nevanlinna/Nevanlinna/logabs.lean

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