198 lines
8.2 KiB
Plaintext
198 lines
8.2 KiB
Plaintext
import Mathlib.Analysis.Complex.TaylorSeries
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import Mathlib.MeasureTheory.Integral.DivergenceTheorem
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import Mathlib.MeasureTheory.Function.LocallyIntegrable
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import Nevanlinna.cauchyRiemann
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import Nevanlinna.partialDeriv
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/-
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variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
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variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F] [CompleteSpace F]
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variable {G : Type*} [NormedAddCommGroup G] [NormedSpace ℂ G] [CompleteSpace G]
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noncomputable def Complex.primitive
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(f : ℂ → F) : ℂ → F :=
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fun z ↦ ∫ t : ℝ in (0)..1, z • f (t * z)
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-/
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theorem MeasureTheory.integral2_divergence_prod_of_hasFDerivWithinAt_off_countable₁
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
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(f : ℝ × ℝ → E)
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(g : ℝ × ℝ → E)
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(f' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E)
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(g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E)
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(a₁ : ℝ)
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(a₂ : ℝ)
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(b₁ : ℝ)
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(b₂ : ℝ)
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(s : Set (ℝ × ℝ))
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(hs : s.Countable)
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(Hcf : ContinuousOn f (Set.uIcc a₁ b₁ ×ˢ Set.uIcc a₂ b₂))
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(Hcg : ContinuousOn g (Set.uIcc a₁ b₁ ×ˢ Set.uIcc a₂ b₂))
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(Hdf : ∀ x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \ s, HasFDerivAt f (f' x) x)
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(Hdg : ∀ x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \ s, HasFDerivAt g (g' x) x)
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(Hi : MeasureTheory.IntegrableOn (fun (x : ℝ × ℝ) => (f' x) (1, 0) + (g' x) (0, 1)) (Set.uIcc a₁ b₁ ×ˢ Set.uIcc a₂ b₂) MeasureTheory.volume) :
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∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, (f' (x, y)) (1, 0) + (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
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exact
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integral2_divergence_prod_of_hasFDerivWithinAt_off_countable f g f' g' a₁ a₂ b₁ b₂ s hs Hcf Hcg
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Hdf Hdg Hi
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theorem MeasureTheory.integral2_divergence_prod_of_hasFDerivWithinAt_off_countable₂
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
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(f : ℝ × ℝ → E)
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(g : ℝ × ℝ → E)
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(f' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E)
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(g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E)
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(a₁ : ℝ)
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(a₂ : ℝ)
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(b₁ : ℝ)
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(b₂ : ℝ)
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(Hcf : ContinuousOn f (Set.uIcc a₁ b₁ ×ˢ Set.uIcc a₂ b₂))
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(Hcg : ContinuousOn g (Set.uIcc a₁ b₁ ×ˢ Set.uIcc a₂ b₂))
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(Hdf : ∀ x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂), HasFDerivAt f (f' x) x)
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(Hdg : ∀ x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂), HasFDerivAt g (g' x) x)
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(Hi : MeasureTheory.IntegrableOn (fun (x : ℝ × ℝ) => (f' x) (1, 0) + (g' x) (0, 1)) (Set.uIcc a₁ b₁ ×ˢ Set.uIcc a₂ b₂) MeasureTheory.volume) :
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∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, (f' (x, y)) (1, 0) + (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
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apply
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integral2_divergence_prod_of_hasFDerivWithinAt_off_countable f g f' g' a₁ a₂ b₁ b₂ ∅
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exact Set.countable_empty
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assumption
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assumption
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rwa [Set.diff_empty]
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rwa [Set.diff_empty]
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assumption
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theorem MeasureTheory.integral2_divergence₃
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
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(f g : ℝ × ℝ → E)
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(h₁f : ContDiff ℝ 1 f)
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(h₁g : ContDiff ℝ 1 g)
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(a₁ : ℝ)
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(a₂ : ℝ)
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(b₁ : ℝ)
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(b₂ : ℝ) :
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∫ (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
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apply integral2_divergence_prod_of_hasFDerivWithinAt_off_countable f g (fderiv ℝ f) (fderiv ℝ g) a₁ a₂ b₁ b₂ ∅
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exact Set.countable_empty
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-- ContinuousOn f (Set.uIcc a₁ b₁ ×ˢ Set.uIcc a₂ b₂)
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exact h₁f.continuous.continuousOn
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--
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exact h₁g.continuous.continuousOn
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--
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rw [Set.diff_empty]
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intro x _
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exact DifferentiableAt.hasFDerivAt ((h₁f.differentiable le_rfl) x)
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--
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rw [Set.diff_empty]
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intro y _
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exact DifferentiableAt.hasFDerivAt ((h₁g.differentiable le_rfl) y)
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--
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apply ContinuousOn.integrableOn_compact
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apply IsCompact.prod
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exact isCompact_uIcc
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exact isCompact_uIcc
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apply ContinuousOn.add
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apply Continuous.continuousOn
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exact Continuous.clm_apply (ContDiff.continuous_fderiv h₁f le_rfl) continuous_const
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apply Continuous.continuousOn
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exact Continuous.clm_apply (ContDiff.continuous_fderiv h₁g le_rfl) continuous_const
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theorem integral_divergence₄
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
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(f g : ℂ → E)
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(h₁f : ContDiff ℝ 1 f)
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(h₁g : ContDiff ℝ 1 g)
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(a₁ : ℝ)
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(a₂ : ℝ)
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(b₁ : ℝ)
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(b₂ : ℝ) :
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∫ (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
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let fr : ℝ × ℝ → E := f ∘ Complex.equivRealProdCLM.symm
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let gr : ℝ × ℝ → E := g ∘ Complex.equivRealProdCLM.symm
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have sfr {x y : ℝ} : f { re := x, im := y } = fr (x, y) := by exact rfl
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have sgr {x y : ℝ} : g { re := x, im := y } = gr (x, y) := by exact rfl
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repeat (conv in f { re := _, im := _ } => rw [sfr])
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repeat (conv in g { re := _, im := _ } => rw [sgr])
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have sfr' {x y : ℝ} {z : ℂ} : (fderiv ℝ f { re := x, im := y }) z = fderiv ℝ fr (x, y) (Complex.equivRealProdCLM z) := by
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rw [fderiv.comp]
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rw [Complex.equivRealProdCLM.symm.fderiv]
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tauto
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apply Differentiable.differentiableAt
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exact h₁f.differentiable le_rfl
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exact Complex.equivRealProdCLM.symm.differentiableAt
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conv in ⇑(fderiv ℝ f { re := _, im := _ }) _ => rw [sfr']
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have sgr' {x y : ℝ} {z : ℂ} : (fderiv ℝ g { re := x, im := y }) z = fderiv ℝ gr (x, y) (Complex.equivRealProdCLM z) := by
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rw [fderiv.comp]
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rw [Complex.equivRealProdCLM.symm.fderiv]
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tauto
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apply Differentiable.differentiableAt
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exact h₁g.differentiable le_rfl
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exact Complex.equivRealProdCLM.symm.differentiableAt
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conv in ⇑(fderiv ℝ g { re := _, im := _ }) _ => rw [sgr']
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apply MeasureTheory.integral2_divergence₃ fr gr _ _ a₁ a₂ b₁ b₂
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-- ContDiff ℝ 1 fr
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exact (ContinuousLinearEquiv.contDiff_comp_iff (ContinuousLinearEquiv.symm Complex.equivRealProdCLM)).mpr h₁f
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-- ContDiff ℝ 1 gr
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exact (ContinuousLinearEquiv.contDiff_comp_iff (ContinuousLinearEquiv.symm Complex.equivRealProdCLM)).mpr h₁g
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theorem integral_divergence₅
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
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(F : ℂ → ℂ)
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(hF : Differentiable ℂ F)
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(a₁ : ℝ)
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(a₂ : ℝ)
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(b₁ : ℝ)
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(b₂ : ℝ) :
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(∫ (x : ℝ) in a₁..b₁, Complex.I • F { re := x, im := b₂ }) +
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(∫ (y : ℝ) in a₂..b₂, F { re := b₁, im := y })
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=
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(∫ (x : ℝ) in a₁..b₁, Complex.I • F { re := x, im := a₂ }) +
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(∫ (y : ℝ) in a₂..b₂, F { re := a₁, im := y }) := by
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let h₁f : ContDiff ℝ 1 F := (hF.contDiff : ContDiff ℂ 1 F).restrict_scalars ℝ
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let g := Complex.I • F
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let h₁g : ContDiff ℝ 1 (Complex.I • F) := by
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have : Complex.I • F = fun x ↦ Complex.I • F x := by rfl
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rw [this]
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apply ContDiff.comp
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exact contDiff_const_smul Complex.I
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exact h₁f
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let A := integral_divergence₄ F g h₁f h₁g a₁ a₂ b₁ b₂
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have {z : ℂ} : fderiv ℝ F z 1 = partialDeriv ℝ 1 F z := by rfl
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conv at A in (fderiv ℝ F _) 1 => rw [this]
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have {z : ℂ} : fderiv ℝ g z Complex.I = partialDeriv ℝ Complex.I g z := by rfl
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conv at A in (fderiv ℝ g _) Complex.I => rw [this]
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have : Differentiable ℂ g := by sorry
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conv at A =>
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left
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arg 1
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intro x
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arg 1
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intro y
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rw [CauchyRiemann₄ this]
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rw [partialDeriv_smul'₂]
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rw [← smul_assoc]
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simp
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simp at A
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sorry
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