
|
||||
congr 1
|
||||
apply Complex.ext <;> simp
|
||||
conv =>
|
||||
left
|
||||
left
|
||||
arg 1
|
||||
intro x
|
||||
rw [this]
|
||||
rw [intervalIntegral.integral_comp_mul_sub g one_ne_zero (t.re)]
|
||||
simp
|
||||
|
||||
congr 1
|
||||
let g : ℝ → E := fun x ↦ f ( {re := z.re - t.re, im := x} )
|
||||
have {x : ℝ} : f ({ re := z.re, im := x} - t) = g (1*x - t.im) := by
|
||||
congr 1
|
||||
apply Complex.ext <;> simp
|
||||
conv =>
|
||||
left
|
||||
arg 1
|
||||
intro x
|
||||
rw [this]
|
||||
rw [intervalIntegral.integral_comp_mul_sub g one_ne_zero (t.im)]
|
||||
simp
|
||||
|
||||
|
||||
theorem primitive_hasDerivAtBasepoint
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
{f : ℂ → E}
|
||||
(hf : Continuous f)
|
||||
(z₀ : ℂ) :
|
||||
HasDerivAt (primitive z₀ f) (f z₀) z₀ := by
|
||||
|
||||
let g := f ∘ fun z ↦ z + z₀
|
||||
have : Continuous g := by continuity
|
||||
let A := primitive_fderivAtBasepointZero g this
|
||||
simp at A
|
||||
|
||||
let B := primitive_translation g z₀ z₀
|
||||
simp at B
|
||||
have : (g ∘ fun z ↦ (z - z₀)) = f := by
|
||||
funext z
|
||||
dsimp [g]
|
||||
simp
|
||||
rw [this] at B
|
||||
rw [B]
|
||||
have : f z₀ = (1 : ℂ) • (f z₀) := by
|
||||
exact (MulAction.one_smul (f z₀)).symm
|
||||
conv =>
|
||||
arg 2
|
||||
rw [this]
|
||||
|
||||
apply HasDerivAt.scomp
|
||||
simp
|
||||
have : g 0 = f z₀ := by simp [g]
|
||||
rw [← this]
|
||||
exact A
|
||||
apply HasDerivAt.sub_const
|
||||
have : (fun (x : ℂ) ↦ x) = id := by
|
||||
funext x
|
||||
simp
|
||||
rw [this]
|
||||
exact hasDerivAt_id z₀
|
||||
|
||||
|
||||
lemma integrability₁
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
(f : ℂ → E)
|
||||
(hf : Differentiable ℂ f)
|
||||
(a₁ a₂ b : ℝ) :
|
||||
IntervalIntegrable (fun x => f { re := x, im := b }) MeasureTheory.volume a₁ a₂ := by
|
||||
apply Continuous.intervalIntegrable
|
||||
apply Continuous.comp
|
||||
exact Differentiable.continuous hf
|
||||
have : ((fun x => { re := x, im := b }) : ℝ → ℂ) = (fun x => Complex.ofRealCLM x + { re := 0, im := b }) := by
|
||||
funext x
|
||||
apply Complex.ext
|
||||
rw [Complex.add_re]
|
||||
simp
|
||||
rw [Complex.add_im]
|
||||
simp
|
||||
rw [this]
|
||||
continuity
|
||||
|
||||
|
||||
lemma integrability₂
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
(f : ℂ → E)
|
||||
(hf : Differentiable ℂ f)
|
||||
(a₁ a₂ b : ℝ) :
|
||||
IntervalIntegrable (fun x => f { re := b, im := x }) MeasureTheory.volume a₁ a₂ := by
|
||||
apply Continuous.intervalIntegrable
|
||||
apply Continuous.comp
|
||||
exact Differentiable.continuous hf
|
||||
have : (Complex.mk b) = (fun x => Complex.I • Complex.ofRealCLM x + { re := b, im := 0 }) := by
|
||||
funext x
|
||||
apply Complex.ext
|
||||
rw [Complex.add_re]
|
||||
simp
|
||||
simp
|
||||
rw [this]
|
||||
apply Continuous.add
|
||||
continuity
|
||||
fun_prop
|
||||
|
||||
|
||||
theorem primitive_additivity
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
(f : ℂ → E)
|
||||
(hf : Differentiable ℂ f)
|
||||
(z₀ z₁ : ℂ) :
|
||||
primitive z₀ f = fun z ↦ (primitive z₁ f) z + (primitive z₀ f z₁) := by
|
||||
funext z
|
||||
unfold primitive
|
||||
|
||||
have : (∫ (x : ℝ) in z₀.re..z.re, f { re := x, im := z₀.im }) = (∫ (x : ℝ) in z₀.re..z₁.re, f { re := x, im := z₀.im }) + (∫ (x : ℝ) in z₁.re..z.re, f { re := x, im := z₀.im }) := by
|
||||
rw [intervalIntegral.integral_add_adjacent_intervals]
|
||||
apply integrability₁ f hf
|
||||
apply integrability₁ f hf
|
||||
rw [this]
|
||||
|
||||
have : (∫ (x : ℝ) in z₀.im..z.im, f { re := z.re, im := x }) = (∫ (x : ℝ) in z₀.im..z₁.im, f { re := z.re, im := x }) + (∫ (x : ℝ) in z₁.im..z.im, f { re := z.re, im := x }) := by
|
||||
rw [intervalIntegral.integral_add_adjacent_intervals]
|
||||
apply integrability₂ f hf
|
||||
apply integrability₂ f hf
|
||||
rw [this]
|
||||
simp
|
||||
|
||||
let A := integral_divergence₅ f hf ⟨z₁.re, z₀.im⟩ ⟨z.re, z₁.im⟩
|
||||
simp at A
|
||||
|
||||
have {a b c d : E} : (b + a) + (c + d) = (a + c) + (b + d) := by
|
||||
abel
|
||||
rw [this]
|
||||
rw [A]
|
||||
abel
|
||||
|
||||
|
||||
theorem primitive_hasDerivAt
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
{f : ℂ → E}
|
||||
(hf : Differentiable ℂ f)
|
||||
(z₀ z : ℂ) :
|
||||
HasDerivAt (primitive z₀ f) (f z) z := by
|
||||
rw [primitive_additivity f hf z₀ z]
|
||||
rw [← add_zero (f z)]
|
||||
apply HasDerivAt.add
|
||||
apply primitive_hasDerivAtBasepoint
|
||||
exact hf.continuous
|
||||
apply hasDerivAt_const
|
||||
|
||||
|
||||
theorem primitive_differentiable
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
{f : ℂ → E}
|
||||
(hf : Differentiable ℂ f)
|
||||
(z₀ : ℂ) :
|
||||
Differentiable ℂ (primitive z₀ f) := by
|
||||
intro z
|
||||
exact (primitive_hasDerivAt hf z₀ z).differentiableAt
|
||||
|
||||
|
||||
theorem primitive_hasFderivAt
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
{f : ℂ → E}
|
||||
(hf : Differentiable ℂ f)
|
||||
(z₀ : ℂ) :
|
||||
∀ z, HasFDerivAt (primitive z₀ f) ((ContinuousLinearMap.lsmul ℂ ℂ).flip (f z)) z := by
|
||||
intro z
|
||||
rw [hasFDerivAt_iff_hasDerivAt]
|
||||
simp
|
||||
exact primitive_hasDerivAt hf z₀ z
|
||||
|
||||
|
||||
theorem primitive_hasFderivAt'
|
||||
{f : ℂ → ℂ}
|
||||
(hf : Differentiable ℂ f)
|
||||
(z₀ : ℂ) :
|
||||
∀ z, HasFDerivAt (primitive z₀ f) (ContinuousLinearMap.lsmul ℂ ℂ (f z)) z := by
|
||||
intro z
|
||||
rw [hasFDerivAt_iff_hasDerivAt]
|
||||
simp
|
||||
exact primitive_hasDerivAt hf z₀ z
|
||||
|
||||
|
||||
theorem primitive_fderiv
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
{f : ℂ → E}
|
||||
(hf : Differentiable ℂ f)
|
||||
(z₀ : ℂ) :
|
||||
∀ z, (fderiv ℂ (primitive z₀ f) z) = (ContinuousLinearMap.lsmul ℂ ℂ).flip (f z) := by
|
||||
intro z
|
||||
apply HasFDerivAt.fderiv
|
||||
exact primitive_hasFderivAt hf z₀ z
|
||||
|
||||
|
||||
theorem primitive_fderiv'
|
||||
{f : ℂ → ℂ}
|
||||
(hf : Differentiable ℂ f)
|
||||
(z₀ : ℂ) :
|
||||
∀ z, (fderiv ℂ (primitive z₀ f) z) = ContinuousLinearMap.lsmul ℂ ℂ (f z) := by
|
||||
intro z
|
||||
apply HasFDerivAt.fderiv
|
||||
exact primitive_hasFderivAt' hf z₀ z
|
||||
|
||||
@ -1,285 +0,0 @@
|
||||
import Mathlib.Analysis.InnerProductSpace.Basic
|
||||
import Mathlib.Analysis.InnerProductSpace.Dual
|
||||
import Mathlib.Analysis.InnerProductSpace.PiL2
|
||||
--import Mathlib.Algebra.BigOperators.Basic
|
||||
import Mathlib.Analysis.Calculus.ContDiff.Bounds
|
||||
import Mathlib.Analysis.Calculus.FDeriv.Symmetric
|
||||
--import Mathlib.LinearAlgebra.Basis
|
||||
--import Mathlib.LinearAlgebra.Contraction
|
||||
|
||||
open BigOperators
|
||||
open Finset
|
||||
|
||||
|
||||
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
|
||||
|
||||
noncomputable def realInnerAsElementOfDualTensorprod
|
||||
{E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
|
||||
: TensorProduct ℝ E E →ₗ[ℝ] ℝ := TensorProduct.lift bilinFormOfRealInner
|
||||
|
||||
|
||||
instance
|
||||
{E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] [FiniteDimensional ℝ E]
|
||||
: NormedAddCommGroup (TensorProduct ℝ E E) where
|
||||
norm := by
|
||||
|
||||
sorry
|
||||
dist_self := by
|
||||
|
||||
sorry
|
||||
|
||||
sorry
|
||||
|
||||
/-
|
||||
instance
|
||||
{E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] [FiniteDimensional ℝ E]
|
||||
: InnerProductSpace ℝ (TensorProduct ℝ E E) where
|
||||
smul := _
|
||||
one_smul := _
|
||||
mul_smul := _
|
||||
smul_zero := _
|
||||
smul_add := _
|
||||
add_smul := _
|
||||
zero_smul := _
|
||||
norm_smul_le := _
|
||||
inner := _
|
||||
norm_sq_eq_inner := _
|
||||
conj_symm := _
|
||||
add_left := _
|
||||
smul_left := _
|
||||
-/
|
||||
|
||||
|
||||
|
||||
noncomputable def dual'
|
||||
{E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] [FiniteDimensional ℝ E]
|
||||
: (TensorProduct ℝ E E →ₗ[ℝ] ℝ) ≃ₗ[ℝ] TensorProduct ℝ E E := by
|
||||
|
||||
let d := InnerProductSpace.toDual ℝ E
|
||||
|
||||
|
||||
let e := d.toLinearEquiv
|
||||
|
||||
let a := TensorProduct.congr e e
|
||||
|
||||
let b := homTensorHomEquiv ℝ E E ℝ ℝ
|
||||
|
||||
sorry
|
||||
|
||||
|
||||
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
|
||||
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
|
||||
|
||||
|
||||
theorem InvariantTensor
|
||||
[Fintype ι]
|
||||
(v : Basis ι ℝ E)
|
||||
(c : ι → ℝ)
|
||||
(L : ContinuousMultilinearMap ℝ (fun (_ : Fin 2) ↦ E) F) :
|
||||
L (fun _ => ∑ j : ι, c j • v j) = ∑ x : Fin 2 → ι, (c (x 0) * c (x 1)) • L ((fun i => v (x i))) := by
|
||||
|
||||
rw [L.map_sum]
|
||||
simp_rw [L.map_smul_univ]
|
||||
sorry
|
||||
|
||||
|
||||
|
||||
theorem BilinearCalc
|
||||
[Fintype ι]
|
||||
(v : Basis ι ℝ E)
|
||||
(c : ι → ℝ)
|
||||
(L : ContinuousMultilinearMap ℝ (fun (_ : Fin 2) ↦ E) F) :
|
||||
L (fun _ => ∑ j : ι, c j • v j) = ∑ x : Fin 2 → ι, (c (x 0) * c (x 1)) • L ((fun i => v (x i))) := by
|
||||
|
||||
rw [L.map_sum]
|
||||
simp_rw [L.map_smul_univ]
|
||||
sorry
|
||||
|
||||
/-
|
||||
lemma c2
|
||||
[Fintype ι]
|
||||
(b : Basis ι ℝ E)
|
||||
(hb : Orthonormal ℝ b)
|
||||
(x y : E) :
|
||||
⟪x, y⟫_ℝ = ∑ i : ι, ⟪x, b i⟫_ℝ * ⟪y, b i⟫_ℝ := by
|
||||
rw [vectorPresentation b hb x]
|
||||
rw [vectorPresentation b hb y]
|
||||
rw [Orthonormal.inner_sum hb]
|
||||
simp
|
||||
conv =>
|
||||
right
|
||||
arg 2
|
||||
intro i'
|
||||
rw [Orthonormal.inner_left_fintype hb]
|
||||
rw [Orthonormal.inner_left_fintype hb]
|
||||
-/
|
||||
|
||||
lemma fin_sum
|
||||
[Fintype ι]
|
||||
(f : ι → ι → F) :
|
||||
∑ r : Fin 2 → ι, f (r 0) (r 1) = ∑ r₀ : ι, (∑ r₁ : ι, f r₀ r₁) := by
|
||||
|
||||
rw [← Fintype.sum_prod_type']
|
||||
apply Fintype.sum_equiv (finTwoArrowEquiv ι)
|
||||
intro x
|
||||
dsimp
|
||||
|
||||
|
||||
theorem LaplaceIndep
|
||||
[Fintype ι] [DecidableEq ι]
|
||||
(v₁ : OrthonormalBasis ι ℝ E)
|
||||
(v₂ : OrthonormalBasis ι ℝ E)
|
||||
(L : ContinuousMultilinearMap ℝ (fun (_ : Fin 2) ↦ E) F) :
|
||||
∑ i, L (fun _ ↦ v₁ i) = ∑ i, L (fun _ => v₂ i) := by
|
||||
|
||||
have vector_vs_function
|
||||
{y : Fin 2 → ι}
|
||||
{v : ι → E}
|
||||
: (fun i => v (y i)) = ![v (y 0), v (y 1)] := by
|
||||
funext i
|
||||
by_cases h : i = 0
|
||||
· rw [h]
|
||||
simp
|
||||
· rw [Fin.eq_one_of_neq_zero i h]
|
||||
simp
|
||||
|
||||
conv =>
|
||||
right
|
||||
arg 2
|
||||
intro i
|
||||
rw [v₁.sum_repr' (v₂ i)]
|
||||
rw [BilinearCalc]
|
||||
rw [Finset.sum_comm]
|
||||
conv =>
|
||||
right
|
||||
arg 2
|
||||
intro y
|
||||
rw [← Finset.sum_smul]
|
||||
rw [← c2 v₂ hv₂ (v₁ (y 0)) (v₁ (y 1))]
|
||||
rw [vector_vs_function]
|
||||
simp
|
||||
|
||||
rw [fin_sum (fun i₀ ↦ (fun i₁ ↦ ⟪v₁ i₀, v₁ i₁⟫_ℝ • L ![v₁ i₀, v₁ i₁]))]
|
||||
|
||||
have xx {r₀ : ι} : ∀ r₁ : ι, r₁ ≠ r₀ → ⟪v₁ r₀, v₁ r₁⟫_ℝ • L ![v₁ r₀, v₁ r₁] = 0 := by
|
||||
intro r₁ hr₁
|
||||
rw [orthonormal_iff_ite.1 hv₁]
|
||||
simp
|
||||
tauto
|
||||
|
||||
conv =>
|
||||
right
|
||||
arg 2
|
||||
intro r₀
|
||||
rw [Fintype.sum_eq_single r₀ xx]
|
||||
rw [orthonormal_iff_ite.1 hv₁]
|
||||
apply sum_congr
|
||||
rfl
|
||||
intro x _
|
||||
rw [vector_vs_function]
|
||||
simp
|
||||
|
||||
|
||||
noncomputable def Laplace_wrt_basis
|
||||
[Fintype ι]
|
||||
(v : Basis ι ℝ E)
|
||||
(_ : Orthonormal ℝ v)
|
||||
(f : E → F) :
|
||||
E → F :=
|
||||
fun z ↦ ∑ i, iteratedFDeriv ℝ 2 f z ![v i, v i]
|
||||
|
||||
|
||||
theorem LaplaceIndep'
|
||||
[Fintype ι] [DecidableEq ι]
|
||||
(v₁ : OrthonormalBasis ι ℝ E)
|
||||
(v₂ : OrthonormalBasis ι ℝ E)
|
||||
(f : E → F)
|
||||
: (Laplace_wrt_basis v₁ hv₁ f) = (Laplace_wrt_basis v₂ hv₂ f) := by
|
||||
|
||||
funext z
|
||||
unfold Laplace_wrt_basis
|
||||
let XX := LaplaceIndep v₁ hv₁ v₂ hv₂ (iteratedFDeriv ℝ 2 f z)
|
||||
have vector_vs_function
|
||||
{v : E}
|
||||
: ![v, v] = (fun _ => v) := by
|
||||
funext i
|
||||
by_cases h : i = 0
|
||||
· rw [h]
|
||||
simp
|
||||
· rw [Fin.eq_one_of_neq_zero i h]
|
||||
simp
|
||||
conv =>
|
||||
left
|
||||
arg 2
|
||||
intro i
|
||||
rw [vector_vs_function]
|
||||
conv =>
|
||||
right
|
||||
arg 2
|
||||
intro i
|
||||
rw [vector_vs_function]
|
||||
assumption
|
||||
|
||||
|
||||
theorem LaplaceIndep''
|
||||
[Fintype ι₁] [DecidableEq ι₁]
|
||||
(v₁ : Basis ι₁ ℝ E)
|
||||
(hv₁ : Orthonormal ℝ v₁)
|
||||
[Fintype ι₂] [DecidableEq ι₂]
|
||||
(v₂ : Basis ι₂ ℝ E)
|
||||
(hv₂ : Orthonormal ℝ v₂)
|
||||
(f : E → F)
|
||||
: (Laplace_wrt_basis v₁ hv₁ f) = (Laplace_wrt_basis v₂ hv₂ f) := by
|
||||
|
||||
have b : ι₁ ≃ ι₂ := by
|
||||
apply Fintype.equivOfCardEq
|
||||
rw [← FiniteDimensional.finrank_eq_card_basis v₁]
|
||||
rw [← FiniteDimensional.finrank_eq_card_basis v₂]
|
||||
|
||||
let v'₁ := Basis.reindex v₁ b
|
||||
have hv'₁ : Orthonormal ℝ v'₁ := by
|
||||
let A := Basis.reindex_apply v₁ b
|
||||
have : ⇑v'₁ = v₁ ∘ b.symm := by
|
||||
funext i
|
||||
exact A i
|
||||
rw [this]
|
||||
let B := Orthonormal.comp hv₁ b.symm
|
||||
apply B
|
||||
exact Equiv.injective b.symm
|
||||
rw [← LaplaceIndep' v'₁ hv'₁ v₂ hv₂ f]
|
||||
|
||||
unfold Laplace_wrt_basis
|
||||
simp
|
||||
funext z
|
||||
|
||||
rw [← Equiv.sum_comp b.symm]
|
||||
apply Fintype.sum_congr
|
||||
intro i₂
|
||||
congr
|
||||
rw [Basis.reindex_apply v₁ b i₂]
|
||||
|
||||
|
||||
noncomputable def Laplace
|
||||
(f : E → F)
|
||||
: E → F := by
|
||||
exact Laplace_wrt_basis (stdOrthonormalBasis ℝ E).toBasis (stdOrthonormalBasis ℝ E).orthonormal f
|
||||
|
||||
|
||||
theorem LaplaceIndep'''
|
||||
[Fintype ι] [DecidableEq ι]
|
||||
(v : Basis ι ℝ E)
|
||||
(hv : Orthonormal ℝ v)
|
||||
(f : E → F)
|
||||
: (Laplace f) = (Laplace_wrt_basis v hv f) := by
|
||||
|
||||
unfold Laplace
|
||||
apply LaplaceIndep'' (stdOrthonormalBasis ℝ E).toBasis (stdOrthonormalBasis ℝ E).orthonormal v hv f
|
||||
|
||||
|
||||
theorem Complex.Laplace'
|
||||
(f : ℂ → F)
|
||||
: (Laplace f) = fun z ↦ (iteratedFDeriv ℝ 2 f z) ![1, 1] + (iteratedFDeriv ℝ 2 f z) ![Complex.I, Complex.I] := by
|
||||
|
||||
rw [LaplaceIndep''' Complex.orthonormalBasisOneI.toBasis Complex.orthonormalBasisOneI.orthonormal f]
|
||||
unfold Laplace_wrt_basis
|
||||
simp
|
||||
@ -1,528 +0,0 @@
|
||||
import Mathlib.Analysis.Complex.CauchyIntegral
|
||||
import Mathlib.Analysis.Analytic.IsolatedZeros
|
||||
import Nevanlinna.analyticOnNhd_zeroSet
|
||||
import Nevanlinna.harmonicAt_examples
|
||||
import Nevanlinna.harmonicAt_meanValue
|
||||
import Nevanlinna.specialFunctions_CircleIntegral_affine
|
||||
import Nevanlinna.stronglyMeromorphicOn
|
||||
import Nevanlinna.stronglyMeromorphicOn_eliminate
|
||||
import Nevanlinna.meromorphicOn_divisor
|
||||
|
||||
open Real
|
||||
|
||||
|
||||
|
||||
theorem jensen_case_R_eq_one'
|
||||
(f : ℂ → ℂ)
|
||||
(h₁f : StronglyMeromorphicOn f (Metric.closedBall 0 1))
|
||||
(h₂f : f 0 ≠ 0) :
|
||||
log ‖f 0‖ = -∑ᶠ s, (h₁f.meromorphicOn.divisor s) * log (‖s‖⁻¹) + (2 * π)⁻¹ * ∫ (x : ℝ) in (0)..(2 * π), log ‖f (circleMap 0 1 x)‖ := by
|
||||
|
||||
have h₁U : IsConnected (Metric.closedBall (0 : ℂ) 1) := by
|
||||
constructor
|
||||
· apply Metric.nonempty_closedBall.mpr (by simp)
|
||||
· exact (convex_closedBall (0 : ℂ) 1).isPreconnected
|
||||
|
||||
have h₂U : IsCompact (Metric.closedBall (0 : ℂ) 1) :=
|
||||
isCompact_closedBall 0 1
|
||||
|
||||
have h'₂f : ∃ u : (Metric.closedBall (0 : ℂ) 1), f u ≠ 0 := by
|
||||
use ⟨0, Metric.mem_closedBall_self (by simp)⟩
|
||||
|
||||
have h₃f : Set.Finite (Function.support h₁f.meromorphicOn.divisor) := by
|
||||
exact Divisor.finiteSupport h₂U (StronglyMeromorphicOn.meromorphicOn h₁f).divisor
|
||||
|
||||
have h₄f: Function.support (fun s ↦ (h₁f.meromorphicOn.divisor s) * log (‖s‖⁻¹)) ⊆ h₃f.toFinset := by
|
||||
intro x
|
||||
contrapose
|
||||
simp
|
||||
intro hx
|
||||
rw [hx]
|
||||
simp
|
||||
rw [finsum_eq_sum_of_support_subset _ h₄f]
|
||||
|
||||
obtain ⟨F, h₁F, h₂F, h₃F, h₄F⟩ := MeromorphicOn.decompose₃' h₂U h₁U h₁f h'₂f
|
||||
|
||||
have h₁F : Function.mulSupport (fun u ↦ fun z => (z - u) ^ (h₁f.meromorphicOn.divisor u)) ⊆ h₃f.toFinset := by
|
||||
intro u
|
||||
contrapose
|
||||
simp
|
||||
intro hu
|
||||
rw [hu]
|
||||
simp
|
||||
exact rfl
|
||||
