Update holomorphic_JensenFormula.lean

This commit is contained in:
Stefan Kebekus 2024-08-09 10:31:13 +02:00
parent 6ab6e6e6a9
commit da859defb1
1 changed files with 34 additions and 3 deletions

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@ -33,6 +33,12 @@ lemma int₀
(ha : a ∈ Metric.ball 0 1) :
∫ (x : ) in (0)..2 * Real.pi, Real.log ‖circleMap 0 1 x - a‖ = 0 := by
by_cases h₁a : a = 0
· -- case: a = 0
rw [h₁a]
simp
-- case: a ≠ 0
simp_rw [l₂]
have {x : } : Real.log ‖1 - circleMap 0 1 (-x) * a‖ = (fun w ↦ Real.log ‖1 - circleMap 0 1 (w) * a‖) (-x) := by rfl
conv =>
@ -58,12 +64,37 @@ lemma int₀
simp
dsimp [f₁]
let ρ := ‖a‖⁻¹
have hρ : 1 < ρ := by
apply one_lt_inv_iff.mpr
constructor
· exact norm_pos_iff'.mpr h₁a
· exact mem_ball_zero_iff.mp ha
let F := fun z ↦ Real.log ‖1 - z * a‖
have hf : ∀ x ∈ Metric.ball 0 2 , HarmonicAt F x := by
sorry
have hf : ∀ x ∈ Metric.ball 0 ρ, HarmonicAt F x := by
intro x hx
apply logabs_of_holomorphicAt_is_harmonic
apply Differentiable.holomorphicAt
fun_prop
apply sub_ne_zero_of_ne
by_contra h
have : ‖x * a‖ < 1 := by
calc ‖x * a‖
_ = ‖x‖ * ‖a‖ := by exact NormedField.norm_mul' x a
_ < ρ * ‖a‖ := by
apply (mul_lt_mul_right _).2
exact mem_ball_zero_iff.mp hx
exact norm_pos_iff'.mpr h₁a
_ = 1 := by
dsimp [ρ]
apply inv_mul_cancel
exact (AbsoluteValue.ne_zero_iff Complex.abs).mpr h₁a
rw [← h] at this
simp at this
let A := harmonic_meanValue 2 1 Real.zero_lt_one one_lt_two hf
let A := harmonic_meanValue ρ 1 Real.zero_lt_one hρ hf
dsimp [F] at A
simp at A
exact A