Update complexHarmonic.lean

Nevanlinna/complexHarmonic.lean

@@ -17,13 +17,47 @@ import Nevanlinna.laplace
 import Nevanlinna.partialDeriv
 
 variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
+variable {F₁ : Type*} [NormedAddCommGroup F₁] [NormedSpace ℂ F₁] [CompleteSpace F₁]
+variable {G : Type*} [NormedAddCommGroup G] [NormedSpace ℝ G]
+variable {G₁ : Type*} [NormedAddCommGroup G₁] [NormedSpace ℂ G₁] [CompleteSpace G₁]
 
 
 def Harmonic (f : ℂ → F) : Prop :=
   (ContDiff ℝ 2 f) ∧ (∀ z, Complex.laplace f z = 0)
 
 
-theorem holomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) :
+theorem harmonic_comp_CLM_is_harmonic {f : ℂ → F₁} {l : F₁ →L[ℝ] G}  (h : Harmonic f) :
+  Harmonic (l ∘ f) := by
+
+  constructor
+  · -- Continuous differentiability
+    apply ContDiff.comp
+    exact ContinuousLinearMap.contDiff l
+    exact ContDiff.restrict_scalars ℝ (Differentiable.contDiff h)
+  · rw [laplace_compContLin]
+    simp
+    intro z
+    rw [h.2 z]
+    simp
+    exact ContDiff.restrict_scalars ℝ (Differentiable.contDiff h)
+
+
+theorem harmonic_iff_comp_CLE_is_harmonic {f : ℂ → F₁} {l : F₁ ≃L[ℝ] G₁} :
+  Harmonic f ↔ Harmonic (l ∘ f) := by
+
+  constructor
+  · have : l ∘ f = (l : F₁ →L[ℝ] G₁) ∘ f := by rfl
+    rw [this]
+    exact harmonic_comp_CLM_is_harmonic
+  · have : f = (l.symm : G₁ →L[ℝ] F₁) ∘ l ∘ f := by 
+      funext z
+      unfold Function.comp
+      simp
+    nth_rewrite 2 [this]
+    exact harmonic_comp_CLM_is_harmonic
+
+
+theorem holomorphic_is_harmonic {f : ℂ → F₁} (h : Differentiable ℂ f) :
   Harmonic f := by
 
   -- f is real C²
@@ -85,34 +119,168 @@ theorem holomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) :
 
 theorem re_of_holomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) :
   Harmonic (Complex.reCLM ∘ f) := by
-
-  constructor
-  · -- Continuous differentiability
-    apply ContDiff.comp
-    exact ContinuousLinearMap.contDiff Complex.reCLM
-    exact ContDiff.restrict_scalars ℝ (Differentiable.contDiff h)
-  · rw [laplace_compContLin]
-    simp
-    intro z
-    rw [(holomorphic_is_harmonic h).right z]
-    simp
-    exact ContDiff.restrict_scalars ℝ (Differentiable.contDiff h)
+  apply harmonic_comp_CLM_is_harmonic
+  exact holomorphic_is_harmonic h
 
 
 theorem im_of_holomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) :
   Harmonic (Complex.imCLM ∘ f) := by
+  apply harmonic_comp_CLM_is_harmonic
+  exact holomorphic_is_harmonic h
+
+
+theorem log_normSq_of_holomorphic_is_harmonic
+  {f : ℂ → ℂ}
+  (h₁ : Differentiable ℂ f)
+  (h₂ : ∀ z, f z ≠ 0)
+  (h₃ : ∀ z, f z ∈ Complex.slitPlane) :
+  Harmonic (Real.log ∘ Complex.normSq ∘ f) := by
+
+
+
+  /- We start with a number of lemmas on regularity of all the functions involved -/
+
+  -- The norm square is real C²
+  have normSq_is_real_C2 : ContDiff ℝ 2 Complex.normSq := by
+    unfold Complex.normSq
+    simp
+    conv =>
+      arg 3
+      intro x
+      rw [← Complex.reCLM_apply, ← Complex.imCLM_apply]
+    apply ContDiff.add
+    apply ContDiff.mul
+    apply ContinuousLinearMap.contDiff Complex.reCLM
+    apply ContinuousLinearMap.contDiff Complex.reCLM
+    apply ContDiff.mul
+    apply ContinuousLinearMap.contDiff Complex.imCLM
+    apply ContinuousLinearMap.contDiff Complex.imCLM
+
+  -- f is real C²
+  have f_is_real_C2 : ContDiff ℝ 2 f :=
+    ContDiff.restrict_scalars ℝ (Differentiable.contDiff h₁)
+
+  -- Complex.log ∘ f is real C²
+  have log_f_is_holomorphic : Differentiable ℂ (Complex.log ∘ f) := by
+    intro z
+    apply DifferentiableAt.comp
+    exact Complex.differentiableAt_log (h₃ z)
+    exact h₁ z
+
+  -- Real.log |f|² is real C²
+  have t₄ : ContDiff ℝ 2 (Real.log ∘ ⇑Complex.normSq ∘ f) := by
+    rw [contDiff_iff_contDiffAt]
+    intro z
+    apply ContDiffAt.comp
+    apply Real.contDiffAt_log.mpr
+    simp
+    exact h₂ z
+    apply ContDiff.comp_contDiffAt z normSq_is_real_C2
+    exact ContDiff.contDiffAt f_is_real_C2
+
+  have t₂ : Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f = Complex.conjCLE ∘ Complex.log ∘ f := by
+    funext z
+    unfold Function.comp
+    rw [Complex.log_conj]
+    rfl
+    exact Complex.slitPlane_arg_ne_pi (h₃ z)
 
   constructor
-  · -- Continuous differentiability
-    apply ContDiff.comp
-    exact ContinuousLinearMap.contDiff Complex.imCLM
-    exact ContDiff.restrict_scalars ℝ (Differentiable.contDiff h)
-  · rw [laplace_compContLin]
+  · -- logabs f is real C²
+    have : (fun z ↦ Real.log ‖f z‖) = (2 : ℝ)⁻¹ • (Real.log ∘ Complex.normSq ∘ f) := by
+      funext z
       simp
+      unfold Complex.abs
+      simp
+      rw [Real.log_sqrt]
+      rw [div_eq_inv_mul (Real.log (Complex.normSq (f z))) 2]
+      exact Complex.normSq_nonneg (f z)
+    rw [this]
+
+    have : (2 : ℝ)⁻¹ • (Real.log ∘ Complex.normSq ∘ f) = (fun z ↦ (2 : ℝ)⁻¹ • ((Real.log ∘ ⇑Complex.normSq ∘ f) z)) := by
+      exact rfl
+    rw [this]
+    apply ContDiff.const_smul
+    exact t₄
+
+  · -- Laplace vanishes
+    have : (fun z ↦ Real.log ‖f z‖) = (2 : ℝ)⁻¹ • (Real.log ∘ Complex.normSq ∘ f) := by
+      funext z
+      simp
+      unfold Complex.abs
+      simp
+      rw [Real.log_sqrt]
+      rw [div_eq_inv_mul (Real.log (Complex.normSq (f z))) 2]
+      exact Complex.normSq_nonneg (f z)
+    rw [this]
+    rw [laplace_smul]
+    simp
+
+    have : ∀ (z : ℂ), Complex.laplace (Real.log ∘ ⇑Complex.normSq ∘ f) z = 0 ↔ Complex.laplace (Complex.ofRealCLM ∘ Real.log ∘ ⇑Complex.normSq ∘ f) z = 0 := by
       intro z
-    rw [(holomorphic_is_harmonic h).right z]
+      rw [laplace_compContLin]
       simp
-    exact ContDiff.restrict_scalars ℝ (Differentiable.contDiff h)
+      -- ContDiff ℝ 2 (Real.log ∘ ⇑Complex.normSq ∘ f)
+      exact t₄
+    conv =>
+      intro z
+      rw [this z]
+
+    have : Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ f = Complex.log ∘ Complex.ofRealCLM ∘ Complex.normSq ∘ f := by
+      unfold Function.comp
+      funext z
+      apply Complex.ofReal_log
+      exact Complex.normSq_nonneg (f z)
+    rw [this]
+
+    have : Complex.ofRealCLM ∘ ⇑Complex.normSq ∘ f = ((starRingEnd ℂ) ∘ f) * f := by
+      funext z
+      simp
+      exact Complex.normSq_eq_conj_mul_self
+    rw [this]
+
+    have : Complex.log ∘ (⇑(starRingEnd ℂ) ∘ f * f) = Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f + Complex.log ∘ f := by
+      unfold Function.comp
+      funext z
+      simp
+      rw [Complex.log_mul_eq_add_log_iff]
+
+      have : Complex.arg ((starRingEnd ℂ) (f z)) = - Complex.arg (f z) := by
+        rw [Complex.arg_conj]
+        have : ¬ Complex.arg (f z) = Real.pi := by
+          exact Complex.slitPlane_arg_ne_pi (h₃ z)
+        simp
+        tauto
+      rw [this]
+      simp
+      constructor
+      · exact Real.pi_pos
+      · exact Real.pi_nonneg
+      exact (AddEquivClass.map_ne_zero_iff starRingAut).mpr (h₂ z)
+      exact h₂ z
+    rw [this]
+    rw [laplace_add]
+
+    rw [t₂, laplace_compCLE]
+    intro z
+    simp
+    rw [(holomorphic_is_harmonic log_f_is_holomorphic).2 z]
+    simp
+
+    -- ContDiff ℝ 2 (Complex.log ∘ f)
+    exact ContDiff.restrict_scalars ℝ (Differentiable.contDiff log_f_is_holomorphic)
+
+    -- ContDiff ℝ 2 (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f)
+    rw [t₂]
+    apply ContDiff.comp
+    exact ContinuousLinearEquiv.contDiff Complex.conjCLE
+    exact ContDiff.restrict_scalars ℝ (Differentiable.contDiff log_f_is_holomorphic)
+
+    -- ContDiff ℝ 2 (Complex.log ∘ f)
+    exact ContDiff.restrict_scalars ℝ (Differentiable.contDiff log_f_is_holomorphic)
+
+    -- ContDiff ℝ 2 (Real.log ∘ ⇑Complex.normSq ∘ f)
+    exact t₄
 
 
 theorem logabs_of_holomorphic_is_harmonic