Theorems · Theorem · complex analysis

HasFPowerSeriesWithinOnBall.iteratedFDerivWithin_eq_sum_of_completeSpace

∀ {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] {f : E → F}
  {p : FormalMultilinearSeries 𝕜 E F} {s : Set E} {x : E} {r : ENNReal} [CompleteSpace F],
  HasFPowerSeriesWithinOnBall f p s x r →
    UniqueDiffOn 𝕜 s →
      x ∈ s → ∀ {n : ℕ} (v : Fin n → E), (iteratedFDerivWithin 𝕜 n f s x) v = ∑ σ, (p n) fun i => v (σ i)

If a function has a power series in a ball, then its n-th iterated derivative is given by (v₁, ..., vₙ) ↦ ∑ pₙ (v_{σ (1)}, ..., v_{σ (n)}) where the sum is over all permutations of {1, ..., n}.

Defined in

Mathlib.Analysis.Analytic.IteratedFDeriv

Cited by

1 results in Mathlib

Foundations

Depth 203 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpace