nevanlinna/Nevanlinna/holomorphic_JensenFormula.lean

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import Nevanlinna.harmonicAt_examples
import Nevanlinna.harmonicAt_meanValue
import Nevanlinna.specialFunctions_CircleIntegral_affine
theorem jensen_case_R_eq_one
(f : )
(h₁f : Differentiable f)
(h₂f : f 0 ≠ 0)
(S : Finset )
(a : S → )
(ha : ∀ s, a s ∈ Metric.ball 0 1)
(F : )
(h₁F : Differentiable F)
(h₂F : ∀ z, F z ≠ 0)
(h₃F : f = fun z ↦ (F z) * ∏ s : S, (z - a s))
:
Real.log ‖f 0‖ = -∑ s, Real.log (‖a s‖⁻¹) + (2 * Real.pi)⁻¹ * ∫ (x : ) in (0)..2 * Real.pi, Real.log ‖f (circleMap 0 1 x)‖ := by
let logAbsF := fun w ↦ Real.log ‖F w‖
have t₀ : ∀ z ∈ Metric.ball 0 2, HarmonicAt logAbsF z := by
intro z _
apply logabs_of_holomorphicAt_is_harmonic
apply h₁F.holomorphicAt
exact h₂F z
have t₁ : (∫ (x : ) in (0)..2 * Real.pi, logAbsF (circleMap 0 1 x)) = 2 * Real.pi * logAbsF 0 := by
have hR : (0 : ) < (1 : ) := by apply Real.zero_lt_one
have hρ : (1 : ) < (2 : ) := by linarith
apply harmonic_meanValue 2 1 hR hρ t₀
have t₂ : ∀ s, f (a s) = 0 := by
intro s
rw [h₃F]
simp
right
apply Finset.prod_eq_zero_iff.2
use s
simp
let logAbsf := fun w ↦ Real.log ‖f w‖
have s₀ : ∀ z, f z ≠ 0 → logAbsf z = logAbsF z + ∑ s, Real.log ‖z - a s‖ := by
intro z hz
dsimp [logAbsf]
rw [h₃F]
simp_rw [Complex.abs.map_mul]
rw [Complex.abs_prod]
rw [Real.log_mul]
rw [Real.log_prod]
rfl
intro s hs
simp
by_contra ha'
rw [ha'] at hz
exact hz (t₂ s)
-- Complex.abs (F z) ≠ 0
simp
exact h₂F z
-- ∏ I : { x // x ∈ S }, Complex.abs (z - a I) ≠ 0
by_contra h'
obtain ⟨s, h's, h''⟩ := Finset.prod_eq_zero_iff.1 h'
simp at h''
rw [h''] at hz
let A := t₂ s
exact hz A
have s₁ : ∀ z, f z ≠ 0 → logAbsF z = logAbsf z - ∑ s, Real.log ‖z - a s‖ := by
intro z hz
rw [s₀ z hz]
simp
rw [s₁ 0 h₂f] at t₁
have h₀ {x : } : f (circleMap 0 1 x) ≠ 0 := by
rw [h₃F]
simp
constructor
· exact h₂F (circleMap 0 1 x)
· by_contra h'
obtain ⟨s, _, h₂s⟩ := Finset.prod_eq_zero_iff.1 h'
have : circleMap 0 1 x = a s := by
rw [← sub_zero (circleMap 0 1 x)]
nth_rw 2 [← h₂s]
simp
let A := ha s
rw [← this] at A
simp at A
simp_rw [s₁ (circleMap 0 1 _) h₀] at t₁
rw [intervalIntegral.integral_sub] at t₁
rw [intervalIntegral.integral_finset_sum] at t₁
simp_rw [int₀ (ha _)] at t₁
simp at t₁
rw [t₁]
simp
have {w : } : Real.pi⁻¹ * 2⁻¹ * (2 * Real.pi * w) = w := by
ring_nf
simp [mul_inv_cancel Real.pi_ne_zero]
rw [this]
simp
rfl
-- ∀ i ∈ Finset.univ, IntervalIntegrable (fun x => Real.log ‖circleMap 0 1 x - a i‖) MeasureTheory.volume 0 (2 * Real.pi)
intro i _
apply Continuous.intervalIntegrable
apply continuous_iff_continuousAt.2
intro x
have : (fun x => Real.log ‖circleMap 0 1 x - a i‖) = Real.log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 1 x - a i) :=
rfl
rw [this]
apply ContinuousAt.comp
apply Real.continuousAt_log
simp
by_contra ha'
let A := ha i
rw [← ha'] at A
simp at A
apply ContinuousAt.comp
apply Complex.continuous_abs.continuousAt
fun_prop
-- IntervalIntegrable (fun x => logAbsf (circleMap 0 1 x)) MeasureTheory.volume 0 (2 * Real.pi)
apply Continuous.intervalIntegrable
apply continuous_iff_continuousAt.2
intro x
have : (fun x => logAbsf (circleMap 0 1 x)) = Real.log ∘ Complex.abs ∘ f ∘ (fun x ↦ circleMap 0 1 x) :=
rfl
rw [this]
apply ContinuousAt.comp
simp
exact h₀
apply ContinuousAt.comp
apply Complex.continuous_abs.continuousAt
apply ContinuousAt.comp
apply h₁f.continuous.continuousAt
let A := continuous_circleMap 0 1
apply A.continuousAt
-- IntervalIntegrable (fun x => ∑ s : { x // x ∈ S }, Real.log ‖circleMap 0 1 x - a s‖) MeasureTheory.volume 0 (2 * Real.pi)
apply Continuous.intervalIntegrable
apply continuous_finset_sum
intro i _
apply continuous_iff_continuousAt.2
intro x
have : (fun x => Real.log ‖circleMap 0 1 x - a i‖) = Real.log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 1 x - a i) :=
rfl
rw [this]
apply ContinuousAt.comp
apply Real.continuousAt_log
simp
by_contra ha'
let A := ha i
rw [← ha'] at A
simp at A
apply ContinuousAt.comp
apply Complex.continuous_abs.continuousAt
fun_prop