Mathlib theorem · Existing formal mathematics
ProofAtlas

Existing in Mathlib · Complex analysis

Cauchy Integral Formula

For a holomorphic function on and inside a circle, every interior value is encoded by the function's values around the boundary, weighted by the reciprocal displacement from the interior point.

Exact theorem

Exact Mathlib statement

theorem DifferentiableOn.circleIntegral_sub_inv_smul {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {R : ℝ} {c w : ℂ} {f : ℂ → E} (hd : DifferentiableOn ℂ f (Metric.closedBall c R)) (hw : w ∈ Metric.ball c R) : (∮ z in C(c, R), (z - w)⁻¹ • f z) = (2 * Real.pi * Complex.I : ℂ) • f w

The theorem at a glance

Cauchy Integral Formula at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

A differentiable function on the closed disk is recovered at each interior point from its Cauchy-weighted boundary integral. Explanatory diagram.
Detailed visual description

The poster makes the disk, interior point, oriented boundary, Cauchy kernel, and vector-valued endpoint explicit. A central equation connects the boundary integral to 2πi times the interior value, with a footer excluding higher derivatives and general-contour variants.

Statement structure

Statement and scope

Statement map for Cauchy Integral FormulaA complex-Banach-space-valued function differentiable on a closed disk is recovered at any interior point from its circle integral against the Cauchy kernel. Claim boundary: This page indexes Mathlib's circle Cauchy-integral formula for a function from ℂ to a complex Banach space that is complex differentiable on the entire closed ball. For an interior point w, it identifies the boundary circle integral of (z-w)⁻¹ • f(z) with 2πi • f(w). It is not the higher-derivative formula, a general-contour winding-number theorem, or a statement with differentiability only on the open disk. The pinned upstream declaration is DifferentiableOn.circleIntegral_sub_inv_smul. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingAcomplex-Banach-space-valuedfunction differentiableon a closed disk isrecovered at anyinterior point from itscircle integral againstthe Cauchy kernel.Claim boundaryThis page indexesMathlib's circleCauchy-integral formulafor a function from ℂ toa complex Banach spacethat is complexdifferentiable on theentire closed ball. Foran interior point w, itidentifies the boundarycircle integral of(z-w)⁻¹ • f(z) with 2πi• f(w). It is not thehigher-derivativeformula, ageneral-contourwinding-number theorem,or a statement withdifferentiability onlyon the open disk.Pinned declarationmathlib ·DifferentiableOn.circleIntegral_sub_inv_smulStatement map for Cauchy Integral FormulaA complex-Banach-space-valued function differentiable on a closed disk is recovered at any interior point from its circle integral against the Cauchy kernel. Claim boundary: This page indexes Mathlib's circle Cauchy-integral formula for a function from ℂ to a complex Banach space that is complex differentiable on the entire closed ball. For an interior point w, it identifies the boundary circle integral of (z-w)⁻¹ • f(z) with 2πi • f(w). It is not the higher-derivative formula, a general-contour winding-number theorem, or a statement with differentiability only on the open disk. The pinned upstream declaration is DifferentiableOn.circleIntegral_sub_inv_smul. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingAcomplex-Banach-space-valuedfunction differentiableon a closed disk isrecovered at anyinterior point from itscircle integral againstthe Cauchy kernel.Claim boundaryThis page indexesMathlib's circleCauchy-integral formulafor a function from ℂ toa complex Banach spacethat is complexdifferentiable on theentire closed ball. Foran interior point w, itidentifies the boundarycircle integral of(z-w)⁻¹ • f(z) with 2πi• f(w). It is not thehigher-derivativeformula, ageneral-contourwinding-number theorem,or a statement withdifferentiability onlyon the open disk.Pinned declarationmathlib ·DifferentiableOn.circleIntegral_sub_inv_smul

Read the exact Mathlib declaration

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

The Cauchy kernel makes boundary values around a circle reconstruct every interior function value. Explanatory diagram.
Detailed visual description

A precise gold boundary circle surrounds an emerald interior point. Ivory samples around the circumference connect through cobalt reciprocal-displacement rays that combine coherently at the interior point, emphasizing the full boundary integral and not a finite sampling rule.

Why it matters

A mathematical landmark

Cauchy's integral formula is a defining bridge in complex analysis: boundary data controls interior values and leads to analyticity, derivative formulas, and rigidity theorems. Mathlib's selected declaration gives a vector-valued circle form in a complex Banach space.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally
Locally reproducedExact upstream declaration replayed
Reviewed pageCurrent public presentation reviewed
Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This page indexes Mathlib's circle Cauchy-integral formula for a function from ℂ to a complex Banach space that is complex differentiable on the entire closed ball. For an interior point w, it identifies the boundary circle integral of (z-w)⁻¹ • f(z) with 2πi • f(w). It is not the higher-derivative formula, a general-contour winding-number theorem, or a statement with differentiability only on the open disk.

Source and local evidence

Where the theorem comes from

Existing declaration
DifferentiableOn.circleIntegral_sub_inv_smul in mathlib
Relationship
The checked artifact is the upstream declaration itself
Local Lean evidence
artifact.library.mathlib.cauchy-integral-formula.v001
Source
Open the pinned upstream reference