Theorems · Theorem · complex analysis

analyticOrderAt_deriv_ge_iff

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] [CharZero 𝕜] [CompleteSpace E] {z₀ : 𝕜} {f : 𝕜 → E},
  AnalyticAt 𝕜 f z₀ → f z₀ = 0 → ∀ {n : ℕ}, ↑n ≤ analyticOrderAt (deriv f) z₀ ↔ ↑n + 1 ≤ analyticOrderAt f z₀

If an analytic function f vanishes at z₀, then the analytic order of its derivative at z₀ is at least n if and only if the analytic order of f at z₀ is at least n + 1.

Defined in

Mathlib.Analysis.Analytic.Order

Cited by

2 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceCharZeroCompleteSpace