Theorems · Theorem · complex analysis

sum_cauchyPowerSeries_eq_integral

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] {f : ℂ → E} {c : ℂ} {R : ℝ} {w : ℂ},
  CircleIntegrable f c R →
    ‖w‖ < R →
      (cauchyPowerSeries f c R).sum w = (2 * ↑Real.pi * Complex.I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - (c + w))⁻¹ • f z

For any circle integrable function f, the power series cauchyPowerSeries f c R, R > 0, converges to the Cauchy integral (2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - w)⁻¹ • f z on the open disc Metric.ball c R.

Defined in

Mathlib.MeasureTheory.Integral.CircleIntegral

Cited by

0 results in Mathlib

Foundations

Depth 259 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpace