Theorems · Theorem · several complex variables

HasFPowerSeriesWithinAt.isBigO_sub_partialSum_pow

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {f : E → F}
  {p : FormalMultilinearSeries 𝕜 E F} {s : Set E} {x : E},
  HasFPowerSeriesWithinAt f p s x →
    ∀ (n : ℕ),
      (fun y => f (x + y) - p.partialSum n y) =O[nhdsWithin 0 ((fun x_1 => x + x_1) ⁻¹' insert x s)] fun y => ‖y‖ ^ n

Taylor formula for an analytic function within a set, IsBigO version.

Defined in

Mathlib.Analysis.Analytic.Basic

Cited by

1 results in Mathlib

Foundations

Depth 173 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace