Theorems · Theorem · complex analysis

AnalyticOn.hasFPowerSeriesOnSubball

∀ {𝕜 : Type u_1} [inst : RCLike 𝕜] {f : 𝕜 → 𝕜} {x : 𝕜} {r : ENNReal},
  0 < r →
    AnalyticOn 𝕜 f (Metric.eball x r) →
      r ≤ (FormalMultilinearSeries.ofScalars 𝕜 fun n => iteratedDeriv n f x / ↑n.factorial).radius →
        HasFPowerSeriesOnBall f (FormalMultilinearSeries.ofScalars 𝕜 fun n => iteratedDeriv n f x / ↑n.factorial) x r

If f is analytic on Bᵣ(x₀) and its Taylor series converges on this ball, then it converges to f.

Defined in

Mathlib.Analysis.Calculus.IteratedDeriv.ConvergenceOnBall

Cited by

2 results in Mathlib

Foundations

Depth 193 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLike