Theorems · Theorem · complex analysis
AnalyticAt.analyticOrderAt_eq_zero
∀ {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 → (analyticOrderAt f z₀ = 0 ↔ f z₀ ≠ 0)The order of an analytic function f at z₀ is zero iff f does not vanish at z₀.
- Defined in
- Mathlib.Analysis.Analytic.Order
- Cited by
- 13 results in Mathlib
- Foundations
- Depth 198 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- NontriviallyNormedFieldstatement and proof · cited by 8,742
- ENatstatement · cited by 4,985
- AnalyticAtstatement and proof · cited by 321
- analyticOrderAtstatement · cited by 69
Cited by13
Results whose statement or proof uses this declaration.
- MeromorphicNFAt.meromorphicOrderAt_eq_zero_iffproof · cited by 14
- AnalyticAt.analyticOrderAt_ne_zeroproof · cited by 2
- AnalyticOnNhd.isClopen_setOfPred_analyticOrderAt_eq_topproof · cited by 2
- MeromorphicOn.codiscrete_setOfPred_meromorphicOrderAt_eq_zero_or_topproof · cited by 2
- MeromorphicAt.meromorphicOrderAt_compproof · cited by 2
- AnalyticAt.analyticOrderAt_deriv_add_oneproof · cited by 2
- AnalyticOnNhd.circleAverage_log_normproof · cited by 1
- AnalyticOnNhd.codiscreteWithin_setOfPred_analyticOrderAt_eq_zero_or_topproof · cited by 1
- AnalyticOnNhd.codiscrete_setOfPred_analyticOrderAt_eq_zero_or_topproof · cited by 1
- MeromorphicAt.MeromorphicOn.codiscreteWithin_setOfPred_ne_zeroproof · cited by 1
- meromorphicOrderAt_smul_of_ne_zeroproof · cited by 1
- analyticOrderAt_eq_nat_iff_iteratedDeriv_eq_zeroproof · cited by 0