rw [finprod_eq_prod_of_mulSupport_subset _ h₁F] at h₄F
|
||||
|
||||
let G := fun z ↦ log ‖F z‖ + ∑ᶠ s, (h₁f.meromorphicOn.divisor s) * log ‖z - s‖
|
||||
|
||||
have h₁G {z : ℂ} : Function.support (fun s ↦ (h₁f.meromorphicOn.divisor s) * log ‖z - s‖) ⊆ h₃f.toFinset := by
|
||||
intro s
|
||||
contrapose
|
||||
simp
|
||||
intro hs
|
||||
rw [hs]
|
||||
simp
|
||||
|
||||
have decompose_f : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, f z ≠ 0 → log ‖f z‖ = G z := by
|
||||
intro z h₁z h₂z
|
||||
|
||||
rw [h₄F]
|
||||
simp only [Pi.mul_apply, norm_mul]
|
||||
simp only [Finset.prod_apply, norm_prod, norm_zpow]
|
||||
rw [Real.log_mul]
|
||||
rw [Real.log_prod]
|
||||
simp_rw [Real.log_zpow]
|
||||
dsimp only [G]
|
||||
rw [finsum_eq_sum_of_support_subset _ h₁G]
|
||||
--
|
||||
intro x hx
|
||||
have : z ≠ x := by
|
||||
by_contra hCon
|
||||
rw [← hCon] at hx
|
||||
simp at hx
|
||||
rw [← StronglyMeromorphicAt.order_eq_zero_iff] at h₂z
|
||||
unfold MeromorphicOn.divisor at hx
|
||||
simp [h₁z] at hx
|
||||
tauto
|
||||
apply zpow_ne_zero
|
||||
simpa
|
||||
-- Complex.abs (F z) ≠ 0
|
||||
simp
|
||||
exact h₃F ⟨z, h₁z⟩
|
||||
--
|
||||
rw [Finset.prod_ne_zero_iff]
|
||||
intro x hx
|
||||
have : z ≠ x := by
|
||||
by_contra hCon
|
||||
rw [← hCon] at hx
|
||||
simp at hx
|
||||
rw [← StronglyMeromorphicAt.order_eq_zero_iff] at h₂z
|
||||
unfold MeromorphicOn.divisor at hx
|
||||
simp [h₁z] at hx
|
||||
tauto
|
||||
apply zpow_ne_zero
|
||||
simpa
|
||||
|
||||
|
||||
have int_logAbs_f_eq_int_G : ∫ (x : ℝ) in (0)..2 * π, log ‖f (circleMap 0 1 x)‖ = ∫ (x : ℝ) in (0)..2 * π, G (circleMap 0 1 x) := by
|
||||
|
||||
rw [intervalIntegral.integral_congr_ae]
|
||||
rw [MeasureTheory.ae_iff]
|
||||
apply Set.Countable.measure_zero
|
||||
simp
|
||||
|
||||
have t₀ : {a | a ∈ Ι 0 (2 * π) ∧ ¬log ‖f (circleMap 0 1 a)‖ = G (circleMap 0 1 a)}
|
||||
⊆ (circleMap 0 1)⁻¹' (h₃f.toFinset) := by
|
||||
intro a ha
|
||||
simp at ha
|
||||
simp
|
||||
by_contra C
|
||||
have t₀ : (circleMap 0 1 a) ∈ Metric.closedBall 0 1 :=
|
||||
circleMap_mem_closedBall 0 (zero_le_one' ℝ) a
|
||||
have t₁ : f (circleMap 0 1 a) ≠ 0 := by
|
||||
let A := h₁f (circleMap 0 1 a) t₀
|
||||
rw [← A.order_eq_zero_iff]
|
||||
unfold MeromorphicOn.divisor at C
|
||||
simp [t₀] at C
|
||||
rcases C with C₁|C₂
|
||||
· assumption
|
||||
· let B := h₁f.meromorphicOn.order_ne_top' h₁U
|
||||
let C := fun q ↦ B q ⟨(circleMap 0 1 a), t₀⟩
|
||||
rw [C₂] at C
|
||||
have : ∃ u : (Metric.closedBall (0 : ℂ) 1), (h₁f u u.2).meromorphicAt.order ≠ ⊤ := by
|
||||
use ⟨(0 : ℂ), (by simp)⟩
|
||||
let H := h₁f 0 (by simp)
|
||||
let K := H.order_eq_zero_iff.2 h₂f
|
||||
rw [K]
|
||||
simp
|
||||
let D := C this
|
||||
tauto
|
||||
exact ha.2 (decompose_f (circleMap 0 1 a) t₀ t₁)
|
||||
|
||||
apply Set.Countable.mono t₀
|
||||
apply Set.Countable.preimage_circleMap
|
||||
apply Set.Finite.countable
|
||||
exact Finset.finite_toSet h₃f.toFinset
|
||||
--
|
||||
simp
|
||||
|
||||
|
||||
have decompose_int_G : ∫ (x : ℝ) in (0)..2 * π, G (circleMap 0 1 x)
|
||||
= (∫ (x : ℝ) in (0)..2 * π, log (Complex.abs (F (circleMap 0 1 x))))
|
||||
+ ∑ᶠ x, (h₁f.meromorphicOn.divisor x) * ∫ (x_1 : ℝ) in (0)..2 * π, log (Complex.abs (circleMap 0 1 x_1 - ↑x)) := by
|
||||
dsimp [G]
|
||||
|
||||
rw [intervalIntegral.integral_add]
|
||||
congr
|
||||
have t₀ {x : ℝ} : Function.support (fun s ↦ (h₁f.meromorphicOn.divisor s) * log (Complex.abs (circleMap 0 1 x - s))) ⊆ h₃f.toFinset := by
|
||||
intro s hs
|
||||
simp at hs
|
||||
simp [hs.1]
|
||||
conv =>
|
||||
left
|
||||
arg 1
|
||||
intro x
|
||||
rw [finsum_eq_sum_of_support_subset _ t₀]
|
||||
rw [intervalIntegral.integral_finset_sum]
|
||||
let G' := fun x ↦ ((h₁f.meromorphicOn.divisor x) : ℂ) * ∫ (x_1 : ℝ) in (0)..2 * π, log (Complex.abs (circleMap 0 1 x_1 - x))
|
||||
have t₁ : (Function.support fun x ↦ (h₁f.meromorphicOn.divisor x) * ∫ (x_1 : ℝ) in (0)..2 * π, log (Complex.abs (circleMap 0 1 x_1 - x))) ⊆ h₃f.toFinset := by
|
||||
simp
|
||||
intro s
|
||||
contrapose!
|
||||
simp
|
||||
tauto
|
||||
conv =>
|
||||
right
|
||||
rw [finsum_eq_sum_of_support_subset _ t₁]
|
||||
simp
|
||||
|
||||
-- ∀ i ∈ (finiteZeros h₁U h₂U h'₁f h'₂f).toFinset,
|
||||
-- IntervalIntegrable (fun x => (h'₁f.order i).toNat *
|
||||
-- log (Complex.abs (circleMap 0 1 x - ↑i))) MeasureTheory.volume 0 (2 * π)
|
||||
intro i _
|
||||
apply IntervalIntegrable.const_mul
|
||||
--simp at this
|
||||
by_cases h₂i : ‖i‖ = 1
|
||||
-- case pos
|
||||
exact int'₂ h₂i
|
||||
-- case neg
|
||||
apply Continuous.intervalIntegrable
|
||||
apply continuous_iff_continuousAt.2
|
||||
intro x
|
||||
have : (fun x => log (Complex.abs (circleMap 0 1 x - ↑i))) = log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 1 x - ↑i) :=
|
||||
rfl
|
||||
rw [this]
|
||||
apply ContinuousAt.comp
|
||||
apply Real.continuousAt_log
|
||||
simp
|
||||
|
||||
by_contra ha'
|
||||
conv at h₂i =>
|
||||
arg 1
|
||||
rw [← ha']
|
||||
rw [Complex.norm_eq_abs]
|
||||
rw [abs_circleMap_zero 1 x]
|
||||
simp
|
||||
tauto
|
||||
apply ContinuousAt.comp
|
||||
apply Complex.continuous_abs.continuousAt
|
||||
fun_prop
|
||||
-- IntervalIntegrable (fun x => log (Complex.abs (F (circleMap 0 1 x)))) MeasureTheory.volume 0 (2 * π)
|
||||
apply Continuous.intervalIntegrable
|
||||
apply continuous_iff_continuousAt.2
|
||||
intro x
|
||||
have : (fun x => log (Complex.abs (F (circleMap 0 1 x)))) = log ∘ Complex.abs ∘ F ∘ (fun x ↦ circleMap 0 1 x) :=
|
||||
rfl
|
||||
rw [this]
|
||||
apply ContinuousAt.comp
|
||||
apply Real.continuousAt_log
|
||||
simp
|
||||
exact h₃F ⟨(circleMap 0 1 x), (by simp)⟩
|
||||
|
||||
-- ContinuousAt (⇑Complex.abs ∘ F ∘ fun x => circleMap 0 1 x) x
|
||||
apply ContinuousAt.comp
|
||||
apply Complex.continuous_abs.continuousAt
|
||||
apply ContinuousAt.comp
|
||||
apply DifferentiableAt.continuousAt (𝕜 := ℂ)
|
||||
apply AnalyticAt.differentiableAt
|
||||
exact h₂F (circleMap 0 1 x) (by simp)
|
||||
-- ContinuousAt (fun x => circleMap 0 1 x) x
|
||||
apply Continuous.continuousAt
|
||||
apply continuous_circleMap
|
||||
-- IntervalIntegrable (fun x => ∑ᶠ (s : ℂ), ↑(↑⋯.divisor s) * log (Complex.abs (circleMap 0 1 x - s))) MeasureTheory.volume 0 (2 * π)
|
||||
--simp? at h₁G
|
||||
have h₁G' {z : ℂ} : (Function.support fun s => (h₁f.meromorphicOn.divisor s) * log (Complex.abs (z - s))) ⊆ ↑h₃f.toFinset := by
|
||||
exact h₁G
|
||||
conv =>
|
||||
arg 1
|
||||
intro z
|
||||
rw [finsum_eq_sum_of_support_subset _ h₁G']
|
||||
conv =>
|
||||
arg 1
|
||||
rw [← Finset.sum_fn]
|
||||
apply IntervalIntegrable.sum
|
||||
intro i _
|
||||
apply IntervalIntegrable.const_mul
|
||||
--have : i.1 ∈ Metric.closedBall (0 : ℂ) 1 := i.2
|
||||
--simp at this
|
||||
by_cases h₂i : ‖i‖ = 1
|
||||
-- case pos
|
||||
exact int'₂ h₂i
|
||||
-- case neg
|
||||
--have : i.1 ∈ Metric.ball (0 : ℂ) 1 := by sorry
|
||||
apply Continuous.intervalIntegrable
|
||||
apply continuous_iff_continuousAt.2
|
||||
intro x
|
||||
have : (fun x => log (Complex.abs (circleMap 0 1 x - ↑i))) = log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 1 x - ↑i) :=
|
||||
rfl
|
||||
rw [this]
|
||||
apply ContinuousAt.comp
|
||||
apply Real.continuousAt_log
|
||||
simp
|
||||
|
||||
by_contra ha'
|
||||
conv at h₂i =>
|
||||
arg 1
|
||||
rw [← ha']
|
||||
rw [Complex.norm_eq_abs]
|
||||
rw [abs_circleMap_zero 1 x]
|
||||
simp
|
||||
tauto
|
||||
apply ContinuousAt.comp
|
||||
apply Complex.continuous_abs.continuousAt
|
||||
fun_prop
|
||||
|
||||
have t₁ : (∫ (x : ℝ) in (0)..2 * Real.pi, log ‖F (circleMap 0 1 x)‖) = 2 * Real.pi * log ‖F 0‖ := by
|
||||
let logAbsF := fun w ↦ Real.log ‖F w‖
|
||||
have t₀ : ∀ z ∈ Metric.closedBall 0 1, HarmonicAt logAbsF z := by
|
||||
intro z hz
|
||||
apply logabs_of_holomorphicAt_is_harmonic
|
||||
exact AnalyticAt.holomorphicAt (h₂F z hz)
|
||||
exact h₃F ⟨z, hz⟩
|
||||
|
||||
apply harmonic_meanValue₁ 1 Real.zero_lt_one t₀
|
||||
|
||||
simp_rw [← Complex.norm_eq_abs] at decompose_int_G
|
||||
rw [t₁] at decompose_int_G
|
||||
|
||||
|
||||
have h₁G' : (Function.support fun s => (h₁f.meromorphicOn.divisor s) * ∫ (x_1 : ℝ) in (0)..(2 * π), log ‖circleMap 0 1 x_1 - s‖) ⊆ ↑h₃f.toFinset := by
|
||||
intro s hs
|
||||
simp at hs
|
||||
simp [hs.1]
|
||||
rw [finsum_eq_sum_of_support_subset _ h₁G'] at decompose_int_G
|
||||
have : ∑ s ∈ h₃f.toFinset, (h₁f.meromorphicOn.divisor s) * ∫ (x_1 : ℝ) in (0)..(2 * π), log ‖circleMap 0 1 x_1 - s‖ = 0 := by
|
||||
apply Finset.sum_eq_zero
|
||||
intro x hx
|
||||
rw [int₃ _]
|
||||
simp
|
||||
simp at hx
|
||||
