Theorems · Theorem · complex analysis

ValueDistribution.logCounting_isBigO_log_iff_finite_support

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] [inst_1 : ProperSpace 𝕜] {E : Type u_2}
  [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace 𝕜 E] {f : 𝕜 → E},
  ValueDistribution.logCounting f ⊤ =O[Filter.atTop] Real.log ↔ (MeromorphicOn.divisor f Set.univ)⁻.support.Finite

A meromorphic function has a finite set of poles if and only if the logarithmic counting function for its pole-divisor is big-O of log.

Defined in: Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Asymptotic
Cited by: 0 results in Mathlib
Foundations: Depth 209 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: NontriviallyNormedField ProperSpace NormedAddCommGroup NormedSpace

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Realstatement and proof · cited by 25,697
NormedAddCommGroupstatement and proof · cited by 15,752
NormedSpacestatement and proof · cited by 12,499
Top.topstatement · cited by 9,680
NontriviallyNormedFieldstatement and proof · cited by 8,742
Set.univstatement and proof · cited by 3,945
WithTopstatement · cited by 3,754
Filter.atTopstatement and proof · cited by 2,405
Set.Finitestatement and proof · cited by 1,814
Real.logstatement and proof · cited by 939
Asymptotics.IsBigOstatement and proof · cited by 506
ProperSpacestatement and proof · cited by 190

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