{
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    "publishableCanonical": false,
    "generatedAt": "2026-09-11T10:19:00Z",
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    "registryId": "vr.tr.trigonometric-series",
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      "XCT.REL"
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    "name": "trigonometric series",
    "purpose": "Let an agent handle trigonometric series by definition, convergence, Fourier coefficients and applications.",
    "definition": "A series of the form a0/2 + sum over n of (an cos nx + bn sin nx); when coefficients come from a function by integration, it is a Fourier series, used to represent periodic functions.",
    "what_it_is_for": "Representing periodic functions in mathematics, physics and engineering.",
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      "discuss convergence",
      "apply to signals and heat equations",
      "relate to Fourier transforms"
    ],
    "distinguishing_features": [
      "Sum of sines and cosines",
      "Represents periodic functions",
      "Convergence can be subtle",
      "Foundation of harmonic analysis"
    ],
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    "visual_identification": [
      "Sums of sin nx and cos nx",
      "Fourier coefficients an and bn",
      "Power series use powers of x instead"
    ],
    "physical_properties": [],
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      "Fourier series",
      "general trigonometric series",
      "Fourier sine and cosine series",
      "Ramanujan sums (related)"
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      {
        "relation": "is a kind of",
        "target": "function series",
        "why": "category"
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      {
        "relation": "includes",
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      {
        "relation": "is studied in",
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      {
        "relation": "is related to",
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        "why": "continuous analogue"
      }
    ],
    "identifiers": [],
    "standards_and_regulation": [
      "ISO 80000-2 notation"
    ],
    "failure_modes_and_hazards": [
      "Assuming pointwise convergence",
      "Gibbs phenomenon surprises"
    ],
    "in_scope": [],
    "out_of_scope": [],
    "characteristics": [],
    "agent_conduct": {
      "may": [
        "Compute and explain trigonometric and Fourier series, stating convergence conditions.",
        "Use them to analyse signals the agent is authorized to process."
      ],
      "must_not": [
        "Claim convergence where it does not hold or has not been shown.",
        "Present truncated series as exact results without stating the error.",
        "Use signal analysis to extract private speech or data without consent."
      ],
      "requires_human": [
        "Relying on series approximations in safety-critical engineering."
      ]
    },
    "ethics": {
      "considerations": [
        "Errors in approximations can matter in engineering and medicine.",
        "Signal analysis can expose private information."
      ],
      "affected_parties": [
        "Students and engineers",
        "People whose signals are analysed"
      ]
    },
    "owners": {
      "steward": "Nobody: a mathematical concept held in common.",
      "master_systems": []
    }
  },
  "sources": [
    {
      "id": "wikidata",
      "title": "Wikidata item Q1149336: trigonometric series",
      "url": "https://www.wikidata.org/wiki/Q1149336",
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        "rationale": "Coefficients define the series.",
        "layers": [
          {
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            "name": "Coefficients",
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            "findings": [
              {
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                "name": "Coefficients",
                "description": "Coefficients.",
                "questions": [
                  {
                    "text": "What are the Fourier coefficients of this function?",
                    "kind": "measurement"
                  },
                  {
                    "text": "Over which interval?",
                    "kind": "definition"
                  }
                ]
              }
            ]
          },
          {
            "id": "form",
            "name": "Form",
            "description": "Sine or cosine series.",
            "findings": [
              {
                "id": "form-finding",
                "name": "Form",
                "description": "Series form.",
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                  {
                    "text": "Is a full, sine or cosine series appropriate?",
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                  },
                  {
                    "text": "Why?",
                    "kind": "definition"
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                ]
              }
            ]
          }
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      {
        "id": "convergence",
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        "description": "Behaviour.",
        "rationale": "Convergence can be subtle.",
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            "id": "conditions",
            "name": "Conditions",
            "description": "Theorems.",
            "findings": [
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                "id": "conditions-finding",
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                "description": "Convergence conditions.",
                "questions": [
                  {
                    "text": "Under which conditions does the series converge to the function?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which theorem applies?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          },
          {
            "id": "gibbs",
            "name": "Gibbs",
            "description": "Overshoot.",
            "findings": [
              {
                "id": "gibbs-finding",
                "name": "Gibbs",
                "description": "Gibbs phenomenon.",
                "questions": [
                  {
                    "text": "Does the Gibbs phenomenon appear near discontinuities?",
                    "kind": "boundary"
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                  {
                    "text": "How large is the overshoot?",
                    "kind": "measurement"
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                ]
              }
            ]
          }
        ]
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      {
        "id": "applications",
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        "description": "Uses.",
        "rationale": "Series solve problems.",
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          {
            "id": "signals",
            "name": "Signals",
            "description": "Spectra.",
            "findings": [
              {
                "id": "signals-finding",
                "name": "Signals",
                "description": "Signal analysis.",
                "questions": [
                  {
                    "text": "How does the series describe the harmonics of a signal?",
                    "kind": "definition"
                  },
                  {
                    "text": "How many terms are needed?",
                    "kind": "measurement"
                  }
                ]
              }
            ]
          },
          {
            "id": "pdes",
            "name": "PDEs",
            "description": "Heat and waves.",
            "findings": [
              {
                "id": "pdes-finding",
                "name": "PDEs",
                "description": "Differential equations.",
                "questions": [
                  {
                    "text": "How is the series used to solve the heat or wave equation?",
                    "kind": "action"
                  },
                  {
                    "text": "What boundary conditions apply?",
                    "kind": "definition"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "computing",
        "name": "Computing",
        "description": "Numerics.",
        "rationale": "Computers approximate series.",
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            "id": "fft",
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              {
                "id": "fft-finding",
                "name": "FFT",
                "description": "Discrete Fourier transform.",
                "questions": [
                  {
                    "text": "How does the discrete Fourier transform relate to the series?",
                    "kind": "definition"
                  },
                  {
                    "text": "Which library computes it?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          },
          {
            "id": "teaching",
            "name": "Teaching",
            "description": "Visualisation.",
            "findings": [
              {
                "id": "teaching-finding",
                "name": "Teaching",
                "description": "Teaching.",
                "questions": [
                  {
                    "text": "How can partial sums be visualised for learners?",
                    "kind": "action"
                  },
                  {
                    "text": "Which examples help?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      }
    ]
  },
  "openQuestions": [
    "Should Fourier series be a separate entry?",
    "How should convergence theorems be linked?",
    "How should numerical methods be linked?"
  ],
  "statistics": {
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    "layers": 8,
    "findings": 8,
    "questions": 16
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}