Theorems · Theorem · complex analysis

Complex.hasSum_deriv_of_summable_norm

∀ {E : Type u_1} {ι : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] {U : Set ℂ} {z : ℂ}
  {F : ι → ℂ → E} [CompleteSpace E] {u : ι → ℝ},
  Summable u →
    (∀ (i : ι), DifferentiableOn ℂ (F i) U) →
      IsOpen U →
        (∀ (i : ι), ∀ w ∈ U, ‖F i w‖ ≤ u i) →
          z ∈ U → HasSum (fun i => deriv (F i) z) (deriv (fun w => ∑' (i : ι), F i w) z)

If the terms in the sum ∑' (i : ι), F i are uniformly bounded on U by a summable function, then the sum of deriv F i at a point in U is the derivative of the sum.

Defined in

Mathlib.Analysis.Complex.LocallyUniformLimit

Cited by

0 results in Mathlib

Foundations

Depth 292 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceCompleteSpace