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Nevanlinna/holomorphic.examples.lean

@@ -0,0 +1,170 @@
+import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
+import Nevanlinna.complexHarmonic
+import Nevanlinna.complexHarmonic
+import Nevanlinna.holomorphicAt
+
+
+theorem CauchyRiemann₆
+  {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] 
+  {F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F] 
+  {f : E → F}
+  {z : E} : 
+  (DifferentiableAt ℂ f z) ↔ (DifferentiableAt ℝ f z) ∧ ∀ e, partialDeriv ℝ (Complex.I • e) f z = Complex.I • partialDeriv ℝ e f z := by
+  constructor
+
+  · -- Direction "→" 
+    intro h
+    constructor
+    · exact DifferentiableAt.restrictScalars ℝ h
+    · unfold partialDeriv
+      conv =>
+        intro e
+        left
+        rw [DifferentiableAt.fderiv_restrictScalars ℝ h]
+        simp
+        rw [← mul_one Complex.I]
+        rw [← smul_eq_mul]
+      conv =>
+        intro e
+        right
+        right
+        rw [DifferentiableAt.fderiv_restrictScalars ℝ h]
+      simp
+
+  · -- Direction "←" 
+    intro ⟨h₁, h₂⟩
+    apply (differentiableAt_iff_restrictScalars ℝ h₁).2
+
+    use {
+      toFun := fderiv ℝ f z
+      map_add' := fun x y => ContinuousLinearMap.map_add (fderiv ℝ f z) x y
+      map_smul' := by
+        simp
+        intro m x
+        have : m = m.re + m.im • Complex.I := by simp
+        rw [this, add_smul, add_smul, ContinuousLinearMap.map_add]
+        congr
+        simp
+        rw [smul_assoc, smul_assoc, ContinuousLinearMap.map_smul (fderiv ℝ f z) m.2]
+        congr
+        exact h₂ x
+    }
+    rfl
+
+
+theorem CauchyRiemann₇
+  {F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F] 
+  {f : ℂ → F}
+  {z : ℂ} : 
+  (DifferentiableAt ℂ f z) ↔ (DifferentiableAt ℝ f z) ∧ partialDeriv ℝ Complex.I f z = Complex.I • partialDeriv ℝ 1 f z := by
+  constructor
+  · intro hf
+    constructor
+    · exact (CauchyRiemann₆.1 hf).1
+    · let A := (CauchyRiemann₆.1 hf).2 1
+      simp at A
+      exact A
+  · intro ⟨h₁, h₂⟩
+    apply CauchyRiemann₆.2
+    constructor
+    · exact h₁
+    · intro e
+      have : Complex.I • e = e • Complex.I := by 
+        rw [smul_eq_mul, smul_eq_mul]
+        exact CommMonoid.mul_comm Complex.I e
+      rw [this]
+      have : e = e.re + e.im • Complex.I := by simp
+      rw [this, add_smul, partialDeriv_add₁, partialDeriv_add₁]
+      simp
+      rw [← smul_eq_mul]
+      have : partialDeriv ℝ ((e.re : ℝ) • Complex.I) f = partialDeriv ℝ ((e.re : ℂ) • Complex.I) f := by rfl
+      rw [← this, partialDeriv_smul₁ ℝ]
+      have : (e.re : ℂ) = (e.re : ℝ) • (1 : ℂ) := by simp
+      rw [this, partialDeriv_smul₁ ℝ]
+      have : partialDeriv ℝ ((e.im : ℂ) * Complex.I) f = partialDeriv ℝ ((e.im : ℝ) • Complex.I) f := by rfl
+      rw [this, partialDeriv_smul₁ ℝ]
+      simp
+      rw [h₂]
+      rw [smul_comm]
+      congr
+      rw [mul_assoc]
+      simp
+      nth_rw 2 [smul_comm]
+      rw [← smul_assoc]
+      simp
+      have : - (e.im : ℂ) = (-e.im : ℝ) • (1 : ℂ) := by simp
+      rw [this, partialDeriv_smul₁ ℝ]
+      simp
+  
+
+
+/-
+
+A harmonic, real-valued function on ℂ is the real part of a suitable holomorphic function.
+
+-/
+
+theorem harmonic_is_realOfHolomorphic
+  {f : ℂ → ℝ}
+  (hf : ∀ z, HarmonicAt f z) :
+  ∃ F : ℂ → ℂ, ∀ z, (HolomorphicAt F z ∧ ((F z).re = f z)) := by
+
+  let f_1 : ℂ → ℂ := Complex.ofRealCLM ∘ (partialDeriv ℝ 1 f)
+  let f_I : ℂ → ℂ := Complex.ofRealCLM ∘ (partialDeriv ℝ Complex.I f)
+
+  let g : ℂ → ℂ := f_1 - Complex.I • f_I
+
+  have reg₀ : Differentiable ℝ g := by 
+    let smulICLM : ℂ ≃L[ℝ] ℂ :=
+    {
+      toFun := fun x ↦ Complex.I • x
+      map_add' := fun x y => DistribSMul.smul_add Complex.I x y
+      map_smul' := fun m x => (smul_comm ((RingHom.id ℝ) m) Complex.I x).symm
+      invFun := fun x ↦ (Complex.I)⁻¹ • x
+      left_inv := by
+        intro x
+        simp
+        rw [← mul_assoc, mul_comm]
+        simp
+      right_inv := by
+        intro x
+        simp
+        rw [← mul_assoc]
+        simp
+      continuous_toFun := continuous_const_smul Complex.I
+      continuous_invFun := continuous_const_smul (Complex.I)⁻¹
+    }
+
+    apply Differentiable.sub
+    apply Differentiable.comp
+    exact ContinuousLinearMap.differentiable Complex.ofRealCLM
+    intro z
+    sorry
+    apply Differentiable.comp
+    sorry
+    sorry
+
+  
+  have reg₁ : Differentiable ℂ g := by 
+    intro z
+    apply CauchyRiemann₇.2
+    constructor
+    · exact reg₀ z
+    · dsimp [g]
+      have : f_1 - Complex.I • f_I = f_1 + (- Complex.I • f_I) := by
+        rw [sub_eq_add_neg]
+        simp
+      rw [this, partialDeriv_add₂, partialDeriv_add₂]
+      simp
+      dsimp [f_1, f_I]
+      
+      sorry
+      sorry
+      sorry
+      sorry
+      sorry
+
+  
+
+
+  sorry

