Theorems · Theorem · several complex variables

HasFPowerSeriesOnBall.changeOrigin

∀ {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] [CompleteSpace F] {f : E → F}
  {p : FormalMultilinearSeries 𝕜 E F} {x y : E} {r : ENNReal},
  HasFPowerSeriesOnBall f p x r → ↑‖y‖₊ < r → HasFPowerSeriesOnBall f (p.changeOrigin y) (x + y) (r - ↑‖y‖₊)

If a function admits a power series expansion p on a ball B (x, r), then it also admits a power series on any subball of this ball (even with a different center), given by p.changeOrigin.

Defined in

Mathlib.Analysis.Analytic.ChangeOrigin

Cited by

2 results in Mathlib

Foundations

Depth 186 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpace