nevanlinna/Nevanlinna/complexHarmonic.examples.lean

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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 {F : Type*} [NormedAddCommGroup F] [NormedSpace F]
variable {F₁ : Type*} [NormedAddCommGroup F₁] [NormedSpace F₁] [CompleteSpace F₁]
variable {G : Type*} [NormedAddCommGroup G] [NormedSpace G]
variable {G₁ : Type*} [NormedAddCommGroup G₁] [NormedSpace G₁] [CompleteSpace G₁]
theorem holomorphic_is_harmonic {f : → F₁} (h : Differentiable f) :
Harmonic f := by
-- f is real C²
have f_is_real_C2 : ContDiff 2 f :=
ContDiff.restrict_scalars (Differentiable.contDiff h)
have fI_is_real_differentiable : Differentiable (partialDeriv 1 f) := by
exact (partialDeriv_contDiff 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 -linear.
have : ∀ v, ∀ s : , ∀ g : → F₁, Differentiable g → partialDeriv v (s • g) = s • (partialDeriv v g) := by
intro v s g hg
-- Present scalar multiplication as a continuous -linear map. This is
-- horrible, there must be better ways to do that.
let sMuls : F₁ →L[] 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 ) 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 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 (Real.partialDeriv 1 f)
exact fI_is_real_differentiable
-- Differentiable (Real.partialDeriv 1 f)
exact fI_is_real_differentiable
theorem holomorphicOn_is_harmonicOn {f : → F₁} {s : Set } (hs : IsOpen s) (h : DifferentiableOn f s) :
HarmonicOn f s := by
-- f is real C²
have f_is_real_C2 : ContDiffOn 2 f s :=
ContDiffOn.restrict_scalars (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 Complex.I f =ᶠ[nhds z] Complex.I • partialDeriv 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 this Complex.I]
rw [partialDeriv_smul'₂]
simp
rw [partialDeriv_commOn hs f_is_real_C2 Complex.I 1 z hz]
have : Complex.I • partialDeriv 1 (partialDeriv Complex.I f) z = Complex.I • (partialDeriv 1 (partialDeriv Complex.I f) z) := by
rfl
rw [this]
have : partialDeriv Complex.I f =ᶠ[nhds z] Complex.I • partialDeriv 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 this 1]
rw [partialDeriv_smul'₂]
simp
rw [← smul_assoc]
simp
theorem re_of_holomorphic_is_harmonic {f : } (h : Differentiable f) :
Harmonic (Complex.reCLM ∘ f) := by
apply harmonic_comp_CLM_is_harmonic
exact holomorphic_is_harmonic h
theorem im_of_holomorphic_is_harmonic {f : } (h : Differentiable f) :
Harmonic (Complex.imCLM ∘ f) := by
apply harmonic_comp_CLM_is_harmonic
exact holomorphic_is_harmonic h
theorem antiholomorphic_is_harmonic {f : } (h : Differentiable 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 : }
{s : Set }
(hs : IsOpen s)
(h₁ : DifferentiableOn 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 ) ∘ f) * f)) s from by
have : Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ f = Complex.log ∘ (((starRingEnd ) ∘ 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 ) ∘ f + Complex.log ∘ f)
-- THIS IS WHERE WE USE h₃
have : ∀ z ∈ s, (Complex.log ∘ (⇑(starRingEnd ) ∘ f * f)) z = (Complex.log ∘ ⇑(starRingEnd ) ∘ 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 ) (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 ) (f x))) = (Complex.log ∘ ⇑(starRingEnd ) ∘ f) := by
rfl
rw [this]
-- HarmonicOn (Complex.log ∘ ⇑(starRingEnd ) ∘ f) s
have : ∀ z ∈ s, (Complex.log ∘ ⇑(starRingEnd ) ∘ 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 ) ∘ f) s
apply holomorphicOn_is_harmonicOn hs
exact DifferentiableOn.clog h₁ h₃
theorem log_normSq_of_holomorphicOn_is_harmonicOn
{f : }
{s : Set }
(hs : IsOpen s)
(h₁ : DifferentiableOn f s)
(h₂ : ∀ z ∈ s, f z ≠ 0) :
HarmonicOn (Real.log ∘ Complex.normSq ∘ f) s := by
have slitPlaneLemma {z : } (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 := { 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 := { z | -f z ∈ Complex.slitPlane} ∩ s
have h₁' : DifferentiableOn (-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 : }
(h₁ : Differentiable 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 ) ∘ f) * f)) from by
have : Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ f = Complex.log ∘ (((starRingEnd ) ∘ 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 ) ∘ f + Complex.log ∘ f)
-- THIS IS WHERE WE USE h₃
have : Complex.log ∘ (⇑(starRingEnd ) ∘ f * f) = Complex.log ∘ ⇑(starRingEnd ) ∘ f + Complex.log ∘ f := by
unfold Function.comp
funext z
simp
rw [Complex.log_mul_eq_add_log_iff]
have : Complex.arg ((starRingEnd ) (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 ) ∘ 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 : }
(h₁ : Differentiable 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 : )⁻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