Theorems · Theorem · complex analysis

Complex.circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable

∀ {E : Type u} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] [CompleteSpace E] {R : ℝ},
  0 < R →
    ∀ {f : ℂ → E} {c : ℂ} {s : Set ℂ},
      s.Countable →
        ContinuousOn f (Metric.closedBall c R) →
          (∀ z ∈ Metric.ball c R \ s, DifferentiableAt ℂ f z) →
            ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z = (2 * ↑Real.pi * Complex.I) • f c

Cauchy integral formula for the value at the center of a disc. If f : ℂ → E is continuous on a closed disc of radius R and center c, and is complex differentiable at all but countably many points of its interior, then the integral $\oint_{|z-c|=R} \frac{f(z)}{z-c}\,dz$ is equal to 2πi • f c.

Defined in

Mathlib.Analysis.Complex.CauchyIntegral

Cited by

1 results in Mathlib

Foundations

Depth 271 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceCompleteSpace