Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: c : ℂ
R : ℝ
f : ℂ → ℂ
hR : R ≠ 0
h₁f : MeromorphicOn f (closedBall c |R|)
⊢ circleAverage (fun x ↦ log ‖f x‖) c R =
    ∑ᶠ (u : ℂ), ↑((divisor f (closedBall c |R|)) u) * log (R * ‖c - u‖⁻¹) +
        ↑((divisor f (closedBall c |R|)) c) * log R +
      log ‖meromorphicTrailingCoeffAt f c‖

Tactic:
by
  -- Shorthand notation to keep line size in check
  let CB := closedBall c |R|
  by_cases h₂f : ∀ u ∈ CB, meromorphicOrderAt f u ≠ ⊤
  · have h₃f := (divisor f CB).finiteSupport (isCompact_closedBall c |R|)
    -- Extract zeros & poles and compute
    obtain ⟨g, h₁g, h₂g, h₃g⟩ := h₁f.extract_zeros_poles (by simp_all) h₃f
    calc circleAverage (log ‖f ·‖) c R
    _ = circleAverage ((∑ᶠ u, (divisor f CB u * log ‖· - u‖)) + (log ‖g ·‖)) c R := by
      have h₄g := extract_zeros_poles_log h₂g h₃g
      rw [circleAverage_congr_codiscreteWithin (codiscreteWithin_mono sphere_subset_closedBall h₄g)
        hR]
    _ = circleAverage (∑ᶠ u, (divisor f CB u * log ‖· - u‖)) c R + circleAverage (log ‖g ·‖) c R :=
      circleAverage_add (circleIntegrable_log_norm_factorizedRational (divisor f CB))
        ((h₁g.mono sphere_subset_closedBall).meromorphicOn.circleIntegrable_log_norm)
    _ = ∑ᶠ u, divisor f CB u * log R + log ‖g c‖ := by
      simp only [circleAverage_log_norm_factorizedRational, add_right_inj]
      rw [h₁g.circleAverage_log_norm_of_ne_zero]
      exact fun u hu ↦ h₂g ⟨u, hu⟩
    _ = ∑ᶠ u, divisor f CB u * log R
      + (log ‖meromorphicTrailingCoeffAt f c‖ - ∑ᶠ u, divisor f CB u * log ‖c - u‖) := by
      have t₀ : c ∈ CB := by simp [CB]
      have t₁ : AccPt c (𝓟 CB) := by
        apply accPt_iff_frequently_nhdsNE.mpr
        apply compl_notMem
        apply mem_nhdsWithin.mpr
        use ball c |R|
        simpa [hR] using! fun _ ⟨h, _⟩ ↦ ball_subset_closedBall h
      simp [MeromorphicOn.log_norm_meromorphicTrailingCoeffAt_extract_zeros_poles h₃f t₀ t₁
        (h₁f c t₀) (h₁g c t₀) (h₂g ⟨c, t₀⟩) h₃g]
    _ = ∑ᶠ u, divisor f CB u * log R - ∑ᶠ u, divisor f CB u * log ‖c - u‖
      + log ‖meromorphicTrailingCoeffAt f c‖ := by
      ring
    _ = (∑ᶠ u, divisor f CB u * (log R - log ‖c - u‖)) + log ‖meromorphicTrailingCoeffAt f c‖ := by
      rw [← finsum_sub_distrib]
      · simp_rw [← mul_sub]
      repeat apply h₃f.subset (fun _ ↦ (by simp_all))
    _ = ∑ᶠ u, divisor f CB u * log (R * ‖c - u‖⁻¹) + divisor f CB c * log R
      + log ‖meromorphicTrailingCoeffAt f c‖ := by
      rw [countingFunction_finsum_eq_finsum_add hR h₃f]
  · -- Trivial case: `f` vanishes on a codiscrete set
    have h₂f : ¬∀ (u : ↑(closedBall c |R|)), meromorphicOrderAt f ↑u ≠ ⊤ := by aesop
    rw [← h₁f.exists_meromorphicOrderAt_ne_top_iff_forall
      ⟨nonempty_closedBall.mpr (abs_nonneg R), (convex_closedBall c |R|).isPreconnected⟩] at h₂f
    push Not at h₂f
    have : divisor f CB = 0 := by
      ext x
      by_cases h : x ∈ CB
      <;> simp_all [CB]
    simp only [CB, this, Function.locallyFinsuppWithin.coe_zero, Pi.zero_apply, Int.cast_zero,
      zero_mul, finsum_zero, add_zero, zero_add]
    rw [MeromorphicAt.meromorphicTrailingCoeffAt_of_order_eq_top (by aesop), norm_zero, log_zero]
    have : f =ᶠ[codiscreteWithin CB] 0 := by
      filter_upwards [h₁f.meromorphicNFAt_mem_codiscreteWithin, self_mem_codiscreteWithin CB]
        with z h₁z h₂z
      simpa [h₂f ⟨z, h₂z⟩] using (not_iff_not.2 h₁z.meromorphicOrderAt_eq_zero_iff)
    rw [circleAverage_congr_codiscreteWithin (f₂ := 0) _ hR]
    · simp only [circleAverage, mul_inv_rev, Pi.zero_apply, intervalIntegral.integral_zero,
        smul_eq_mul, mul_zero]
    apply Filter.codiscreteWithin_mono (U := CB) sphere_subset_closedBall
    filter_upwards [this] with z hz
    simp_all

Goal state after: