"Bernoulli number" meaning in English

{
  "etymology_text": "Named after Swiss mathematician Jacob Bernoulli (1654–1705), who discovered the numbers independently of and at about the same time as Japanese mathematician Seki Kōwa.\nThe numbers first appeared as coefficients in Bernoulli's formulae for the sum of the first n positive integers, each raised to a given power. The sequence was subsequently found in numerous other contexts.",
  "forms": [
    {
      "form": "Bernoulli numbers",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
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      "args": {},
      "expansion": "Bernoulli number (plural Bernoulli numbers)",
      "name": "en-noun"
    }
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
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          "kind": "other",
          "name": "English entries with incorrect language header",
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            "Entries with incorrect language header",
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          "kind": "topical",
          "langcode": "en",
          "name": "Mathematical analysis",
          "orig": "en:Mathematical analysis",
          "parents": [
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            "Formal sciences",
            "Sciences",
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            "Fundamental"
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          "kind": "topical",
          "langcode": "en",
          "name": "Number theory",
          "orig": "en:Number theory",
          "parents": [
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            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
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          "source": "w"
        }
      ],
      "examples": [
        {
          "text": "1993, Serge Lang, Complex Analysis, Springer, 3rd Edition, page 418,\nThe assertion about the value of the zeta function at negative integers then comes immediately from the definition of the Bernoulli numbers in terms of the coefficients of a power series, namely\nt/(eᵗ-1)=∑ₙ₌₀ ᪲B_n(tⁿ)/(n!)"
        },
        {
          "ref": "2002, M. Aslam Chaudhry, Syed M. Zubair, On a Class of Incomplete Gamma Functions with Applications, CRC Press (Chapman & Hall/CRC), page 287:",
          "text": "In the first three sections, we present standard results about the Bernoulli numbers and polynomials and the Riemann zeta functions and its zeros.",
          "type": "quote"
        },
        {
          "ref": "2007, C. Edward Sandifer, How Euler Did It, Mathematical Association of America, page 175:",
          "text": "This done, he extends occurrence of Bernoulli numbers in the expansion of #x5C;textstyle#x5C;frac 1 2#x5C;cot(#x5C;frac 1 2x) to the more general form #x5C;textstyle#x5C;frac#x5C;pin#x5C;cot(#x5C;frac#x7B;m#x5C;pi#x7D;n) and uses that to relate Bernoulli numbers to the values of #x5C;zeta(2n). To end the theoretical parts of his exposition, he gives some of the properties of the Bernoulli polynomials and notes that Bernoulli numbers grow faster than any geometric series.",
          "type": "quote"
        }
      ],
      "glosses": [
        "Any one of the rational numbers in a sequence of such that appears in numerous contexts, including formulae for sums of integer powers and certain power series expansions."
      ],
      "id": "en-Bernoulli_number-en-noun-UhB51MSN",
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          "number theory",
          "number theory"
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          "rational number",
          "rational number"
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      ],
      "raw_glosses": [
        "(mathematical analysis, number theory) Any one of the rational numbers in a sequence of such that appears in numerous contexts, including formulae for sums of integer powers and certain power series expansions."
      ],
      "related": [
        {
          "word": "Bernoulli polynomial"
        }
      ],
      "synonyms": [
        {
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            "dated",
            "plural"
          ],
          "word": "Bernoulli's number"
        }
      ],
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        "mathematical-analysis",
        "mathematics",
        "number-theory",
        "sciences"
      ],
      "wikipedia": [
        "Bernoulli number",
        "Jacob Bernoulli",
        "Seki Takakazu"
      ]
    }
  ],
  "word": "Bernoulli number"
}

{
  "etymology_text": "Named after Swiss mathematician Jacob Bernoulli (1654–1705), who discovered the numbers independently of and at about the same time as Japanese mathematician Seki Kōwa.\nThe numbers first appeared as coefficients in Bernoulli's formulae for the sum of the first n positive integers, each raised to a given power. The sequence was subsequently found in numerous other contexts.",
  "forms": [
    {
      "form": "Bernoulli numbers",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
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      "expansion": "Bernoulli number (plural Bernoulli numbers)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "Bernoulli polynomial"
    }
  ],
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      "examples": [
        {
          "text": "1993, Serge Lang, Complex Analysis, Springer, 3rd Edition, page 418,\nThe assertion about the value of the zeta function at negative integers then comes immediately from the definition of the Bernoulli numbers in terms of the coefficients of a power series, namely\nt/(eᵗ-1)=∑ₙ₌₀ ᪲B_n(tⁿ)/(n!)"
        },
        {
          "ref": "2002, M. Aslam Chaudhry, Syed M. Zubair, On a Class of Incomplete Gamma Functions with Applications, CRC Press (Chapman & Hall/CRC), page 287:",
          "text": "In the first three sections, we present standard results about the Bernoulli numbers and polynomials and the Riemann zeta functions and its zeros.",
          "type": "quote"
        },
        {
          "ref": "2007, C. Edward Sandifer, How Euler Did It, Mathematical Association of America, page 175:",
          "text": "This done, he extends occurrence of Bernoulli numbers in the expansion of #x5C;textstyle#x5C;frac 1 2#x5C;cot(#x5C;frac 1 2x) to the more general form #x5C;textstyle#x5C;frac#x5C;pin#x5C;cot(#x5C;frac#x7B;m#x5C;pi#x7D;n) and uses that to relate Bernoulli numbers to the values of #x5C;zeta(2n). To end the theoretical parts of his exposition, he gives some of the properties of the Bernoulli polynomials and notes that Bernoulli numbers grow faster than any geometric series.",
          "type": "quote"
        }
      ],
      "glosses": [
        "Any one of the rational numbers in a sequence of such that appears in numerous contexts, including formulae for sums of integer powers and certain power series expansions."
      ],
      "links": [
        [
          "mathematical analysis",
          "mathematical analysis"
        ],
        [
          "number theory",
          "number theory"
        ],
        [
          "rational number",
          "rational number"
        ]
      ],
      "raw_glosses": [
        "(mathematical analysis, number theory) Any one of the rational numbers in a sequence of such that appears in numerous contexts, including formulae for sums of integer powers and certain power series expansions."
      ],
      "topics": [
        "mathematical-analysis",
        "mathematics",
        "number-theory",
        "sciences"
      ],
      "wikipedia": [
        "Bernoulli number",
        "Jacob Bernoulli",
        "Seki Takakazu"
      ]
    }
  ],
  "synonyms": [
    {
      "tags": [
        "dated",
        "plural"
      ],
      "word": "Bernoulli's number"
    }
  ],
  "word": "Bernoulli number"
}