Theorems · Theorem · complex analysis

DiffContOnCl.deriv_eq_smul_circleIntegral

∀ {E : Type u} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] [CompleteSpace E] {R : ℝ} {f : ℂ → E} {c : ℂ},
  0 < R →
    DiffContOnCl ℂ f (Metric.ball c R) →
      ∮ (z : ℂ) in C(c, R), (1 / (z - c) ^ 2) • f z = (2 * ↑Real.pi * Complex.I) • deriv f c

Cauchy integral formula for the first order derivative, assuming f is continuous on a closed ball and differentiable on its interior.

Defined in

Mathlib.Analysis.Complex.CauchyIntegral

Cited by

0 results in Mathlib

Foundations

Depth 285 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceCompleteSpace