working
This commit is contained in:
Stefan Kebekus
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5d5d7557c0
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import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
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import Nevanlinna.complexHarmonic
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import Nevanlinna.holomorphic
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variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
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variable {F₁ : Type*} [NormedAddCommGroup F₁] [NormedSpace ℂ F₁] [CompleteSpace F₁]
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variable {G : Type*} [NormedAddCommGroup G] [NormedSpace ℝ G]
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theorem holomorphicAt_is_harmonicAt
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{f : ℂ → F₁}
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{z : ℂ}
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(hf : HolomorphicAt f z) :
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HarmonicAt f z := by
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let t := {x | HolomorphicAt f x}
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have ht : IsOpen t := HolomorphicAt_isOpen f
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have hz : z ∈ t := by exact hf
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constructor
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· -- ContDiffAt ℝ 2 f z
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exact HolomorphicAt_contDiffAt hf
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· -- Δ f =ᶠ[nhds z] 0
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apply Filter.eventuallyEq_iff_exists_mem.2
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use t
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constructor
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· exact IsOpen.mem_nhds ht hz
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· intro w hw
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unfold Complex.laplace
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simp
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rw [partialDeriv_eventuallyEq ℝ (CauchyRiemann'₆ hw) Complex.I]
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rw [partialDeriv_smul'₂]
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simp
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rw [partialDeriv_commAt (HolomorphicAt_contDiffAt hw) Complex.I 1]
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rw [partialDeriv_eventuallyEq ℝ (CauchyRiemann'₆ hw) 1]
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rw [partialDeriv_smul'₂]
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simp
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rw [← smul_assoc]
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simp
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theorem re_of_holomorphicAt_is_harmonicAr
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{f : ℂ → ℂ}
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{z : ℂ}
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(h : HolomorphicAt f z) :
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HarmonicAt (Complex.reCLM ∘ f) z := by
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apply harmonicAt_comp_CLM_is_harmonicAt
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exact holomorphicAt_is_harmonicAt h
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theorem im_of_holomorphicAt_is_harmonicAt
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{f : ℂ → ℂ}
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{z : ℂ}
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(h : HolomorphicAt f z) :
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HarmonicAt (Complex.imCLM ∘ f) z := by
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apply harmonicAt_comp_CLM_is_harmonicAt
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exact holomorphicAt_is_harmonicAt h
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theorem antiholomorphicAt_is_harmonicAt
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{f : ℂ → ℂ}
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{z : ℂ}
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(h : HolomorphicAt f z) :
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HarmonicAt (Complex.conjCLE ∘ f) z := by
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apply harmonicAt_iff_comp_CLE_is_harmonicAt.1
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exact holomorphicAt_is_harmonicAt h
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theorem log_normSq_of_holomorphicAt_is_harmonicAt
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{f : ℂ → ℂ}
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{z : ℂ}
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(h₁f : HolomorphicAt f z)
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(h₂f : f z ≠ 0) :
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HarmonicAt (Real.log ∘ Complex.normSq ∘ f) z := by
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-- For later use
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have slitPlaneLemma {z : ℂ} (hz : z ≠ 0) : z ∈ Complex.slitPlane ∨ -z ∈ Complex.slitPlane := by
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rw [Complex.mem_slitPlane_iff, Complex.mem_slitPlane_iff]
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simp at hz
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rw [Complex.ext_iff] at hz
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push_neg at hz
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simp at hz
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simp
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by_contra contra
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push_neg at contra
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exact hz (le_antisymm contra.1.1 contra.2.1) contra.1.2
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-- First prove the theorem for functions with image in the slitPlane
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have lem₁ : ∀ g : ℂ → ℂ, (HolomorphicAt g z) → (g z ≠ 0) → (g z ∈ Complex.slitPlane) → HarmonicAt (Real.log ∘ Complex.normSq ∘ g) z := by
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intro g h₁g h₂g h₃g
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-- Rewrite the log |g|² as Complex.log (g * gc)
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suffices hyp : HarmonicAt (Complex.log ∘ ((Complex.conjCLE ∘ g) * g)) z from by
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have : Real.log ∘ Complex.normSq ∘ g = Complex.reCLM ∘ Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ g := by
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funext x
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simp
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rw [this]
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have : Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ g = Complex.log ∘ ((Complex.conjCLE ∘ g) * g) := by
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funext x
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simp
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rw [Complex.ofReal_log]
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rw [Complex.normSq_eq_conj_mul_self]
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exact Complex.normSq_nonneg (g x)
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rw [← this] at hyp
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apply harmonicAt_comp_CLM_is_harmonicAt hyp
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-- Locally around z, rewrite Complex.log (g * gc) as Complex.log g + Complex.log.gc
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-- This uses the assumption that g z is in Complex.slitPlane
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have : (Complex.log ∘ (Complex.conjCLE ∘ g * g)) =ᶠ[nhds z] (Complex.log ∘ Complex.conjCLE ∘ g + Complex.log ∘ g) := by
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apply Filter.eventuallyEq_iff_exists_mem.2
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use g⁻¹' (Complex.slitPlane ∩ {0}ᶜ)
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constructor
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· apply ContinuousAt.preimage_mem_nhds
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· exact (HolomorphicAt_differentiableAt h₁g).continuousAt
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· apply IsOpen.mem_nhds
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apply IsOpen.inter Complex.isOpen_slitPlane isOpen_ne
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constructor
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· exact h₃g
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· exact h₂g
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· intro x hx
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simp
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rw [Complex.log_mul_eq_add_log_iff _ hx.2]
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rw [Complex.arg_conj]
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simp [Complex.slitPlane_arg_ne_pi hx.1]
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constructor
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· exact Real.pi_pos
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· exact Real.pi_nonneg
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simp
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apply hx.2
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-- Locally around z, rewrite Complex.log (g * gc) as Complex.log g + Complex.log.gc
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-- This uses the assumption that g z is in Complex.slitPlane
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have : (Complex.log ∘ (Complex.conjCLE ∘ g * g)) =ᶠ[nhds z] (Complex.conjCLE ∘ Complex.log ∘ g + Complex.log ∘ g) := by
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apply Filter.eventuallyEq_iff_exists_mem.2
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use g⁻¹' (Complex.slitPlane ∩ {0}ᶜ)
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constructor
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· apply ContinuousAt.preimage_mem_nhds
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· exact (HolomorphicAt_differentiableAt h₁g).continuousAt
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· apply IsOpen.mem_nhds
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apply IsOpen.inter Complex.isOpen_slitPlane isOpen_ne
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constructor
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· exact h₃g
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· exact h₂g
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· intro x hx
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simp
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rw [← Complex.log_conj]
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rw [Complex.log_mul_eq_add_log_iff _ hx.2]
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rw [Complex.arg_conj]
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simp [Complex.slitPlane_arg_ne_pi hx.1]
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constructor
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· exact Real.pi_pos
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· exact Real.pi_nonneg
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simp
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apply hx.2
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apply Complex.slitPlane_arg_ne_pi hx.1
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rw [HarmonicAt_eventuallyEq this]
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apply harmonicAt_add_harmonicAt_is_harmonicAt
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· rw [← harmonicAt_iff_comp_CLE_is_harmonicAt]
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apply holomorphicAt_is_harmonicAt
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apply HolomorphicAt_comp
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use Complex.slitPlane
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constructor
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· apply IsOpen.mem_nhds
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exact Complex.isOpen_slitPlane
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assumption
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· exact fun z a => Complex.differentiableAt_log a
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exact h₁g
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· apply holomorphicAt_is_harmonicAt
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apply HolomorphicAt_comp
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use Complex.slitPlane
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constructor
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· apply IsOpen.mem_nhds
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exact Complex.isOpen_slitPlane
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assumption
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· exact fun z a => Complex.differentiableAt_log a
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exact h₁g
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by_cases h₃f : f z ∈ Complex.slitPlane
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· exact lem₁ f h₁f h₂f h₃f
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· have : Complex.normSq ∘ f = Complex.normSq ∘ (-f) := by funext; simp
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rw [this]
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apply lem₁ (-f)
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· exact HolomorphicAt_neg h₁f
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· simpa
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· exact (slitPlaneLemma h₂f).resolve_left h₃f
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theorem holomorphic_is_harmonic {f : ℂ → F₁} (h : Differentiable ℂ f) :
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Harmonic f := by
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-- f is real C²
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have f_is_real_C2 : ContDiff ℝ 2 f :=
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ContDiff.restrict_scalars ℝ (Differentiable.contDiff h)
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have fI_is_real_differentiable : Differentiable ℝ (partialDeriv ℝ 1 f) := by
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exact (partialDeriv_contDiff ℝ f_is_real_C2 1).differentiable (Submonoid.oneLE.proof_2 ℕ∞)
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constructor
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· -- f is two times real continuously differentiable
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exact f_is_real_C2
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· -- Laplace of f is zero
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unfold Complex.laplace
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rw [CauchyRiemann₄ h]
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-- This lemma says that partial derivatives commute with complex scalar
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-- multiplication. This is a consequence of partialDeriv_compContLin once we
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-- note that complex scalar multiplication is continuous ℝ-linear.
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have : ∀ v, ∀ s : ℂ, ∀ g : ℂ → F₁, Differentiable ℝ g → partialDeriv ℝ v (s • g) = s • (partialDeriv ℝ v g) := by
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intro v s g hg
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-- Present scalar multiplication as a continuous ℝ-linear map. This is
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-- horrible, there must be better ways to do that.
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let sMuls : F₁ →L[ℝ] F₁ :=
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{
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toFun := fun x ↦ s • x
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map_add' := by exact fun x y => DistribSMul.smul_add s x y
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map_smul' := by exact fun m x => (smul_comm ((RingHom.id ℝ) m) s x).symm
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cont := continuous_const_smul s
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}
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-- Bring the goal into a form that is recognized by
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-- partialDeriv_compContLin.
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have : s • g = sMuls ∘ g := by rfl
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rw [this]
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rw [partialDeriv_compContLin ℝ hg]
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rfl
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rw [this]
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rw [partialDeriv_comm f_is_real_C2 Complex.I 1]
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rw [CauchyRiemann₄ h]
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rw [this]
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rw [← smul_assoc]
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simp
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-- Subgoals coming from the application of 'this'
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-- Differentiable ℝ (Real.partialDeriv 1 f)
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exact fI_is_real_differentiable
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-- Differentiable ℝ (Real.partialDeriv 1 f)
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exact fI_is_real_differentiable
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theorem holomorphicOn_is_harmonicOn {f : ℂ → F₁} {s : Set ℂ} (hs : IsOpen s) (h : DifferentiableOn ℂ f s) :
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HarmonicOn f s := by
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-- f is real C²
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have f_is_real_C2 : ContDiffOn ℝ 2 f s :=
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ContDiffOn.restrict_scalars ℝ (DifferentiableOn.contDiffOn h hs)
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constructor
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· -- f is two times real continuously differentiable
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exact f_is_real_C2
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· -- Laplace of f is zero
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unfold Complex.laplace
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intro z hz
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simp
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have : partialDeriv ℝ Complex.I f =ᶠ[nhds z] Complex.I • partialDeriv ℝ 1 f := by
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unfold Filter.EventuallyEq
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unfold Filter.Eventually
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simp
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refine mem_nhds_iff.mpr ?_
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use s
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constructor
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· intro x hx
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simp
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apply CauchyRiemann₅
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apply DifferentiableOn.differentiableAt h
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exact IsOpen.mem_nhds hs hx
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· constructor
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· exact hs
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· exact hz
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rw [partialDeriv_eventuallyEq ℝ this Complex.I]
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rw [partialDeriv_smul'₂]
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simp
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rw [partialDeriv_commOn hs f_is_real_C2 Complex.I 1 z hz]
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have : partialDeriv ℝ Complex.I f =ᶠ[nhds z] Complex.I • partialDeriv ℝ 1 f := by
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unfold Filter.EventuallyEq
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unfold Filter.Eventually
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simp
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refine mem_nhds_iff.mpr ?_
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use s
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constructor
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· intro x hx
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simp
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apply CauchyRiemann₅
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apply DifferentiableOn.differentiableAt h
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exact IsOpen.mem_nhds hs hx
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· constructor
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· exact hs
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· exact hz
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rw [partialDeriv_eventuallyEq ℝ this 1]
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rw [partialDeriv_smul'₂]
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simp
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rw [← smul_assoc]
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simp
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theorem re_of_holomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) :
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Harmonic (Complex.reCLM ∘ f) := by
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apply harmonic_comp_CLM_is_harmonic
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exact holomorphic_is_harmonic h
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theorem im_of_holomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) :
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Harmonic (Complex.imCLM ∘ f) := by
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apply harmonic_comp_CLM_is_harmonic
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exact holomorphic_is_harmonic h
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theorem antiholomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) :
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Harmonic (Complex.conjCLE ∘ f) := by
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apply harmonic_iff_comp_CLE_is_harmonic.1
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exact holomorphic_is_harmonic h
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theorem log_normSq_of_holomorphicOn_is_harmonicOn'
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{f : ℂ → ℂ}
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{s : Set ℂ}
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(hs : IsOpen s)
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(h₁ : DifferentiableOn ℂ f s)
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(h₂ : ∀ z ∈ s, f z ≠ 0)
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(h₃ : ∀ z ∈ s, f z ∈ Complex.slitPlane) :
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HarmonicOn (Real.log ∘ Complex.normSq ∘ f) s := by
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suffices hyp : HarmonicOn (⇑Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ f) s from
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(harmonicOn_comp_CLM_is_harmonicOn hs hyp : HarmonicOn (Complex.reCLM ∘ Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ f) s)
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suffices hyp : HarmonicOn (Complex.log ∘ (((starRingEnd ℂ) ∘ f) * f)) s from by
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have : Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ f = Complex.log ∘ (((starRingEnd ℂ) ∘ f) * f) := by
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funext z
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simp
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rw [Complex.ofReal_log (Complex.normSq_nonneg (f z))]
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rw [Complex.normSq_eq_conj_mul_self]
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rw [this]
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exact hyp
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-- Suffices to show Harmonic (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f + Complex.log ∘ f)
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-- THIS IS WHERE WE USE h₃
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|
||||||
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
|
|
||||||
|
|
@ -163,6 +163,20 @@ theorem harmonic_smul_const_is_harmonic {f : ℂ → F} {c : ℝ} (h : Harmonic
|
||||||
simp
|
simp
|
||||||
|
|
||||||
|
|
||||||
|
theorem harmonicAt_smul_const_is_harmonicAt
|
||||||
|
{f : ℂ → F}
|
||||||
|
{x : ℂ}
|
||||||
|
{c : ℝ}
|
||||||
|
(h : HarmonicAt f x) :
|
||||||
|
HarmonicAt (c • f) x := by
|
||||||
|
constructor
|
||||||
|
· exact ContDiffAt.const_smul c h.1
|
||||||
|
· rw [laplace_smul]
|
||||||
|
have A := Filter.EventuallyEq.const_smul h.2 c
|
||||||
|
simp at A
|
||||||
|
assumption
|
||||||
|
|
||||||
|
|
||||||
theorem harmonic_iff_smul_const_is_harmonic {f : ℂ → F} {c : ℝ} (hc : c ≠ 0) :
|
theorem harmonic_iff_smul_const_is_harmonic {f : ℂ → F} {c : ℝ} (hc : c ≠ 0) :
|
||||||
Harmonic f ↔ Harmonic (c • f) := by
|
Harmonic f ↔ Harmonic (c • f) := by
|
||||||
constructor
|
constructor
|
||||||
|
|
@ -171,6 +185,18 @@ theorem harmonic_iff_smul_const_is_harmonic {f : ℂ → F} {c : ℝ} (hc : c
|
||||||
exact fun a => harmonic_smul_const_is_harmonic a
|
exact fun a => harmonic_smul_const_is_harmonic a
|
||||||
|
|
||||||
|
|
||||||
|
theorem harmonicAt_iff_smul_const_is_harmonicAt
|
||||||
|
{f : ℂ → F}
|
||||||
|
{x : ℂ}
|
||||||
|
{c : ℝ}
|
||||||
|
(hc : c ≠ 0) :
|
||||||
|
HarmonicAt f x ↔ HarmonicAt (c • f) x := by
|
||||||
|
constructor
|
||||||
|
· exact harmonicAt_smul_const_is_harmonicAt
|
||||||
|
· nth_rewrite 2 [((eq_inv_smul_iff₀ hc).mpr rfl : f = c⁻¹ • c • f)]
|
||||||
|
exact fun a => harmonicAt_smul_const_is_harmonicAt a
|
||||||
|
|
||||||
|
|
||||||
theorem harmonic_comp_CLM_is_harmonic {f : ℂ → F₁} {l : F₁ →L[ℝ] G} (h : Harmonic f) :
|
theorem harmonic_comp_CLM_is_harmonic {f : ℂ → F₁} {l : F₁ →L[ℝ] G} (h : Harmonic f) :
|
||||||
Harmonic (l ∘ f) := by
|
Harmonic (l ∘ f) := by
|
||||||
|
|
||||||
|
|
|
||||||
|
|
@ -0,0 +1,211 @@
|
||||||
|
import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
|
||||||
|
import Nevanlinna.complexHarmonic
|
||||||
|
import Nevanlinna.holomorphicAt
|
||||||
|
|
||||||
|
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F] [CompleteSpace F]
|
||||||
|
|
||||||
|
|
||||||
|
theorem holomorphicAt_is_harmonicAt
|
||||||
|
{f : ℂ → F}
|
||||||
|
{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 hf.contDiffAt
|
||||||
|
|
||||||
|
· -- 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 hw.CauchyRiemannAt Complex.I]
|
||||||
|
rw [partialDeriv_smul'₂]
|
||||||
|
simp
|
||||||
|
|
||||||
|
rw [partialDeriv_commAt hw.contDiffAt Complex.I 1]
|
||||||
|
rw [partialDeriv_eventuallyEq ℝ hw.CauchyRiemannAt 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 h₁g.differentiableAt.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 h₁g.differentiableAt.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 logabs_of_holomorphicAt_is_harmonic
|
||||||
|
{f : ℂ → ℂ}
|
||||||
|
{z : ℂ}
|
||||||
|
(h₁f : HolomorphicAt f z)
|
||||||
|
(h₂f : f z ≠ 0) :
|
||||||
|
HarmonicAt (fun w ↦ Real.log ‖f w‖) 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 (harmonicAt_iff_smul_const_is_harmonicAt (inv_ne_zero two_ne_zero)).1
|
||||||
|
exact log_normSq_of_holomorphicAt_is_harmonicAt h₁f h₂f
|
||||||
|
|
@ -0,0 +1,159 @@
|
||||||
|
import Mathlib.Analysis.Complex.TaylorSeries
|
||||||
|
import Nevanlinna.cauchyRiemann
|
||||||
|
|
||||||
|
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
