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Nevanlinna/complexHarmonic.examples.lean

@@ -23,7 +23,6 @@ variable {F₁ : Type*} [NormedAddCommGroup F₁] [NormedSpace ℂ F₁] [Comple
 variable {G : Type*} [NormedAddCommGroup G] [NormedSpace ℝ G]
 variable {G₁ : Type*} [NormedAddCommGroup G₁] [NormedSpace ℂ G₁] [CompleteSpace G₁]
 
-#check DifferentiableOn.contDiffOn
 
 theorem holomorphicAt_is_harmonicAt
   {f : ℂ → F₁}
@@ -67,6 +66,33 @@ theorem holomorphicAt_is_harmonicAt
       simp
 
 
+theorem re_of_holomorphicAt_is_harmonicAr
+  {f : ℂ → ℂ} 
+  {z : ℂ}
+  (h : HolomorphicAt f z) :
+  HarmonicAt (Complex.reCLM ∘ f) z := by
+  apply harmonicAt_comp_CLM_is_harmonicAt
+  exact holomorphicAt_is_harmonicAt h
+
+
+theorem im_of_holomorphicAt_is_harmonicAt
+  {f : ℂ → ℂ}
+  {z : ℂ}
+  (h : HolomorphicAt f z) :
+  HarmonicAt (Complex.imCLM ∘ f) z := by
+  apply harmonicAt_comp_CLM_is_harmonicAt
+  exact holomorphicAt_is_harmonicAt h
+
+
+theorem antiholomorphicAt_is_harmonicAt
+  {f : ℂ → ℂ}
+  {z : ℂ}
+  (h : HolomorphicAt f z) :
+  HarmonicAt (Complex.conjCLE ∘ f) z := by
+  apply harmonicAt_iff_comp_CLE_is_harmonicAt.1
+  exact holomorphicAt_is_harmonicAt h
+
+
 theorem holomorphic_is_harmonic {f : ℂ → F₁} (h : Differentiable ℂ f) :
   Harmonic f := by
 

Nevanlinna/complexHarmonic.lean

@@ -195,6 +195,39 @@ theorem harmonicOn_comp_CLM_is_harmonicOn {f : ℂ → F₁} {s : Set ℂ} {l :
     exact IsOpen.mem_nhds hs zHyp
 
 
+theorem harmonicAt_comp_CLM_is_harmonicAt
+  {f : ℂ → F₁}
+  {z : ℂ}
+  {l : F₁ →L[ℝ] G}
+  (h : HarmonicAt f z) :
+  HarmonicAt (l ∘ f) z := by
+
+  constructor
+  · -- ContDiffAt ℝ 2 (⇑l ∘ f) z
+    apply ContDiffAt.continuousLinearMap_comp
+    exact h.1
+  · -- Δ (⇑l ∘ f) =ᶠ[nhds z] 0
+    obtain ⟨r, h₁r, h₂r⟩ := h.1.contDiffOn le_rfl
+    obtain ⟨s, h₁s, h₂s, h₃s⟩ := mem_nhds_iff.1 h₁r
+    obtain ⟨t, h₁t, h₂t⟩ := Filter.eventuallyEq_iff_exists_mem.1 h.2
+    obtain ⟨u, h₁u, h₂u, h₃u⟩ := mem_nhds_iff.1 h₁t
+    apply Filter.eventuallyEq_iff_exists_mem.2
+    use s ∩ u
+    constructor
+    · apply IsOpen.mem_nhds
+      exact IsOpen.inter h₂s h₂u
+      constructor
+      · exact h₃s
+      · exact h₃u
+    · intro x xHyp
+      rw [laplace_compCLMAt]
+      simp
+      rw [h₂t]
+      simp
+      exact h₁u xHyp.2
+      apply (h₂r.mono h₁s).contDiffAt (IsOpen.mem_nhds h₂s xHyp.1)
+
+
 theorem harmonic_iff_comp_CLE_is_harmonic {f : ℂ → F₁} {l : F₁ ≃L[ℝ] G₁} :
   Harmonic f ↔ Harmonic (l ∘ f) := by
 
@@ -210,6 +243,24 @@ theorem harmonic_iff_comp_CLE_is_harmonic {f : ℂ → F₁} {l : F₁ ≃L[ℝ]
     exact harmonic_comp_CLM_is_harmonic
 
 
+theorem harmonicAt_iff_comp_CLE_is_harmonicAt 
+  {f : ℂ → F₁} 
+  {z : ℂ}
+  {l : F₁ ≃L[ℝ] G₁} :
+  HarmonicAt f z ↔ HarmonicAt (l ∘ f) z := by
+
+  constructor
+  · have : l ∘ f = (l : F₁ →L[ℝ] G₁) ∘ f := by rfl
+    rw [this]
+    exact harmonicAt_comp_CLM_is_harmonicAt
+  · have : f = (l.symm : G₁ →L[ℝ] F₁) ∘ l ∘ f := by
+      funext z
+      unfold Function.comp
+      simp
+    nth_rewrite 2 [this]
+    exact harmonicAt_comp_CLM_is_harmonicAt
+
+
 theorem harmonicOn_iff_comp_CLE_is_harmonicOn {f : ℂ → F₁} {s : Set ℂ} {l : F₁ ≃L[ℝ] G₁} (hs : IsOpen s) :
   HarmonicOn f s ↔ HarmonicOn (l ∘ f) s := by
 

