64 lines
1.6 KiB
Plaintext
64 lines
1.6 KiB
Plaintext
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import Mathlib.Analysis.Complex.Basic
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import Mathlib.Analysis.Calculus.ContDiff.Defs
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import Nevanlinna.partialDeriv
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variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
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variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F]
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def HolomorphicAt (f : E → F) (x : E) : Prop :=
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∃ s ∈ nhds x, ∀ z ∈ s, DifferentiableAt ℂ f z
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theorem HolomorphicAt_iff
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{f : E → F}
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{x : E} :
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HolomorphicAt f x ↔ ∃ s : Set E, IsOpen s ∧ x ∈ s ∧ (∀ z ∈ s, DifferentiableAt ℂ f z) := by
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constructor
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· intro hf
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obtain ⟨t, h₁t, h₂t⟩ := hf
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obtain ⟨s, h₁s, h₂s, h₃s⟩ := mem_nhds_iff.1 h₁t
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use s
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constructor
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· assumption
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· constructor
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· assumption
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· intro z hz
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exact h₂t z (h₁s hz)
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· intro hyp
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obtain ⟨s, h₁s, h₂s, hf⟩ := hyp
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use s
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constructor
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· apply (IsOpen.mem_nhds_iff h₁s).2 h₂s
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· assumption
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theorem CauchyRiemann₅
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{f : ℂ → F}
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{z : ℂ} :
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(DifferentiableAt ℂ f z) → partialDeriv ℝ Complex.I f z = Complex.I • partialDeriv ℝ 1 f z := by
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intro h
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unfold partialDeriv
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conv =>
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left
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rw [DifferentiableAt.fderiv_restrictScalars ℝ h]
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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 z) Complex.I 1]
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conv =>
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right
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right
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rw [DifferentiableAt.fderiv_restrictScalars ℝ h]
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theorem CauchyRiemann₆
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{f : ℂ → ℂ}
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{z : ℂ} :
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(HolomorphicAt f z) → partialDeriv ℝ Complex.I f =ᶠ[nhds z] Complex.I • partialDeriv ℝ 1 f := by
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sorry
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