|
||||||
|
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F]
|
||||||
|
variable {G : Type*} [NormedAddCommGroup G] [NormedSpace ℂ G]
|
||||||
|
|
||||||
|
def HolomorphicAt (f : E → F) (x : E) : Prop :=
|
||||||
|
∃ s ∈ nhds x, ∀ z ∈ s, DifferentiableAt ℂ f z
|
||||||
|
|
||||||
|
|
||||||
|
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
|
||||||
|
constructor
|
||||||
|
· intro hf
|
||||||
|
obtain ⟨t, h₁t, h₂t⟩ := hf
|
||||||
|
obtain ⟨s, h₁s, h₂s, h₃s⟩ := mem_nhds_iff.1 h₁t
|
||||||
|
use s
|
||||||
|
constructor
|
||||||
|
· assumption
|
||||||
|
· constructor
|
||||||
|
· assumption
|
||||||
|
· intro z hz
|
||||||
|
exact h₂t z (h₁s hz)
|
||||||
|
· intro hyp
|
||||||
|
obtain ⟨s, h₁s, h₂s, hf⟩ := hyp
|
||||||
|
use s
|
||||||
|
constructor
|
||||||
|
· apply (IsOpen.mem_nhds_iff h₁s).2 h₂s
|
||||||
|
· assumption
|
||||||
|
|
||||||
|
|
||||||
|
theorem HolomorphicAt_isOpen
|
||||||
|
(f : E → F) :
|
||||||
|
IsOpen { x : E | HolomorphicAt f x } := by
|
||||||
|
|
||||||
|
rw [← subset_interior_iff_isOpen]
|
||||||
|
intro x hx
|
||||||
|
simp at hx
|
||||||
|
obtain ⟨s, h₁s, h₂s, h₃s⟩ := HolomorphicAt_iff.1 hx
|
||||||
|
use s
|
||||||
|
constructor
|
||||||
|
· simp
|
||||||
|
constructor
|
||||||
|
· exact h₁s
|
||||||
|
· intro x hx
|
||||||
|
simp
|
||||||
|
use s
|
||||||
|
constructor
|
||||||
|
· exact IsOpen.mem_nhds h₁s hx
|
||||||
|
· exact h₃s
|
||||||
|
· exact h₂s
|
||||||
|
|
||||||
|
|
||||||
|
theorem HolomorphicAt_comp
|
||||||
|
{g : E → F}
|
||||||
|
{f : F → G}
|
||||||
|
{z : E}
|
||||||
|
(hf : HolomorphicAt f (g z))
|
||||||
|
(hg : HolomorphicAt g z) :
|
||||||
|
HolomorphicAt (f ∘ g) z := by
|
||||||
|
obtain ⟨UE, h₁UE, h₂UE⟩ := hg
|
||||||
|
obtain ⟨UF, h₁UF, h₂UF⟩ := hf
|
||||||
|
use UE ∩ g⁻¹' UF
|
||||||
|
constructor
|
||||||
|
· simp
|
||||||
|
constructor
|
||||||
|
· assumption
|
||||||
|
· apply ContinuousAt.preimage_mem_nhds
|
||||||
|
apply (h₂UE z (mem_of_mem_nhds h₁UE)).continuousAt
|
||||||
|
assumption
|
||||||
|
· intro x hx
|
||||||
|
apply DifferentiableAt.comp
|
||||||
|
apply h₂UF
|
||||||
|
exact hx.2
|
||||||
|
apply h₂UE
|
||||||
|
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
|
||||||
|
[CompleteSpace F]
|
||||||
|
{f : ℂ → F}
|
||||||
|
{z : ℂ}
|
||||||
|
{n : ℕ}
|
||||||
|
(hf : HolomorphicAt f z) :
|
||||||
|
ContDiffAt ℝ n f z := by
|
||||||
|
|
||||||
|
let t := {x | HolomorphicAt f x}
|
||||||
|
have ht : IsOpen t := HolomorphicAt_isOpen f
|
||||||
|
have hz : z ∈ t := by exact hf
|
||||||
|
|
||||||
|
-- ContDiffAt ℝ _ f z
|
||||||
|
apply ContDiffOn.contDiffAt _ ((IsOpen.mem_nhds_iff ht).2 hz)
|
||||||
|
suffices h : ContDiffOn ℂ n f t from by
|
||||||
|
apply ContDiffOn.restrict_scalars ℝ h
|
||||||
|
apply DifferentiableOn.contDiffOn _ ht
|
||||||
|
intro w hw
|
||||||
|
apply DifferentiableAt.differentiableWithinAt
|
||||||
|
-- DifferentiableAt ℂ f w
|
||||||
|
let hfw : HolomorphicAt f w := hw
|
||||||
|
obtain ⟨s, _, h₂s, h₃s⟩ := HolomorphicAt_iff.1 hfw
|
||||||
|
exact h₃s w h₂s
|
||||||
|
|
||||||
|
|
||||||
|
theorem HolomorphicAt.differentiableAt
|
||||||
|
{f : ℂ → F}
|
||||||
|
{z : ℂ}
|
||||||
|
(hf : HolomorphicAt f z) :
|
||||||
|
DifferentiableAt ℂ f z := by
|
||||||
|
obtain ⟨s, _, h₂s, h₃s⟩ := HolomorphicAt_iff.1 hf
|
||||||
|
exact h₃s z h₂s
|
||||||
|
|
||||||
|
|
||||||
|
theorem HolomorphicAt.CauchyRiemannAt
|
||||||
|
{f : ℂ → F}
|
||||||
|
{z : ℂ}
|
||||||
|
(h : HolomorphicAt f z) :
|
||||||
|
partialDeriv ℝ Complex.I f =ᶠ[nhds z] Complex.I • partialDeriv ℝ 1 f := by
|
||||||
|
|
||||||
|
obtain ⟨s, h₁s, hz, h₂f⟩ := HolomorphicAt_iff.1 h
|
||||||
|
|
||||||
|
apply Filter.eventuallyEq_iff_exists_mem.2
|
||||||
|
use s
|
||||||
|
constructor
|
||||||
|
· exact IsOpen.mem_nhds h₁s hz
|
||||||
|
· intro w hw
|
||||||
|
let h := h₂f w hw
|
||||||
|
-- WARNING This should go to partialDeriv
|
||||||
|
unfold partialDeriv
|
||||||
|
simp
|
||||||
|
conv =>
|
||||||
|
left
|
||||||
|
rw [DifferentiableAt.fderiv_restrictScalars ℝ h]
|
||||||
|
simp
|
||||||
|
rw [← mul_one Complex.I]
|
||||||
|
rw [← smul_eq_mul]
|
||||||
|
rw [ContinuousLinearMap.map_smul_of_tower (fderiv ℂ f w) Complex.I 1]
|
||||||
|
conv =>
|
||||||
|
right
|
||||||
|
right
|
||||||
|
rw [DifferentiableAt.fderiv_restrictScalars ℝ h]
|
||||||
Loading…
Reference in New Issue