let ZZ := h₁f.meromorphicOn.divisor.supportInU
|
||||
simp at ZZ
|
||||
let UU := ZZ x hx
|
||||
simpa
|
||||
rw [this] at decompose_int_G
|
||||
|
||||
|
||||
simp at decompose_int_G
|
||||
|
||||
rw [int_logAbs_f_eq_int_G]
|
||||
rw [decompose_int_G]
|
||||
let X := h₄F
|
||||
nth_rw 1 [h₄F]
|
||||
simp
|
||||
have {l : ℝ} : π⁻¹ * 2⁻¹ * (2 * π * l) = l := by
|
||||
calc π⁻¹ * 2⁻¹ * (2 * π * l)
|
||||
_ = π⁻¹ * (2⁻¹ * 2) * π * l := by ring
|
||||
_ = π⁻¹ * π * l := by ring
|
||||
_ = (π⁻¹ * π) * l := by ring
|
||||
_ = 1 * l := by
|
||||
rw [inv_mul_cancel₀]
|
||||
exact pi_ne_zero
|
||||
_ = l := by simp
|
||||
rw [this]
|
||||
rw [log_mul]
|
||||
rw [log_prod]
|
||||
simp
|
||||
rw [add_comm]
|
||||
--
|
||||
intro x hx
|
||||
simp at hx
|
||||
rw [zpow_ne_zero_iff]
|
||||
by_contra hCon
|
||||
simp at hCon
|
||||
rw [← (h₁f 0 (by simp)).order_eq_zero_iff] at h₂f
|
||||
rw [hCon] at hx
|
||||
unfold MeromorphicOn.divisor at hx
|
||||
simp at hx
|
||||
rw [h₂f] at hx
|
||||
tauto
|
||||
assumption
|
||||
--
|
||||
simp
|
||||
|
||||
by_contra hCon
|
||||
nth_rw 1 [h₄F] at h₂f
|
||||
simp at h₂f
|
||||
tauto
|
||||
--
|
||||
rw [Finset.prod_ne_zero_iff]
|
||||
intro x hx
|
||||
simp at hx
|
||||
rw [zpow_ne_zero_iff]
|
||||
by_contra hCon
|
||||
simp at hCon
|
||||
rw [← (h₁f 0 (by simp)).order_eq_zero_iff] at h₂f
|
||||
rw [hCon] at hx
|
||||
unfold MeromorphicOn.divisor at hx
|
||||
simp at hx
|
||||
rw [h₂f] at hx
|
||||
tauto
|
||||
assumption
|
||||
|
||||
|
||||
|
||||
lemma const_mul_circleMap_zero'
|
||||
{R θ : ℝ} :
|
||||
circleMap 0 R θ = R * circleMap 0 1 θ := by
|
||||
rw [circleMap_zero, circleMap_zero]
|
||||
simp
|
||||
|
||||
|
||||
|
||||
theorem jensen'
|
||||
{R : ℝ}
|
||||
(hR : 0 < R)
|
||||
(f : ℂ → ℂ)
|
||||
(h₁f : StronglyMeromorphicOn f (Metric.closedBall 0 R))
|
||||
(h₂f : f 0 ≠ 0) :
|
||||
log ‖f 0‖ = -∑ᶠ s, (h₁f.meromorphicOn.divisor s) * log (R * ‖s‖⁻¹) + (2 * π)⁻¹ * ∫ (x : ℝ) in (0)..(2 * π), log ‖f (circleMap 0 R x)‖ := by
|
||||
|
||||
|
||||
let ℓ : ℂ ≃L[ℂ] ℂ :=
|
||||
{
|
||||
toFun := fun x ↦ R * x
|
||||
map_add' := fun x y => DistribSMul.smul_add R x y
|
||||
map_smul' := fun m x => mul_smul_comm m (↑R) x
|
||||
invFun := fun x ↦ R⁻¹ * x
|
||||
left_inv := by
|
||||
intro x
|
||||
simp
|
||||
rw [← mul_assoc, mul_comm, inv_mul_cancel₀, mul_one]
|
||||
simp
|
||||
exact ne_of_gt hR
|
||||
right_inv := by
|
||||
intro x
|
||||
simp
|
||||
rw [← mul_assoc, mul_inv_cancel₀, one_mul]
|
||||
simp
|
||||
exact ne_of_gt hR
|
||||
continuous_toFun := continuous_const_smul R
|
||||
continuous_invFun := continuous_const_smul R⁻¹
|
||||
}
|
||||
|
||||
|
||||
let F := f ∘ ℓ
|
||||
|
||||
have h₁F : StronglyMeromorphicOn F (Metric.closedBall 0 1) := by
|
||||
sorry
|
||||
/-
|
||||
apply AnalyticOnNhd.comp (t := Metric.closedBall 0 R)
|
||||
exact h₁f
|
||||
intro x _
|
||||
apply ℓ.toContinuousLinearMap.analyticAt x
|
||||
|
||||
intro x hx
|
||||
have : ℓ x = R * x := by rfl
|
||||
rw [this]
|
||||
simp
|
||||
simp at hx
|
||||
rw [abs_of_pos hR]
|
||||
calc R * Complex.abs x
|
||||
_ ≤ R * 1 := by exact (mul_le_mul_iff_of_pos_left hR).mpr hx
|
||||
_ = R := by simp
|
||||
-/
|
||||
have h₂F : F 0 ≠ 0 := by
|
||||
dsimp [F]
|
||||
have : ℓ 0 = R * 0 := by rfl
|
||||
rw [this]
|
||||
simpa
|
||||
|
||||
let A := jensen_case_R_eq_one' F h₁F h₂F
|
||||
|
||||
dsimp [F] at A
|
||||
have {x : ℂ} : ℓ x = R * x := by rfl
|
||||
repeat
|
||||
simp_rw [this] at A
|
||||
simp at A
|
||||
simp
|
||||
rw [A]
|
||||
simp_rw [← const_mul_circleMap_zero']
|
||||
simp
|
||||
|
||||
let e : (Metric.closedBall (0 : ℂ) 1) → (Metric.closedBall (0 : ℂ) R) := by
|
||||
intro ⟨x, hx⟩
|
||||
have hy : R • x ∈ Metric.closedBall (0 : ℂ) R := by
|
||||
simp
|
||||
simp at hx
|
||||
have : R = |R| := by exact Eq.symm (abs_of_pos hR)
|
||||
rw [← this]
|
||||
norm_num
|
||||
calc R * Complex.abs x
|
||||
_ ≤ R * 1 := by exact (mul_le_mul_iff_of_pos_left hR).mpr hx
|
||||
_ = R := by simp
|
||||
exact ⟨R • x, hy⟩
|
||||
|
||||
let e' : (Metric.closedBall (0 : ℂ) R) → (Metric.closedBall (0 : ℂ) 1) := by
|
||||
intro ⟨x, hx⟩
|
||||
have hy : R⁻¹ • x ∈ Metric.closedBall (0 : ℂ) 1 := by
|
||||
simp
|
||||
simp at hx
|
||||
have : R = |R| := by exact Eq.symm (abs_of_pos hR)
|
||||
rw [← this]
|
||||
norm_num
|
||||
calc R⁻¹ * Complex.abs x
|
||||
_ ≤ R⁻¹ * R := by
|
||||
apply mul_le_mul_of_nonneg_left hx
|
||||
apply inv_nonneg.mpr
|
||||
exact abs_eq_self.mp (id (Eq.symm this))
|
||||
_ = 1 := by
|
||||
apply inv_mul_cancel₀
|
||||
exact Ne.symm (ne_of_lt hR)
