Theorems · Theorem · complex analysis

MeromorphicOn.logDeriv_mul_eventuallyEq

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {𝕜' : Type u_2} [inst_1 : NontriviallyNormedField 𝕜']
  [inst_2 : NormedAlgebra 𝕜 𝕜'] {f g : 𝕜 → 𝕜'} {U : Set 𝕜},
  MeromorphicOn f U →
    MeromorphicOn g U →
      (∀ x ∈ U, meromorphicOrderAt f x ≠ ⊤) →
        (∀ x ∈ U, meromorphicOrderAt g x ≠ ⊤) → logDeriv (f * g) =ᶠ[Filter.codiscreteWithin U] logDeriv f + logDeriv g

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

Defined in

Mathlib.Analysis.Meromorphic.LogDeriv

Cited by

2 results in Mathlib

Foundations

Depth 205 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNontriviallyNormedFieldNormedAlgebra