48 lines
1.5 KiB
Plaintext
48 lines
1.5 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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import Nevanlinna.partialDeriv
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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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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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have fI_is_real_differentiable : Differentiable ℝ (Real.partialDeriv 1 f) := by
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exact (partialDeriv_contDiff 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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rw [partialDeriv_smul fI_is_real_differentiable]
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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 [partialDeriv_smul fI_is_real_differentiable]
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rw [← smul_assoc]
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simp
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