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Nevanlinna/complexHarmonic.examples.lean

@@ -93,9 +93,7 @@ theorem log_normSq_of_holomorphicAt_is_harmonicAt
   (h₂f : f z ≠ 0) :
   HarmonicAt (Real.log ∘ Complex.normSq ∘ f) z := by
 
-  /- First prove the theorem under the additional assumption that
-
-  -/
+  -- First prove the theorem for functions with image in the slitPlane
   have lem₁ : ∀ g : ℂ → ℂ, (HolomorphicAt g z) → (g z ≠ 0) → (g z ∈ Complex.slitPlane) → HarmonicAt (Real.log ∘ Complex.normSq ∘ g) z := by
     intro g h₁g h₂g h₃g
 
@@ -139,6 +137,11 @@ theorem log_normSq_of_holomorphicAt_is_harmonicAt
         simp
         apply hx.2
 
+    rw [HarmonicAt_eventuallyEq this]
+
+
+
+
     sorry
 
 

Nevanlinna/complexHarmonic.lean

@@ -64,6 +64,18 @@ theorem HarmonicAt_iff
       · exact h₂f
 
 
+theorem HarmonicAt_eventuallyEq {f₁ f₂ : ℂ → F} {x : ℂ} (h : f₁ =ᶠ[nhds x] f₂) : HarmonicAt f₁ x ↔ HarmonicAt f₂ x := by
+  constructor
+  · intro h₁
+    constructor
+    · exact ContDiffAt.congr_of_eventuallyEq h₁.1 (Filter.EventuallyEq.symm h)
+    · exact Filter.EventuallyEq.trans (laplace_eventuallyEq' (Filter.EventuallyEq.symm h)) h₁.2
+  · intro h₁
+    constructor
+    · exact ContDiffAt.congr_of_eventuallyEq h₁.1 h
+    · exact Filter.EventuallyEq.trans (laplace_eventuallyEq' h) h₁.2
+
+
 theorem HarmonicOn_of_locally_HarmonicOn {f : ℂ → F} {s : Set ℂ} (h : ∀ x ∈ s, ∃ (u : Set ℂ), IsOpen u ∧ x ∈ u ∧ HarmonicOn f (s ∩ u)) :
   HarmonicOn f s := by
   constructor
@@ -142,6 +154,21 @@ theorem harmonicOn_add_harmonicOn_is_harmonicOn {f₁ f₂ : ℂ → F} {s : Set
     simp
 
 
+theorem harmonicAt_add_harmonicAt_is_harmonicAt
+  {f₁ f₂ : ℂ → F}
+  {x : ℂ}
+  (h₁ : HarmonicAt f₁ x)
+  (h₂ : HarmonicAt f₂ x) :
+  HarmonicAt (f₁ + f₂) x := by
+  constructor
+  · exact ContDiffAt.add h₁.1 h₂.1
+  · rw [laplace_add_ContDiffAt]
+    intro z hz
+    rw [laplace_add_ContDiffOn hs h₁.1 h₂.1 z hz]
+    rw [h₁.2 z hz, h₂.2 z hz]
+    simp
+
+
 theorem harmonic_smul_const_is_harmonic {f : ℂ → F} {c : ℝ} (h : Harmonic f) :
   Harmonic (c • f) := by
   constructor

Nevanlinna/laplace.lean

@@ -33,6 +33,14 @@ theorem laplace_eventuallyEq {f₁ f₂ : ℂ → F} {x : ℂ} (h : f₁ =ᶠ[nh
   rw [partialDeriv_eventuallyEq ℝ (partialDeriv_eventuallyEq' ℝ h Complex.I) Complex.I]
 
 
+
+theorem laplace_eventuallyEq' {f₁ f₂ : ℂ → F} {x : ℂ} (h : f₁ =ᶠ[nhds x] f₂) : Δ f₁ =ᶠ[nhds x] Δ f₂ := by
+  unfold Complex.laplace
+  apply Filter.EventuallyEq.add
+  exact partialDeriv_eventuallyEq' ℝ (partialDeriv_eventuallyEq' ℝ h 1) 1
+  exact partialDeriv_eventuallyEq' ℝ (partialDeriv_eventuallyEq' ℝ h Complex.I) Complex.I
+
+
 theorem laplace_add
   {f₁ f₂ : ℂ → F}
   (h₁ : ContDiff ℝ 2 f₁)
@@ -161,6 +169,57 @@ theorem laplace_add_ContDiffAt
     apply fun v ↦ (partialDeriv_contDiffAt ℝ h₁ v).differentiableAt le_rfl
     apply fun v ↦ (partialDeriv_contDiffAt ℝ h₂ v).differentiableAt le_rfl
 
+theorem laplace_add_ContDiffAt'
+  {f₁ f₂ : ℂ → F}
+  {x : ℂ}
+  (h₁ : ContDiffAt ℝ 2 f₁ x)
+  (h₂ : ContDiffAt ℝ 2 f₂ x) :
+  Δ (f₁ + f₂) =ᶠ[nhds x] (Δ f₁) + (Δ f₂):= by
+
+  unfold Complex.laplace
+  have : partialDeriv ℝ 1 (partialDeriv ℝ 1 f₁) + partialDeriv ℝ Complex.I (partialDeriv ℝ Complex.I f₁) +
+    (partialDeriv ℝ 1 (partialDeriv ℝ 1 f₂) + partialDeriv ℝ Complex.I (partialDeriv ℝ Complex.I f₂)) =
+    (partialDeriv ℝ 1 (partialDeriv ℝ 1 f₁) + (partialDeriv ℝ 1 (partialDeriv ℝ 1 f₂)) +
+    (partialDeriv ℝ Complex.I (partialDeriv ℝ Complex.I f₁) + partialDeriv ℝ Complex.I (partialDeriv ℝ Complex.I f₂))) := by
+    group
+  rw [this]
+  apply Filter.EventuallyEq.add
+
+  suffices hyp : partialDeriv ℝ 1 (partialDeriv ℝ 1 (f₁ + f₂)) =ᶠ[nhds x] partialDeriv ℝ 1 (partialDeriv ℝ 1 f₁) + partialDeriv ℝ 1 (partialDeriv ℝ 1 f₂) from by
+    have : (fun x => partialDeriv ℝ 1 (partialDeriv ℝ 1 (f₁ + f₂)) x) = partialDeriv ℝ 1 (partialDeriv ℝ 1 (f₁ + f₂)) := by
+      rfl
+    rw [this]
+    have : (fun x => (partialDeriv ℝ 1 (partialDeriv ℝ 1 f₁) + partialDeriv ℝ 1 (partialDeriv ℝ 1 f₂)) x) = (partialDeriv ℝ 1 (partialDeriv ℝ 1 f₁)) + (partialDeriv ℝ 1 (partialDeriv ℝ 1 f₂)) := by
+      rfl
+    rwa [this]
+
+  simp
+
+  have h₁₁ : ContDiffAt ℝ 1 f₁ x := h₁.of_le one_le_two
+  have h₂₁ : ContDiffAt ℝ 1 f₂ x := h₂.of_le one_le_two
+  let A : partialDeriv ℝ 1 (f₁ + f₂)  =ᶠ[nhds x] (partialDeriv ℝ 1 f₁) + (partialDeriv ℝ 1 f₂) := by
+    exact partialDeriv_add₂_contDiffAt ℝ h₁₁ h₂₁
+  let B : partialDeriv ℝ 1 (partialDeriv ℝ 1 (f₁ + f₂)) =ᶠ[nhds x] (partialDeriv ℝ 1 (partialDeriv ℝ 1 f₁)) + (partialDeriv ℝ 1 (partialDeriv ℝ 1 f₂)) := by
+    sorry
+
+  sorry
+  repeat
+    rw [partialDeriv_eventuallyEq ℝ (partialDeriv_add₂_contDiffAt ℝ h₁₁ h₂₁)]
+    rw [partialDeriv_add₂_differentiableAt]
+
+  -- I am super confused at this point because the tactic 'ring' does not work.
+  -- I do not understand why. So, I need to do things by hand.
+  rw [add_assoc]
+  rw [add_assoc]
+  rw [add_right_inj (partialDeriv ℝ 1 (partialDeriv ℝ 1 f₁) x)]
+  rw [add_comm]
+  rw [add_assoc]
+  rw [add_right_inj (partialDeriv ℝ Complex.I (partialDeriv ℝ Complex.I f₁) x)]
+  rw [add_comm]
+
+  repeat
+    apply fun v ↦ (partialDeriv_contDiffAt ℝ h₁ v).differentiableAt le_rfl
+    apply fun v ↦ (partialDeriv_contDiffAt ℝ h₂ v).differentiableAt le_rfl
 
 theorem laplace_smul {f : ℂ → F} : ∀ v : ℝ, Δ (v • f) = v • (Δ f) := by
   intro v