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Nevanlinna/complexHarmonic.examples.lean

@@ -86,6 +86,123 @@ theorem antiholomorphicAt_is_harmonicAt
   exact holomorphicAt_is_harmonicAt h
 
 
+theorem log_normSq_of_holomorphicAt_is_harmonicAt
+  {f : ℂ → ℂ}
+  {z : ℂ}
+  (h₁f : HolomorphicAt f z)
+  (h₂f : f z ≠ 0) :
+  HarmonicAt (Real.log ∘ Complex.normSq ∘ f) z := by
+
+  /- First prove the theorem under the additional assumption that
+
+  -/
+  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
+
+    -- Rewrite the log |g|² as Complex.log (g * gc)
+    suffices hyp : HarmonicAt (Complex.log ∘ ((Complex.conjCLE ∘ g) * g)) z from by
+      have : Real.log ∘ Complex.normSq ∘ g = Complex.reCLM ∘ Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ g := by
+        funext x
+        simp
+      rw [this]
+
+      have : Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ g = Complex.log ∘ ((Complex.conjCLE ∘ g) * g) := by
+        funext x
+        simp
+        rw [Complex.ofReal_log]
+        rw [Complex.normSq_eq_conj_mul_self]
+        exact Complex.normSq_nonneg (g x)
+      rw [← this] at hyp
+      apply harmonicAt_comp_CLM_is_harmonicAt hyp
+
+    -- Locally around z, rewrite Complex.log (g * gc) as Complex.log g + Complex.log.gc
+    -- This uses the assumption that g z is in Complex.slitPlane
+    have : (Complex.log ∘ (Complex.conjCLE ∘ g * g)) =ᶠ[nhds z] (Complex.log ∘ Complex.conjCLE ∘ g + Complex.log ∘ g) := by
+      apply Filter.eventuallyEq_iff_exists_mem.2
+      use g⁻¹' (Complex.slitPlane ∩ {0}ᶜ)
+      constructor
+      · apply ContinuousAt.preimage_mem_nhds
+        · exact (HolomorphicAt_differentiableAt h₁g).continuousAt
+        · apply IsOpen.mem_nhds
+          apply IsOpen.inter Complex.isOpen_slitPlane isOpen_ne
+          constructor
+          · exact h₃g
+          · exact h₂g
+      · intro x hx
+        simp
+        rw [Complex.log_mul_eq_add_log_iff _ hx.2]
+        rw [Complex.arg_conj]
+        simp [Complex.slitPlane_arg_ne_pi hx.1]
+        constructor
+        · exact Real.pi_pos
+        · exact Real.pi_nonneg
+        simp
+        apply hx.2
+
+    sorry
+
+
+
+  -- Suffices to show Harmonic (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f + Complex.log ∘ f)
+  -- THIS IS WHERE WE USE h₃
+  have : (Complex.log ∘ (Complex.conjCLE ∘ f * f)) z = (Complex.log ∘ Complex.conjCLE ∘ f + Complex.log ∘ f) z := by
+    unfold Function.comp
+    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 hz)
+      simp
+      tauto
+    rw [this]
+    simp
+    constructor
+    · exact Real.pi_pos
+    · exact Real.pi_nonneg
+    exact (AddEquivClass.map_ne_zero_iff starRingAut).mpr (h₂ z hz)
+    exact h₂ z hz
+
+  rw [HarmonicOn_congr hs this]
+  simp
+
+  apply harmonicOn_add_harmonicOn_is_harmonicOn hs
+
+  have : (fun x => Complex.log ((starRingEnd ℂ) (f x))) = (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f) := by
+    rfl
+  rw [this]
+
+  -- HarmonicOn (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f) s
+  have : ∀ z ∈ s, (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f) z = (Complex.conjCLE ∘ Complex.log ∘ f) z := by
+    intro z hz
+    unfold Function.comp
+    rw [Complex.log_conj]
+    rfl
+    exact Complex.slitPlane_arg_ne_pi (h₃ z hz)
+  rw [HarmonicOn_congr hs this]
+
+  rw [← harmonicOn_iff_comp_CLE_is_harmonicOn]
+
+  apply holomorphicOn_is_harmonicOn
+  exact hs
+
+  intro z hz
+  apply DifferentiableAt.differentiableWithinAt
+  apply DifferentiableAt.comp
+
+
+
+  exact Complex.differentiableAt_log (h₃ z hz)
+  apply DifferentiableOn.differentiableAt h₁ -- (h₁ z hz)
+  exact IsOpen.mem_nhds hs hz
+  exact hs
+
+  -- HarmonicOn (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f) s
+  apply holomorphicOn_is_harmonicOn hs
+  exact DifferentiableOn.clog h₁ h₃
+
+
 theorem holomorphic_is_harmonic {f : ℂ → F₁} (h : Differentiable ℂ f) :
   Harmonic f := by