Theorems · Theorem · complex analysis

natCast_le_analyticOrderAt

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {f : 𝕜 → E} {z₀ : 𝕜},
  AnalyticAt 𝕜 f z₀ →
    ∀ {n : ℕ}, ↑n ≤ analyticOrderAt f z₀ ↔ ∃ g, AnalyticAt 𝕜 g z₀ ∧ ∀ᶠ (z : 𝕜) in nhds z₀, f z = (z - z₀) ^ n • g z

Characterization of which natural numbers are ≤ hf.order. Useful for avoiding case splits, since it applies whether or not the order is ∞.

Defined in

Mathlib.Analysis.Analytic.Order

Cited by

2 results in Mathlib

Foundations

Depth 195 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpace