Theorems · Theorem · complex analysis

MeromorphicOn.divisor_prod

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {U : Set 𝕜} {ι : Type u_3} {s : Finset ι} {f : ι → 𝕜 → 𝕜},
  (∀ i ∈ s, MeromorphicOn (f i) U) →
    (∀ i ∈ s, ∀ z ∈ U, meromorphicOrderAt (f i) z ≠ ⊤) →
      MeromorphicOn.divisor (∏ i ∈ s, f i) U = ∑ i ∈ s, MeromorphicOn.divisor (f i) U

If orders are finite, the divisor of a product of meromorphic functions is the sum of the divisors.

Defined in

Mathlib.Analysis.Meromorphic.Divisor

Cited by

1 results in Mathlib

Foundations

Depth 210 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedField