{
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  "category": "Complex analysis",
  "claimBoundary": "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.",
  "collectionSlug": "landmark-theorems-in-mathlib",
  "exactFormalStatement": "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",
  "family": "Contour integration",
  "id": "library.mathlib.cauchy-integral-formula.v001",
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    "entry": "/library-theorems/cauchy-integral-formula/",
    "entryData": "/data/library-theorems/cauchy-integral-formula.json",
    "evidence": "/proofs/artifact.library.mathlib.cauchy-integral-formula.v001.html",
    "evidenceData": "/data/proofs/artifact.library.mathlib.cauchy-integral-formula.v001.evidence.json",
    "source": "/sources/upstream/cauchy-integral-formula/"
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  "landmark": {
    "completeFormalType": "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",
    "concepts": [
      "holomorphic functions",
      "circle integrals",
      "Cauchy kernel",
      "boundary reconstruction",
      "complex Banach spaces"
    ],
    "plainLanguage": "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.",
    "poster": {
      "alt": "An ivory complex-analysis poster shows a closed disk, an interior point w, boundary values f(z), reciprocal-displacement weighting, and the exact circle-integral identity.",
      "byteSize": 2630584,
      "caption": "A differentiable function on the closed disk is recovered at each interior point from its Cauchy-weighted boundary integral.",
      "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.",
      "derivatives": [
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          "width": 1024
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      ],
      "generatedAt": "2026-07-25T06:01:22.119158525Z",
      "height": 1536,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/cauchy-integral-formula/theorem-poster-v1.png",
      "sha256": "sha256:2114a204054ef4924c3f4fb2eb1dba9889cb7b3de8afba0eb719c51ad17ab693",
      "title": "Cauchy Integral Formula at a glance",
      "width": 1024,
      "kind": "theorem_poster",
      "transcript": "COMPLEX ANALYSIS · BOUNDARY RECONSTRUCTION\nCAUCHY INTEGRAL FORMULA\nf : ℂ → E · E a complex Banach space\nf differentiable on the closed disk\nw inside the open disk\n∮ (z − w)⁻¹ • f(z) dz = 2πi • f(w)\nTHE BOUNDARY DETERMINES THE INTERIOR VALUE\nEXACT SCOPE\nCircle form on a closed disk. No higher-derivative or general-contour claim."
    },
    "visual": {
      "alt": "A closed complex disk surrounds an interior point, with weighted rays from the boundary circle coherently focusing into the function value at that point.",
      "byteSize": 2925532,
      "caption": "The Cauchy kernel makes boundary values around a circle reconstruct every interior function value.",
      "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.",
      "derivatives": [
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          "byteSize": 31430,
          "height": 427,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/cauchy-integral-formula/theorem-schematic-v1-640.webp",
          "sha256": "sha256:2acfd20e76b92f7003bf300b5b0c5a876bae1a1e57df8dd41243948b0428675a",
          "width": 640
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        {
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          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/cauchy-integral-formula/theorem-schematic-v1-1200.webp",
          "sha256": "sha256:37a9ed3ba8f387da3e9f78acbf6b36b55bf8c1633d265080d18bb70bdb31cf8e",
          "width": 1200
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      ],
      "generatedAt": "2026-07-25T05:25:50.950229048Z",
      "height": 1024,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/cauchy-integral-formula/theorem-schematic-v1.png",
      "sha256": "sha256:07cddf77471caee54f1f89a5ec494dd50a0344c55d25ebe4829c05bf0f2144d6",
      "title": "Boundary-to-interior Cauchy reconstruction schematic",
      "width": 1536
    },
    "walkthrough": null,
    "whyLandmark": "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."
  },
  "nonClaims": [
    "Proof Atlas did not originate Cauchy's integral formula or Mathlib's declaration.",
    "The selected declaration is the closed-disk, circle-integral, complex-Banach-space-valued form with its exact differentiability hypothesis.",
    "The generated explanation and visuals are not proof evidence."
  ],
  "publicExportEligible": true,
  "relationship": "upstream_direct",
  "schemaVersion": "library-theorem-entry-data.v1",
  "statementId": "statement.library.mathlib.cauchy-integral-formula.v001",
  "statusAxes": {
    "acceptedAtlasResult": "Not recorded for the preferred artifact",
    "locallyReproduced": "Exact upstream declaration replayed",
    "reviewedPage": "Current public presentation reviewed",
    "upstreamIndexed": "Pinned source bytes verified locally"
  },
  "summary": "A complex-Banach-space-valued function differentiable on a closed disk is recovered at any interior point from its circle integral against the Cauchy kernel.",
  "targetId": "target.library.mathlib.cauchy-integral-formula.v001",
  "title": "Cauchy Integral Formula",
  "upstreamOrigin": {
    "declarationName": "DifferentiableOn.circleIntegral_sub_inv_smul",
    "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/Analysis/Complex/CauchyIntegral.lean#L568",
    "packageName": "mathlib",
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    "sourceFile": "Mathlib/Analysis/Complex/CauchyIntegral.lean",
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  "publicPresentationReview": {
    "status": "reviewed"
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}