|
||||
exact ⟨R⁻¹ • x, hy⟩
|
||||
|
||||
apply finsum_eq_of_bijective e
|
||||
|
||||
|
||||
apply Function.bijective_iff_has_inverse.mpr
|
||||
use e'
|
||||
constructor
|
||||
· apply Function.leftInverse_iff_comp.mpr
|
||||
funext x
|
||||
dsimp only [e, e', id_eq, eq_mp_eq_cast, Function.comp_apply]
|
||||
conv =>
|
||||
left
|
||||
arg 1
|
||||
rw [← smul_assoc, smul_eq_mul]
|
||||
rw [inv_mul_cancel₀ (Ne.symm (ne_of_lt hR))]
|
||||
rw [one_smul]
|
||||
· apply Function.rightInverse_iff_comp.mpr
|
||||
funext x
|
||||
dsimp only [e, e', id_eq, eq_mp_eq_cast, Function.comp_apply]
|
||||
conv =>
|
||||
left
|
||||
arg 1
|
||||
rw [← smul_assoc, smul_eq_mul]
|
||||
rw [mul_inv_cancel₀ (Ne.symm (ne_of_lt hR))]
|
||||
rw [one_smul]
|
||||
|
||||
intro x
|
||||
simp
|
||||
by_cases hx : x = (0 : ℂ)
|
||||
rw [hx]
|
||||
simp
|
||||
|
||||
rw [log_mul, log_mul, log_inv, log_inv]
|
||||
have : R = |R| := by exact Eq.symm (abs_of_pos hR)
|
||||
rw [← this]
|
||||
simp
|
||||
left
|
||||
congr 1
|
||||
|
||||
dsimp [AnalyticOnNhd.order]
|
||||
rw [← AnalyticAt.order_comp_CLE ℓ]
|
||||
|
||||
--
|
||||
simpa
|
||||
--
|
||||
have : R = |R| := by exact Eq.symm (abs_of_pos hR)
|
||||
rw [← this]
|
||||
apply inv_ne_zero
|
||||
exact Ne.symm (ne_of_lt hR)
|
||||
--
|
||||
exact Ne.symm (ne_of_lt hR)
|
||||
--
|
||||
simp
|
||||
constructor
|
||||
· assumption
|
||||
· exact Ne.symm (ne_of_lt hR)
|
||||
@ -1,65 +0,0 @@
|
||||
import Mathlib.Analysis.SpecialFunctions.Integrals
|
||||
import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
|
||||
import Mathlib.MeasureTheory.Integral.CircleIntegral
|
||||
import Mathlib.MeasureTheory.Measure.Restrict
|
||||
|
||||
open scoped Interval Topology
|
||||
open Real Filter MeasureTheory intervalIntegral
|
||||
|
||||
-- The following theorem was suggested by Gareth Ma on Zulip
|
||||
|
||||
|
||||
theorem logInt
|
||||
{t : ℝ}
|
||||
(ht : 0 < t) :
|
||||
∫ x in (0 : ℝ)..t, log x = t * log t - t := by
|
||||
rw [← integral_add_adjacent_intervals (b := 1)]
|
||||
trans (-1) + (t * log t - t + 1)
|
||||
· congr
|
||||
· -- ∫ x in 0..1, log x = -1, same proof as before
|
||||
rw [integral_eq_sub_of_hasDerivAt_of_tendsto (f := fun x ↦ x * log x - x) (fa := 0) (fb := -1)]
|
||||
· simp
|
||||
· simp
|
||||
· intro x hx
|
||||
norm_num at hx
|
||||
convert (hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x) using 1
|
||||
norm_num
|
||||
· rw [← neg_neg log]
|
||||
apply IntervalIntegrable.neg
|
||||
apply intervalIntegrable_deriv_of_nonneg (g := fun x ↦ -(x * log x - x))
|
||||
· exact (continuous_mul_log.continuousOn.sub continuous_id.continuousOn).neg
|
||||
· intro x hx
|
||||
norm_num at hx
|
||||
convert ((hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x)).neg using 1
|
||||
norm_num
|
||||
· intro x hx
|
||||
norm_num at hx
|
||||
rw [Pi.neg_apply, Left.nonneg_neg_iff]
|
||||
exact (log_nonpos_iff hx.left).mpr hx.right.le
|
||||
· have := tendsto_log_mul_rpow_nhds_zero zero_lt_one
|
||||
simp_rw [rpow_one, mul_comm] at this
|
||||
-- tendsto_nhdsWithin_of_tendsto_nhds should be under Tendsto namespace
|
||||
convert this.sub (tendsto_nhdsWithin_of_tendsto_nhds tendsto_id)
|
||||
norm_num
|
||||
· rw [(by simp : -1 = 1 * log 1 - 1)]
|
||||
apply tendsto_nhdsWithin_of_tendsto_nhds
|
||||
exact (continuousAt_id.mul (continuousAt_log one_ne_zero)).sub continuousAt_id
|
||||
· -- ∫ x in 1..t, log x = t * log t + 1, just use integral_log_of_pos
|
||||
rw [integral_log_of_pos zero_lt_one ht]
|
||||
norm_num
|
||||
· abel
|
||||
· -- log is integrable on [[0, 1]]
|
||||
rw [← neg_neg log]
|
||||
apply IntervalIntegrable.neg
|
||||
apply intervalIntegrable_deriv_of_nonneg (g := fun x ↦ -(x * log x - x))
|
||||
· exact (continuous_mul_log.continuousOn.sub continuous_id.continuousOn).neg
|
||||
· intro x hx
|
||||
norm_num at hx
|
||||
convert ((hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x)).neg using 1
|
||||
norm_num
|
||||
· intro x hx
|
||||
norm_num at hx
|
||||
rw [Pi.neg_apply, Left.nonneg_neg_iff]
|
||||
exact (log_nonpos_iff hx.left).mpr hx.right.le
|
||||
· -- log is integrable on [[0, t]]
|
||||
simp [Set.mem_uIcc, ht]
|
||||
@ -5,7 +5,7 @@
|
||||
"type": "git",
|
||||
"subDir": null,
|
||||
"scope": "",
|
||||
"rev": "504007f6f772de93ea967b17320baeac5a21cb55",
|
||||
"rev": "0571c82877b7c2b508cbc9a39e041eb117d050b7",
|
||||
"name": "mathlib",
|
||||
"manifestFile": "lake-manifest.json",
|
||||
"inputRev": null,
|
||||
|
||||
Loading…
Reference in New Issue
Block a user