57 lines
1.6 KiB
Plaintext
57 lines
1.6 KiB
Plaintext
import Mathlib.Analysis.Complex.CauchyIntegral
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open ComplexConjugate
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/- logAbs of a product is sum of logAbs of factors -/
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lemma logAbs_mul : ∀ z₁ z₂ : ℂ, z₁ ≠ 0 → z₂ ≠ 0 → Real.log (Complex.abs (z₁ * z₂)) = Real.log (Complex.abs z₁) + Real.log (Complex.abs z₂) := by
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intro z₁ z₂ z₁Hyp z₂Hyp
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rw [Complex.instNormedFieldComplex.proof_2 z₁ z₂]
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exact Real.log_mul ((AbsoluteValue.ne_zero_iff Complex.abs).mpr z₁Hyp) ((AbsoluteValue.ne_zero_iff Complex.abs).mpr z₂Hyp)
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lemma absAndProd : ∀ z : ℂ, Complex.abs z = Real.sqrt ( (z * conj z).re ) := by
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intro z
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simp
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rfl
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#check Complex.log_mul_eq_add_log_iff
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#check Complex.arg_eq_pi_iff
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lemma logAbsXX : ∀ z : ℂ, z ≠ 0 → Real.log (Complex.abs z) = (1 / 2) * Complex.log z + (1 / 2) * Complex.log (conj z) := by
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intro z z₁Hyp
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by_cases argHyp : Complex.arg z = Real.pi
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-- Show pos: Complex.arg z = Real.pi
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have : conj z = z := by
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apply Complex.conj_eq_iff_im.2
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rw [Complex.arg_eq_pi_iff] at argHyp
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exact argHyp.right
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rw [this]
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sorry
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-- Show pos: Complex.arg z ≠ Real.pi
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have t₁ : Complex.abs z = Real.sqrt (Complex.normSq z) := by
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exact rfl
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rw [t₁]
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have t₂ : 0 ≤ Complex.normSq z := by
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exact Complex.normSq_nonneg z
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rw [ Real.log_sqrt t₂ ]
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have t₃ : Real.log (Complex.normSq z) = Complex.log (Complex.normSq z) := by
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apply Complex.ofReal_log
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exact t₂
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simp
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rw [t₃]
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rw [Complex.normSq_eq_conj_mul_self]
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have t₄ : conj z ≠ 0 := by
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exact (AddEquivClass.map_ne_zero_iff starRingAut).mpr z₁Hyp
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let XX := Complex.log_mul_eq_add_log_iff this z₁Hyp
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sorry
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