Theorems · Theorem · complex analysis

meromorphicOrderAt_comp_of_deriv_ne_zero

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {x : 𝕜} {f : 𝕜 → E} {g : 𝕜 → 𝕜},
  AnalyticAt 𝕜 g x →
    deriv g x ≠ 0 → ∀ [CompleteSpace 𝕜] [CharZero 𝕜], meromorphicOrderAt (f ∘ g) x = meromorphicOrderAt f (g x)

If g is analytic at x, and g' x ≠ 0, then the meromorphic order of f ∘ g at x is the meromorphic order of f at g x (even if f is not meromorphic).

Defined in

Mathlib.Analysis.Meromorphic.Order

Cited by

2 results in Mathlib

Foundations

Depth 201 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceCompleteSpaceCharZero