nevanlinna/Nevanlinna/complexHarmonic.examples.lean

import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
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]


theorem holomorphicAt_is_harmonicAt
  {f : ℂ → F₁}
  {z : ℂ}
  (hf : HolomorphicAt f z) :
  HarmonicAt f z := by

  let t := {x | HolomorphicAt f x}
  have ht : IsOpen t := HolomorphicAt_isOpen f
  have hz : z ∈ t := by exact hf

  constructor
  · -- ContDiffAt ℝ 2 f z
    exact HolomorphicAt_contDiffAt hf

  · -- Δ f =ᶠ[nhds z] 0
    apply Filter.eventuallyEq_iff_exists_mem.2
    use t
    constructor
    · exact IsOpen.mem_nhds ht hz
    · intro w hw
      unfold Complex.laplace
      simp
      rw [partialDeriv_eventuallyEq ℝ (CauchyRiemann'₆ hw) Complex.I]
      rw [partialDeriv_smul'₂]
      simp

      rw [partialDeriv_commAt (HolomorphicAt_contDiffAt hw) Complex.I 1]
      rw [partialDeriv_eventuallyEq ℝ (CauchyRiemann'₆ hw) 1]
      rw [partialDeriv_smul'₂]
      simp

      rw [← smul_assoc]
      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 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

  -- 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

    -- 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

    -- 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
    · 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
      constructor
      · apply IsOpen.mem_nhds
        exact Complex.isOpen_slitPlane
        assumption
      · exact fun z a => Complex.differentiableAt_log a
      exact h₁g

  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]
    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) :
  Harmonic f := by

  -- f is real C²
  have f_is_real_C2 : ContDiff ℝ 2 f :=
    ContDiff.restrict_scalars ℝ (Differentiable.contDiff h)

  have fI_is_real_differentiable : Differentiable ℝ (partialDeriv ℝ 1 f) := by
    exact (partialDeriv_contDiff ℝ f_is_real_C2 1).differentiable (Submonoid.oneLE.proof_2 ℕ∞)

  constructor
  · -- f is two times real continuously differentiable
    exact f_is_real_C2

  · -- Laplace of f is zero
    unfold Complex.laplace
    rw [CauchyRiemann₄ h]

    -- This lemma says that partial derivatives commute with complex scalar
    -- multiplication. This is a consequence of partialDeriv_compContLin once we
    -- note that complex scalar multiplication is continuous ℝ-linear.
    have : ∀ v, ∀ s : ℂ, ∀ g : ℂ → F₁, Differentiable ℝ g → partialDeriv ℝ v (s • g) = s • (partialDeriv ℝ v g) := by
      intro v s g hg

      -- Present scalar multiplication as a continuous ℝ-linear map. This is
      -- horrible, there must be better ways to do that.
      let sMuls : F₁ →L[ℝ] F₁ :=
      {
        toFun := fun x ↦ s • x
        map_add' := by exact fun x y => DistribSMul.smul_add s x y
        map_smul' := by exact fun m x => (smul_comm ((RingHom.id ℝ) m) s x).symm
        cont := continuous_const_smul s
      }

      -- Bring the goal into a form that is recognized by
      -- partialDeriv_compContLin.
      have : s • g = sMuls ∘ g := by rfl
      rw [this]

      rw [partialDeriv_compContLin ℝ hg]
      rfl

    rw [this]
    rw [partialDeriv_comm f_is_real_C2 Complex.I 1]
    rw [CauchyRiemann₄ h]
    rw [this]
    rw [← smul_assoc]
    simp

    -- Subgoals coming from the application of 'this'
    -- Differentiable ℝ (Real.partialDeriv 1 f)
    exact fI_is_real_differentiable
    -- Differentiable ℝ (Real.partialDeriv 1 f)
    exact fI_is_real_differentiable


theorem holomorphicOn_is_harmonicOn {f : ℂ → F₁} {s : Set ℂ} (hs : IsOpen s) (h : DifferentiableOn ℂ f s) :
  HarmonicOn f s := by

  -- f is real C²
  have f_is_real_C2 : ContDiffOn ℝ 2 f s :=
    ContDiffOn.restrict_scalars ℝ (DifferentiableOn.contDiffOn h hs)

  constructor
  · -- f is two times real continuously differentiable
    exact f_is_real_C2

  · -- Laplace of f is zero
    unfold Complex.laplace
    intro z hz
    simp
    have : partialDeriv ℝ Complex.I f =ᶠ[nhds z] Complex.I • partialDeriv ℝ 1 f := by
      unfold Filter.EventuallyEq
      unfold Filter.Eventually
      simp
      refine mem_nhds_iff.mpr ?_
      use s
      constructor
      · intro x hx
        simp
        apply CauchyRiemann₅
        apply DifferentiableOn.differentiableAt h
        exact IsOpen.mem_nhds hs hx
      · constructor
        · exact hs
        · exact hz
    rw [partialDeriv_eventuallyEq ℝ this Complex.I]
    rw [partialDeriv_smul'₂]

    simp
    rw [partialDeriv_commOn hs f_is_real_C2 Complex.I 1 z hz]

    have : partialDeriv ℝ Complex.I f =ᶠ[nhds z] Complex.I • partialDeriv ℝ 1 f := by
      unfold Filter.EventuallyEq
      unfold Filter.Eventually
      simp
      refine mem_nhds_iff.mpr ?_
      use s
      constructor
      · intro x hx
        simp
        apply CauchyRiemann₅
        apply DifferentiableOn.differentiableAt h
        exact IsOpen.mem_nhds hs hx
      · constructor
        · exact hs
        · exact hz
    rw [partialDeriv_eventuallyEq ℝ this 1]
    rw [partialDeriv_smul'₂]
    simp
    rw [← smul_assoc]
    simp


theorem re_of_holomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) :
  Harmonic (Complex.reCLM ∘ f) := by
  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 antiholomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) :
  Harmonic (Complex.conjCLE ∘ f) := by
  apply harmonic_iff_comp_CLE_is_harmonic.1
  exact holomorphic_is_harmonic h


theorem log_normSq_of_holomorphicOn_is_harmonicOn'
  {f : ℂ → ℂ}
  {s : Set ℂ}
  (hs : IsOpen s)
  (h₁ : DifferentiableOn ℂ f s)
  (h₂ : ∀ z ∈ s, f z ≠ 0)
  (h₃ : ∀ z ∈ s, f z ∈ Complex.slitPlane) :
  HarmonicOn (Real.log ∘ Complex.normSq ∘ f) s := by

  suffices hyp : HarmonicOn (⇑Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ f) s from
    (harmonicOn_comp_CLM_is_harmonicOn hs hyp : HarmonicOn (Complex.reCLM ∘ Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ f) s)

  suffices hyp : HarmonicOn (Complex.log ∘ (((starRingEnd ℂ) ∘ f) * f)) s from by
    have : Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ f = Complex.log ∘ (((starRingEnd ℂ) ∘ f) * f) := by
      funext z
      simp
      rw [Complex.ofReal_log (Complex.normSq_nonneg (f z))]
      rw [Complex.normSq_eq_conj_mul_self]
    rw [this]
    exact hyp


  -- Suffices to show Harmonic (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f + Complex.log ∘ f)
  -- THIS IS WHERE WE USE h₃
  have : ∀ z ∈ s, (Complex.log ∘ (⇑(starRingEnd ℂ) ∘ f * f)) z = (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f + Complex.log ∘ f) z := by
    intro z hz
    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 log_normSq_of_holomorphicOn_is_harmonicOn
  {f : ℂ → ℂ}
  {s : Set ℂ}
  (hs : IsOpen s)
  (h₁ : DifferentiableOn ℂ f s)
  (h₂ : ∀ z ∈ s, f z ≠ 0) :
  HarmonicOn (Real.log ∘ Complex.normSq ∘ f) s := by

  have slitPlaneLemma {z : ℂ} (hz : z ≠ 0) : z ∈ Complex.slitPlane ∨ -z ∈ Complex.slitPlane := by
    rw [Complex.mem_slitPlane_iff]
    rw [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

  let s₁ : Set ℂ := { z | f z ∈ Complex.slitPlane} ∩ s

  have hs₁ : IsOpen s₁ := by
    let A := DifferentiableOn.continuousOn h₁
    let B := continuousOn_iff'.1 A
    obtain ⟨u, hu₁, hu₂⟩ := B Complex.slitPlane Complex.isOpen_slitPlane
    have : u ∩ s = s₁ := by
      rw [← hu₂]
      tauto
    rw [← this]
    apply IsOpen.inter hu₁ hs

  have harm₁ : HarmonicOn (Real.log ∘ Complex.normSq ∘ f) s₁ := by
    apply log_normSq_of_holomorphicOn_is_harmonicOn'
    exact hs₁
    apply DifferentiableOn.mono h₁ (Set.inter_subset_right {z | f z ∈ Complex.slitPlane} s)
    -- ∀ z ∈ s₁, f z ≠ 0
    exact fun z hz ↦ h₂ z (Set.mem_of_mem_inter_right hz)
    -- ∀ z ∈ s₁, f z ∈ Complex.slitPlane
    intro z hz
    apply hz.1

