Theorems · Theorem · real analysis

HasFTaylorSeriesUpToOn.comp

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type u_3} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {G : Type u_4}
  [inst_5 : NormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G] {s : Set E} {t : Set F}
  {q : F → FormalMultilinearSeries 𝕜 F G} {p : E → FormalMultilinearSeries 𝕜 E F} {n : WithTop ℕ∞} {g : F → G}
  {f : E → F},
  HasFTaylorSeriesUpToOn n g q t →
    HasFTaylorSeriesUpToOn n f p s →
      Set.MapsTo f s t → HasFTaylorSeriesUpToOn n (g ∘ f) (fun x => (q (f x)).taylorComp (p x)) s

Faa di Bruno formula: If two functions g and f have Taylor series up to n given by q and p, then g ∘ f also has a Taylor series, given by q.taylorComp p.

Defined in

Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Cited by

2 results in Mathlib

Foundations

Depth 191 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace