Theorems · Theorem · complex analysis

MeromorphicAt.meromorphicTrailingCoeffAt_add_eq_add

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {x : 𝕜} {f₁ f₂ : 𝕜 → E},
  MeromorphicAt f₁ x →
    MeromorphicAt f₂ x →
      meromorphicOrderAt f₁ x = meromorphicOrderAt f₂ x →
        meromorphicTrailingCoeffAt f₁ x + meromorphicTrailingCoeffAt f₂ x ≠ 0 →
          meromorphicTrailingCoeffAt (f₁ + f₂) x = meromorphicTrailingCoeffAt f₁ x + meromorphicTrailingCoeffAt f₂ x

If f₁ and f₂ have equal order at x and if their trailing coefficients do not cancel, then the trailing coefficient of f₁ + f₂ at x is the sum of the trailing coefficients.

Defined in

Mathlib.Analysis.Meromorphic.TrailingCoefficient

Cited by

2 results in Mathlib

Foundations

Depth 204 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpace