Theorems · Theorem · complex analysis

MeromorphicOn.log_norm_meromorphicTrailingCoeffAt_extract_zeros_poles

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {U : Set 𝕜} {x : 𝕜} {f g : 𝕜 → E} {D : 𝕜 → ℤ},
  Function.HasFiniteSupport D →
    x ∈ U →
      AccPt x (Filter.principal U) →
        MeromorphicAt f x →
          AnalyticAt 𝕜 g x →
            g x ≠ 0 →
              f =ᶠ[Filter.codiscreteWithin U] (∏ᶠ (u : 𝕜), (fun x => x - u) ^ D u) • g →
                Real.log ‖meromorphicTrailingCoeffAt f x‖ = ∑ᶠ (u : 𝕜), ↑(D u) * Real.log ‖x - u‖ + Real.log ‖g x‖

In the setting of MeromorphicOn.extract_zeros_poles, compute the log of the norm of the trailing coefficient of f in terms of divisor f U and g x.

Defined in

Mathlib.Analysis.Meromorphic.FactorizedRational

Cited by

1 results in Mathlib

Foundations

Depth 208 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpace