155 lines
4.1 KiB
Plaintext
155 lines
4.1 KiB
Plaintext
import Mathlib.Data.Fin.Tuple.Basic
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import Mathlib.Analysis.Complex.Basic
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import Mathlib.Analysis.Complex.TaylorSeries
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import Mathlib.Analysis.Calculus.LineDeriv.Basic
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import Mathlib.Analysis.Calculus.ContDiff.Defs
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import Mathlib.Analysis.Calculus.FDeriv.Basic
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import Mathlib.Analysis.Calculus.FDeriv.Symmetric
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import Nevanlinna.cauchyRiemann
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noncomputable def Real.partialDeriv : ℂ → (ℂ → ℂ) → (ℂ → ℂ) := by
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intro v
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intro f
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exact fun w ↦ (fderiv ℝ f w) v
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theorem CauchyRiemann₄ {f : ℂ → ℂ} : (Differentiable ℂ f)
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→ Real.partialDeriv Complex.I f = Complex.I • Real.partialDeriv 1 f := by
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intro h
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unfold Real.partialDeriv
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conv =>
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left
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intro w
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rw [DifferentiableAt.fderiv_restrictScalars ℝ (h w)]
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simp
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rw [← mul_one Complex.I]
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rw [← smul_eq_mul]
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rw [ContinuousLinearMap.map_smul_of_tower (fderiv ℂ f w) Complex.I 1]
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conv =>
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right
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right
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intro w
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rw [DifferentiableAt.fderiv_restrictScalars ℝ (h w)]
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theorem partialDeriv_smul {f : ℂ → ℂ } {a v : ℂ } : Real.partialDeriv v (a • f) = a • Real.partialDeriv v f := by
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unfold Real.partialDeriv
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have : a • f = fun y ↦ a • f y := by rfl
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conv =>
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left
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intro w
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rw [this]
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rw [fderiv_const_smul]
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sorry
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noncomputable def Complex.laplace : (ℂ → ℂ) → (ℂ → ℂ) := by
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intro f
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let fx := Real.partialDeriv 1 f
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let fxx := Real.partialDeriv 1 fx
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let fy := Real.partialDeriv Complex.I f
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let fyy := Real.partialDeriv Complex.I fy
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exact fxx + fyy
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def Harmonic (f : ℂ → ℂ) : Prop :=
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(ContDiff ℝ 2 f) ∧ (∀ z, Complex.laplace f z = 0)
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lemma derivSymm (f : ℂ → ℂ) (hf : ContDiff ℝ 2 f) :
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∀ z a b : ℂ, (fderiv ℝ (fun w => fderiv ℝ f w) z) a b = (fderiv ℝ (fun w => fderiv ℝ f w) z) b a := by
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intro z a b
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let f' := fderiv ℝ f
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have h₀ : ∀ y, HasFDerivAt f (f' y) y := by
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have h : Differentiable ℝ f := by
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exact (contDiff_succ_iff_fderiv.1 hf).left
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exact fun y => DifferentiableAt.hasFDerivAt (h y)
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let f'' := (fderiv ℝ f' z)
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have h₁ : HasFDerivAt f' f'' z := by
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apply DifferentiableAt.hasFDerivAt
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let A := (contDiff_succ_iff_fderiv.1 hf).right
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let B := (contDiff_succ_iff_fderiv.1 A).left
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simp at B
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exact B z
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let A := second_derivative_symmetric h₀ h₁ a b
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dsimp [f'', f'] at A
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apply A
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lemma l₂ {f : ℂ → ℂ} (hf : ContDiff ℝ 2 f) (z a b : ℂ) :
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fderiv ℝ (fderiv ℝ f) z b a = fderiv ℝ (fun w ↦ fderiv ℝ f w a) z b := by
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rw [fderiv_clm_apply]
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· simp
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· exact (contDiff_succ_iff_fderiv.1 hf).2.differentiable le_rfl z
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· simp
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theorem holomorphic_is_harmonic {f : ℂ → ℂ} (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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-- f' is real C¹
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have f'_is_real_C1 : ContDiff ℝ 1 (fderiv ℝ f) :=
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(contDiff_succ_iff_fderiv.1 f_is_real_C2).right
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-- f' is real differentiable
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have f'_is_differentiable : Differentiable ℝ (fderiv ℝ f) :=
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(contDiff_succ_iff_fderiv.1 f'_is_real_C1).left
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-- Partial derivative in direction 1
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let f_1 := fun w ↦ (fderiv ℝ f w) 1
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-- Partial derivative in direction I
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let f_I := fun w ↦ (fderiv ℝ f w) Complex.I
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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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intro z
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unfold Complex.laplace
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rw [CauchyRiemann₄ h]
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have t₁a : (fderiv ℝ (fun w ↦ Complex.I * (fderiv ℝ f w) 1) z)
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= Complex.I • (fderiv ℝ f_1 z) := by
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rw [fderiv_const_mul]
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fun_prop
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rw [t₁a]
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have t₂ : (fderiv ℝ f_1 z) Complex.I = (fderiv ℝ f_I z) 1 := by
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let B := l₂ f_is_real_C2 z Complex.I 1
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rw [← B]
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let A := derivSymm f f_is_real_C2 z 1 Complex.I
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rw [A]
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let C := l₂ f_is_real_C2 z 1 Complex.I
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rw [C]
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simp
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rw [t₂]
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conv =>
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left
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right
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arg 2
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arg 1
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arg 2
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intro z
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simp [f_I]
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rw [CauchyRiemann₁ (h z)]
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rw [t₁a]
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simp
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rw [← mul_assoc]
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simp
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