Done with examples for holomorphicAt

Nevanlinna/complexHarmonic.examples.lean

@@ -1,27 +1,9 @@
-import Mathlib.Analysis.Complex.Basic
-import Mathlib.Analysis.Complex.TaylorSeries
-import Mathlib.Analysis.Calculus.LineDeriv.Basic
-import Mathlib.Analysis.Calculus.ContDiff.Defs
-import Mathlib.Analysis.Calculus.FDeriv.Basic
-import Mathlib.Analysis.Calculus.FDeriv.Symmetric
-import Mathlib.Analysis.RCLike.Basic
-import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
-import Mathlib.Data.Complex.Module
-import Mathlib.Data.Complex.Order
-import Mathlib.Data.Complex.Exponential
-import Mathlib.Data.Fin.Tuple.Basic
-import Mathlib.Topology.Algebra.InfiniteSum.Module
-import Mathlib.Topology.Defs.Filter
-import Mathlib.Topology.Instances.RealVectorSpace
-import Nevanlinna.cauchyRiemann
-import Nevanlinna.laplace
 import Nevanlinna.complexHarmonic
 import Nevanlinna.holomorphic
 
 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]
 
 
 theorem holomorphicAt_is_harmonicAt
@@ -93,6 +75,18 @@ theorem log_normSq_of_holomorphicAt_is_harmonicAt
   (h₂f : f z ≠ 0) :
   HarmonicAt (Real.log ∘ Complex.normSq ∘ f) z := by
 
+  -- For later use
+  have slitPlaneLemma {z : ℂ} (hz : z ≠ 0) : z ∈ Complex.slitPlane ∨ -z ∈ Complex.slitPlane := by
+    rw [Complex.mem_slitPlane_iff, Complex.mem_slitPlane_iff]
+    simp at hz
+    rw [Complex.ext_iff] at hz
+    push_neg at hz
+    simp at hz
+    simp
+    by_contra contra
+    push_neg at contra
+    exact hz (le_antisymm contra.1.1 contra.2.1) contra.1.2
+
   -- 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
@@ -137,10 +131,44 @@ theorem log_normSq_of_holomorphicAt_is_harmonicAt
         simp
         apply hx.2
 
+    -- 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.conjCLE ∘ Complex.log ∘ 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_conj]
+        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
+        apply Complex.slitPlane_arg_ne_pi hx.1
+
     rw [HarmonicAt_eventuallyEq this]
     apply harmonicAt_add_harmonicAt_is_harmonicAt
-    ·
-      sorry
+    · rw [← harmonicAt_iff_comp_CLE_is_harmonicAt]
+      apply holomorphicAt_is_harmonicAt
+      apply HolomorphicAt_comp
+      use Complex.slitPlane
+      constructor
+      · apply IsOpen.mem_nhds
+        exact Complex.isOpen_slitPlane
+        assumption
+      · exact fun z a => Complex.differentiableAt_log a
+      exact h₁g
     · apply holomorphicAt_is_harmonicAt
       apply HolomorphicAt_comp
       use Complex.slitPlane
@@ -149,78 +177,16 @@ theorem log_normSq_of_holomorphicAt_is_harmonicAt
         exact Complex.isOpen_slitPlane
         assumption
       · exact fun z a => Complex.differentiableAt_log a
+      exact h₁g
 
-
-
-
-
-      sorry
-      assumption
-
-
-
-    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
+  by_cases h₃f : f z ∈ Complex.slitPlane
+  · exact lem₁ f h₁f h₂f h₃f
+  · have : Complex.normSq ∘ f = Complex.normSq ∘ (-f) := by funext; simp
     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
+    apply lem₁ (-f)
+    · exact HolomorphicAt_neg h₁f
+    · simpa
+    · exact (slitPlaneLemma h₂f).resolve_left h₃f
 
 
 theorem holomorphic_is_harmonic {f : ℂ → F₁} (h : Differentiable ℂ f) :

Nevanlinna/holomorphic.lean

@@ -1,21 +1,5 @@
-import Mathlib.Analysis.Complex.Basic
 import Mathlib.Analysis.Complex.TaylorSeries
-import Mathlib.Analysis.Calculus.LineDeriv.Basic
-import Mathlib.Analysis.Calculus.ContDiff.Defs
-import Mathlib.Analysis.Calculus.FDeriv.Basic
-import Mathlib.Analysis.Calculus.FDeriv.Symmetric
-import Mathlib.Analysis.RCLike.Basic
-import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
-import Mathlib.Data.Complex.Module
-import Mathlib.Data.Complex.Order
-import Mathlib.Data.Complex.Exponential
-import Mathlib.Data.Fin.Tuple.Basic
-import Mathlib.Topology.Algebra.InfiniteSum.Module
-import Mathlib.Topology.Defs.Filter
-import Mathlib.Topology.Instances.RealVectorSpace
 import Nevanlinna.cauchyRiemann
-import Nevanlinna.laplace
-import Nevanlinna.complexHarmonic
 
 variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
 variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F] [CompleteSpace F]
@@ -28,7 +12,8 @@ def HolomorphicAt (f : E → F) (x : E) : Prop :=
 theorem HolomorphicAt_iff
   {f : E → F}
   {x : E} :
-  HolomorphicAt f x ↔ ∃ s : Set E, IsOpen s ∧ x ∈ s ∧ (∀ z ∈ s, DifferentiableAt ℂ f z) := by
+  HolomorphicAt f x ↔ ∃ s :
+  Set E, IsOpen s ∧ x ∈ s ∧ (∀ z ∈ s, DifferentiableAt ℂ f z) := by
   constructor
   · intro hf
     obtain ⟨t, h₁t, h₂t⟩ := hf
@@ -104,6 +89,21 @@ theorem HolomorphicAt_comp
     exact hx.1
 
 
+theorem HolomorphicAt_neg
+  {f : E → F}
+  {z : E}
+  (hf : HolomorphicAt f z) :
+  HolomorphicAt (-f) z := by
+  obtain ⟨UF, h₁UF, h₂UF⟩ := hf
+  use UF
+  constructor
+  · assumption
+  · intro z hz
+    apply differentiableAt_neg_iff.mp
+    simp
+    exact h₂UF z hz
+
+
 theorem HolomorphicAt_contDiffAt
   {f : ℂ → F}
   {z : ℂ}