Nevanlinna/holomorphic.primitive.lean

@@ -25,6 +25,7 @@ theorem partialDeriv_compContLinAt
   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]
@@ -46,6 +47,7 @@ theorem partialDeriv_compCLE
     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]
@@ -82,6 +84,7 @@ theorem partialDeriv_smul'₂
     rw [partialDeriv_compCLE]
     tauto
 
+
 theorem CauchyRiemann₄
   {F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F] 
   {f : ℂ → F} : 
@@ -104,6 +107,7 @@ theorem CauchyRiemann₄
   funext w
   simp
 
+
 theorem MeasureTheory.integral2_divergence₃
   {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
   (f g : ℝ × ℝ → E)
@@ -231,7 +235,6 @@ theorem integral_divergence₅
   exact B
 
 
-
 noncomputable def primitive
   {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] :
   ℂ  → (ℂ → E) → (ℂ → E) := by
@@ -303,7 +306,6 @@ theorem primitive_fderivAtBasepointZero
     apply Continuous.add
     continuity
     fun_prop
-    
 
   have t₃ {a : ℝ} : IntervalIntegrable (fun _ => f 0) MeasureTheory.volume 0 a := by
     apply Continuous.intervalIntegrable

Nevanlinna/partialDeriv.lean

@@ -38,7 +38,11 @@ theorem partialDeriv_eventuallyEq
   exact fun v => rfl
 
 
-theorem partialDeriv_smul₁ {f : E → F} {a : 𝕜} {v : E} : partialDeriv 𝕜 (a • v) f = a • partialDeriv 𝕜 v f := by
+theorem partialDeriv_smul₁ 
+  {f : E → F}
+  {a : 𝕜}
+  {v : E} : 
+  partialDeriv 𝕜 (a • v) f = a • partialDeriv 𝕜 v f := by
   unfold partialDeriv
   conv =>
     left