Nevanlinna/holomorphic.lean

@@ -1,5 +1,4 @@
 import Mathlib.Analysis.Complex.Basic
-import Mathlib.Analysis.Calculus.ContDiff.Defs
 import Nevanlinna.partialDeriv
 
 variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
@@ -33,6 +32,28 @@ theorem HolomorphicAt_iff
     · assumption
 
 
+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_isOpen
   {f : E → F}
   {x : E} :
@@ -62,7 +83,7 @@ theorem HolomorphicAt_isOpen
     · intro z hz
       obtain ⟨t, ht⟩ := (hf z hz)
       exact ht.2 z (mem_of_mem_nhds ht.1)
-
+-/
 
 theorem CauchyRiemann'₅
   {f : ℂ → F}

Nevanlinna/laplace.lean

@@ -170,7 +170,11 @@ theorem laplace_smul {f : ℂ → F} : ∀ v : ℝ, Δ (v • f) = v • (Δ f)
   simp
 
 
-theorem laplace_compCLMAt {f : ℂ → F} {l : F →L[ℝ] G} {x : ℂ} (h : ContDiffAt ℝ 2 f x) :
+theorem laplace_compCLMAt
+  {f : ℂ → F}
+  {l : F →L[ℝ] G}
+  {x : ℂ}
+  (h : ContDiffAt ℝ 2 f x) :
   Δ (l ∘ f) x = (l ∘ (Δ f)) x := by
 
   obtain ⟨v, hv₁, hv₂, hv₃, hv₄⟩ : ∃ v ∈ nhds x, (IsOpen v) ∧ (x ∈ v) ∧ (ContDiffOn ℝ 2 f v) := by

Nevanlinna/partialDeriv.lean

@@ -361,7 +361,7 @@ theorem partialDeriv_commOn
     (fderiv ℝ (fun w => fderiv ℝ f w) z) v₁ v₂ = (fderiv ℝ (fun w => fderiv ℝ f w) z) v₂ v₁ := by
 
     let f' := fderiv ℝ f
-    have h₀ : ∀ y ∈ s, HasFDerivAt f (f' y) y := by
+    have h₀1 : ∀ y ∈ s, HasFDerivAt f (f' y) y := by
       intro y hy
       apply DifferentiableAt.hasFDerivAt
       apply DifferentiableOn.differentiableAt _ (IsOpen.mem_nhds hs hy)
@@ -393,31 +393,8 @@ theorem partialDeriv_commAt
   (h : ContDiffAt ℝ 2 f z) :
   ∀ v₁ v₂ : E, partialDeriv ℝ v₁ (partialDeriv ℝ v₂ f) z = partialDeriv ℝ v₂ (partialDeriv ℝ v₁ f) z := by
 
+  obtain ⟨u, hu₁, hu₂⟩ := h.contDiffOn le_rfl
+  obtain ⟨v, hv₁, hv₂, hv₃⟩ := mem_nhds_iff.1 hu₁
+
   intro v₁ v₂
-
+  exact partialDeriv_commOn hv₂ (hu₂.mono hv₁) v₁ v₂ z hv₃
-  have derivSymm :
-    (fderiv ℝ (fun w => fderiv ℝ f w) z) v₁ v₂ = (fderiv ℝ (fun w => fderiv ℝ f w) z) v₂ v₁ := by
-
-    let f' := fderiv ℝ f
-    have h₀ : ∀ y ∈ s, HasFDerivAt f (f' y) y := by
-      intro y hy
-      apply DifferentiableAt.hasFDerivAt
-      apply DifferentiableOn.differentiableAt _ (IsOpen.mem_nhds hs hy)
-      apply h.differentiableOn one_le_two
-
-    let f'' := (fderiv ℝ f' z)
-    have h₁ : HasFDerivAt f' f'' z := by
-      apply DifferentiableAt.hasFDerivAt
-      apply DifferentiableOn.differentiableAt _ (IsOpen.mem_nhds hs hz)
-      apply ContDiffOn.differentiableOn _ (Submonoid.oneLE.proof_2 ℕ∞)
-      exact ((contDiffOn_succ_iff_fderiv_of_isOpen hs).1 h).2
-
-    have h₀' : ∀ᶠ (y : E) in nhds z, HasFDerivAt f (f' y) y := by
-      apply eventually_nhds_iff.mpr
-      use s
-
-    exact second_derivative_symmetric_of_eventually h₀' h₁ v₁ v₂
-
-  rw [← partialDeriv_fderivAt ℝ hs h v₂ v₁ z hz]
-  rw [derivSymm]
-  rw [← partialDeriv_fderivOn ℝ hs h v₁ v₂ z hz]