  let s₂ : Set ℂ := { z | -f z ∈ Complex.slitPlane} ∩ s

  have h₁' : DifferentiableOn ℂ (-f) s := by
    rw [← differentiableOn_neg_iff]
    simp
    exact h₁

  have hs₂ : IsOpen s₂ := by
    let A := DifferentiableOn.continuousOn h₁'
    let B := continuousOn_iff'.1 A
    obtain ⟨u, hu₁, hu₂⟩ := B Complex.slitPlane Complex.isOpen_slitPlane
    have : u ∩ s = s₂ := by
      rw [← hu₂]
      tauto
    rw [← this]
    apply IsOpen.inter hu₁ hs

  have harm₂ : HarmonicOn (Real.log ∘ Complex.normSq ∘ (-f)) s₂ := by
    apply log_normSq_of_holomorphicOn_is_harmonicOn'
    exact hs₂
    apply DifferentiableOn.mono h₁' (Set.inter_subset_right {z | -f z ∈ Complex.slitPlane} s)
    -- ∀ z ∈ s₁, f z ≠ 0
    intro z hz
    simp
    exact h₂ z (Set.mem_of_mem_inter_right hz)
    -- ∀ z ∈ s₁, f z ∈ Complex.slitPlane
    intro z hz
    apply hz.1

  apply HarmonicOn_of_locally_HarmonicOn
  intro z hz
  by_cases hfz : f z ∈ Complex.slitPlane
  · use s₁
    constructor
    · exact hs₁
    · constructor
      · tauto
      · have : s₁ = s ∩ s₁ := by
          apply Set.right_eq_inter.mpr
          exact Set.inter_subset_right {z | f z ∈ Complex.slitPlane} s
        rw [← this]
        exact harm₁
  · use s₂
    constructor
    · exact hs₂
    · constructor
      · constructor
        · apply Or.resolve_left (slitPlaneLemma (h₂ z hz)) hfz
        · exact hz
      · have : s₂ = s ∩ s₂ := by
          apply Set.right_eq_inter.mpr
          exact Set.inter_subset_right {z | -f z ∈ Complex.slitPlane} s
        rw [← this]
        have : Real.log ∘ ⇑Complex.normSq ∘ f = Real.log ∘ ⇑Complex.normSq ∘ (-f) := by
          funext x
          simp
        rw [this]
        exact harm₂


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

  suffices hyp : Harmonic (⇑Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ f) from
    (harmonic_comp_CLM_is_harmonic hyp : Harmonic (Complex.reCLM ∘ Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ f))

  suffices hyp : Harmonic (Complex.log ∘ (((starRingEnd ℂ) ∘ f) * f)) from by
    have : Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ f = Complex.log ∘ (((starRingEnd ℂ) ∘ f) * f) := by
      funext z
      simp
      rw [Complex.ofReal_log (Complex.normSq_nonneg (f z))]
      rw [Complex.normSq_eq_conj_mul_self]
    rw [this]
    exact hyp


  -- Suffices to show Harmonic (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f + Complex.log ∘ f)
  -- THIS IS WHERE WE USE h₃
  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]

  apply harmonic_add_harmonic_is_harmonic
  have : 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)
  rw [this]
  rw [← harmonic_iff_comp_CLE_is_harmonic]

  repeat
    apply holomorphic_is_harmonic
    intro z
    apply DifferentiableAt.comp
    exact Complex.differentiableAt_log (h₃ z)
    exact h₁ z


theorem logabs_of_holomorphic_is_harmonic
  {f : ℂ → ℂ}
  (h₁ : Differentiable ℂ f)
  (h₂ : ∀ z, f z ≠ 0)
  (h₃ : ∀ z, f z ∈ Complex.slitPlane) :
  Harmonic (fun z ↦ Real.log ‖f z‖) := by

  -- Suffices: Harmonic (2⁻¹ • Real.log ∘ ⇑Complex.normSq ∘ f)
  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]

  -- Suffices: Harmonic (Real.log ∘ ⇑Complex.normSq ∘ f)
  apply (harmonic_iff_smul_const_is_harmonic (inv_ne_zero two_ne_zero)).1

  exact log_normSq_of_holomorphic_is_harmonic h₁ h₂ h₃