Theorems · Theorem · global analysis

HasFPowerSeriesOnBall.factorial_smul

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type v} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F]
  {p : FormalMultilinearSeries 𝕜 E F} {f : E → F} {x : E} {r : ENNReal},
  HasFPowerSeriesOnBall f p x r →
    ∀ (y : E) [CompleteSpace F] (n : ℕ), (n.factorial • (p n) fun x => y) = (iteratedFDeriv 𝕜 n f x) fun x => y

The iterated derivative of an analytic function, on vectors (y, ..., y), is given by n! times the n-th term in the power series. For a more general result giving the full iterated derivative as a sum over the permutations of Fin n, see HasFPowerSeriesOnBall.iteratedFDeriv_eq_sum.

Defined in

Mathlib.Analysis.Calculus.FDeriv.Analytic

Cited by

3 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpace