Theorems · Theorem · complex analysis

MeromorphicOn.logDeriv_finprod_eventuallyEq

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {𝕜' : Type u_2} [inst_1 : NontriviallyNormedField 𝕜']
  [inst_2 : NormedAlgebra 𝕜 𝕜'] {U : Set 𝕜} {ι : Type u_3} {F : ι → 𝕜 → 𝕜'},
  (Function.mulSupport F).Finite →
    (∀ (i : ι), MeromorphicOn (F i) U) →
      (∀ (i : ι), ∀ x ∈ U, meromorphicOrderAt (F i) x ≠ ⊤) →
        logDeriv (∏ᶠ (i : ι), F i) =ᶠ[Filter.codiscreteWithin U] ∑ᶠ (i : ι), logDeriv (F i)

The logarithmic derivative converts products into sums: away from a codiscrete subset of U, the logarithmic derivative of a finite product of meromorphic functions is the sum of the logarithmic derivatives.

Defined in

Mathlib.Analysis.Meromorphic.LogDeriv

Cited by

1 results in Mathlib

Foundations

Depth 206 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNontriviallyNormedFieldNormedAlgebra