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This is an overview table with the 95 benchmark entries showing the TeX of the formula and if the translation to semantic LaTeX and Mathematica (CAS) was correct. If the translation was wrong, a more detailed explanation and categorization of the error is given. This table does not contain the actual translation results. You can reach this information by clicking the link in the most left column, e.g., [[Gold 1]]. | |||
We provide the entire benchmark as a JSON file too: [[MediaWiki:Gold-data.json|gold-data.json]]. | |||
{| class="wikitable sortable" | {| class="wikitable sortable" | ||
Latest revision as of 13:58, 1 September 2021
This is an overview table with the 95 benchmark entries showing the TeX of the formula and if the translation to semantic LaTeX and Mathematica (CAS) was correct. If the translation was wrong, a more detailed explanation and categorization of the error is given. This table does not contain the actual translation results. You can reach this information by clicking the link in the most left column, e.g., Gold 1.
We provide the entire benchmark as a JSON file too: gold-data.json.
| Entry Info | Translations | Reason For Failure | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Formula | Title | Semantic LaTeX | CAS Translations | Definition / Substitution | Pattern Matching | Derivatives / Primes | Missing Infos | Untranslatable Macro | Explanation | Evaluation Data |
| Gold 1 | \begin{align}J_{-(m+\frac{1}{2})}(x) &= (-1)^{m+1} Y_{m+\frac{1}{2}}(x), \\Y_{-(m+\frac{1}{2})}(x) &= (-1)^m J_{m+\frac{1}{2}}(x).\end{align}
|
Bessel function | 
|
|
- | - | - | - | - | - | Full data:
{
"id": 1,
"pid": 51,
"eid": "math.51.18",
"title": "Bessel function",
"formulae": [
{
"id": "FORMULA_0f521573a47e7fd187dafed615b0ecce",
"formula": "\\begin{align}J_{-(m+\\frac{1}{2})}(x) &= (-1)^{m+1} Y_{m+\\frac{1}{2}}(x), \\\\Y_{-(m+\\frac{1}{2})}(x) &= (-1)^m J_{m+\\frac{1}{2}}(x).\\end{align}",
"semanticFormula": "\\begin{align}\\BesselJ{- (m + \\frac{1}{2})}@{x} &= (- 1)^{m+1} \\BesselY{m+\\frac{1}{2}}@{x} , \\\\ \\BesselY{- (m + \\frac{1}{2})}@{x} &= (-1)^m \\BesselJ{m+\\frac{1}{2}}@{x} .\\end{align}",
"confidence": 0.8803349492974287,
"translations": {
"Mathematica": {
"translation": "BesselJ[- (m +Divide[1,2]), x] == (- 1)^(m + 1)* BesselY[m +Divide[1,2], x]\nBesselY[- (m +Divide[1,2]), x] == (- 1)^(m)* BesselJ[m +Divide[1,2], x]",
"translationInformation": {
"subEquations": [
"BesselJ[- (m +Divide[1,2]), x] = (- 1)^(m + 1)* BesselY[m +Divide[1,2], x]",
"BesselY[- (m +Divide[1,2]), x] = (- 1)^(m)* BesselJ[m +Divide[1,2], x]"
],
"freeVariables": [
"m",
"x"
],
"constraints": [],
"tokenTranslations": {
"\\pgcd": "Greatest common divisor; Example: \\pgcd{m,n}\nWill be translated to: GCD[$0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/27.1#p2.t1.r3\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/GCD.html",
"\\BesselY": "Bessel function second kind; Example: \\BesselY{v}@{z}\nWill be translated to: BesselY[$0, $1]\nBranch Cuts: (-\\infty, 0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/10.2#E3\nMathematica: https:\/\/",
"\\BesselJ": "Bessel function first kind; Example: \\BesselJ{v}@{z}\nWill be translated to: BesselJ[$0, $1]\nBranch Cuts: if v \\notin \\Integers: (-\\infty, 0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/10.2#E2\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/BesselJ.html"
}
}
},
"Maple": {
"translation": "BesselJ(- (m +(1)\/(2)), x) = (- 1)^(m + 1)* BesselY(m +(1)\/(2), x); BesselY(- (m +(1)\/(2)), x) = (- 1)^(m)* BesselJ(m +(1)\/(2), x)",
"translationInformation": {
"subEquations": [
"BesselJ(- (m +(1)\/(2)), x) = (- 1)^(m + 1)* BesselY(m +(1)\/(2), x)",
"BesselY(- (m +(1)\/(2)), x) = (- 1)^(m)* BesselJ(m +(1)\/(2), x)"
],
"freeVariables": [
"m",
"x"
],
"constraints": [],
"tokenTranslations": {
"\\pgcd": "Greatest common divisor; Example: \\pgcd{m,n}\nWill be translated to: gcd($0)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/27.1#p2.t1.r3\nMaple: https:\/\/www.maplesoft.com\/support\/help\/Maple\/view.aspx?path=gcd",
"\\BesselY": "Bessel function second kind; Example: \\BesselY{v}@{z}\nWill be translated to: BesselY($0, $1)\nBranch Cuts: (-\\infty, 0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/10.2#E3\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=Bessel",
"\\BesselJ": "Bessel function first kind; Example: \\BesselJ{v}@{z}\nWill be translated to: BesselJ($0, $1)\nBranch Cuts: if v \\notin \\Integers: (-\\infty, 0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/10.2#E2\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=Bessel"
}
}
}
},
"positions": [
{
"section": 8,
"sentence": 8,
"word": 32
}
],
"includes": [
"Y_{\\alpha}",
"J_{-\\alpha}(x)",
"J",
"J_{\\alpha}(x)",
"Y_{n}",
"J_{n}(x)",
"m",
"Y_{\\alpha}(x)",
"J_{\\alpha}",
"x",
"(-1)^{m}",
"J_{n}",
"J_{\\alpha}(z)",
"J_{\\alpha}(k)",
"Y",
"J_{n + m}(x)"
],
"isPartOf": [],
"definiens": [
{
"definition": "Bessel function first kind",
"score": 2
},
{
"definition": "Bessel function second kind",
"score": 2
},
{
"definition": "above relation",
"score": 0
},
{
"definition": "spherical Bessel",
"score": 1
},
{
"definition": "integer",
"score": 1
},
{
"definition": "nonnegative integer",
"score": 1
},
{
"definition": "relationship",
"score": 0
},
{
"definition": "function",
"score": 1
},
{
"definition": "recurrence relation",
"score": 1
},
{
"definition": "Bessel",
"score": 1
},
{
"definition": "large number of other known integral",
"score": 0
},
{
"definition": "positive zero",
"score": 0
},
{
"definition": "entire function of genus",
"score": 0
},
{
"definition": "identity",
"score": 0
},
{
"definition": "orthogonality relation",
"score": 0
},
{
"definition": "Bessel function",
"score": 2
},
{
"definition": "term",
"score": 0
},
{
"definition": "real zero",
"score": 0
},
{
"definition": "similar relation",
"score": 0
},
{
"definition": "Hankel",
"score": 1
},
{
"definition": "Bessel function of the second kind",
"score": 2
},
{
"definition": "limit",
"score": 0
},
{
"definition": "ordinary Bessel function",
"score": 1
},
{
"definition": "case",
"score": 0
},
{
"definition": "negative integer",
"score": 0
},
{
"definition": "integral formula",
"score": 0
},
{
"definition": "small argument",
"score": 0
},
{
"definition": "average",
"score": 0
},
{
"definition": "Bessel function of the first kind",
"score": 2
},
{
"definition": "reference",
"score": 0
},
{
"definition": "series expansion",
"score": 0
},
{
"definition": "spherical Bessel function",
"score": 1
},
{
"definition": "Abel 's identity",
"score": 0
},
{
"definition": "important property of Bessel 's equation",
"score": 1
},
{
"definition": "particular Bessel",
"score": 1
},
{
"definition": "solution of Bessel 's equation",
"score": 0
},
{
"definition": "Wronskian of the solution",
"score": 0
},
{
"definition": "series",
"score": 0
},
{
"definition": "closure equation",
"score": 0
}
]
}
]
}
|
| Gold 2 | E(e) \,=\, \int_0^{\pi/2}\sqrt {1 - e^2 \sin^2\theta}\ d\theta
|
Ellipse | 
|
|
- | 
|
- | - | - | e was interpreted as Euler's number
|
Full data:
{
"id": 2,
"pid": 52,
"eid": "math.52.404",
"title": "Ellipse",
"formulae": [
{
"id": "FORMULA_d3e28ddd096754fb8e1e52aaaa4f7770",
"formula": "E(e) \\,=\\, \\int_0^{\\pi\/2}\\sqrt {1 - e^2 \\sin^2\\theta}\\ d\\theta",
"semanticFormula": "\\compellintEk@{e} = \\int_0^{\\cpi \/ 2} \\sqrt{1 - e^2 \\sin^2 \\theta} \\diff{\\theta}",
"confidence": 0.8896531556938116,
"translations": {
"Mathematica": {
"translation": "EllipticE[(e)^2] == Integrate[Sqrt[1 - (e)^(2)*(Sin[\\[Theta]])^(2)], {\\[Theta], 0, Pi\/2}, GenerateConditions->None]",
"translationInformation": {
"subEquations": [
"EllipticE[(e)^2] = Integrate[Sqrt[1 - (e)^(2)*(Sin[\\[Theta]])^(2)], {\\[Theta], 0, Pi\/2}, GenerateConditions->None]"
],
"freeVariables": [],
"constraints": [],
"tokenTranslations": {
"\\cpi": "Pi was translated to: Pi",
"\\expe": "Recognizes e with power as the exponential function. It was translated as a function.",
"\\compellintEk": "Legendre's complete elliptic integral of the second kind; Example: \\compellintEk@{k}\nWill be translated to: EllipticE[($0)^2]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/19.2#E8\nMathematica: https:\/\/",
"\\sin": "Sine; Example: \\sin@@{z}\nWill be translated to: Sin[$0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/4.14#E1\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/Sin.html"
}
}
},
"Maple": {
"translation": "EllipticE(e) = int(sqrt(1 - (e)^(2)*(sin(theta))^(2)), theta = 0..Pi\/2)",
"translationInformation": {
"subEquations": [
"EllipticE(e) = int(sqrt(1 - (e)^(2)*(sin(theta))^(2)), theta = 0..Pi\/2)"
],
"freeVariables": [],
"constraints": [],
"tokenTranslations": {
"\\cpi": "Pi was translated to: Pi",
"\\expe": "Recognizes e with power as the exponential function. It was translated as a function.",
"\\compellintEk": "Legendre's complete elliptic integral of the second kind; Example: \\compellintEk@{k}\nWill be translated to: EllipticE($0)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/19.2#E8\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=EllipticE",
"\\sin": "Sine; Example: \\sin@@{z}\nWill be translated to: sin($0)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/4.14#E1\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=sin"
}
}
}
},
"positions": [
{
"section": 37,
"sentence": 0,
"word": 39
}
],
"includes": [
"\\theta",
"E",
"\\pi a b",
"\\pi",
"e",
"E(e)"
],
"isPartOf": [],
"definiens": [
{
"definition": "complete elliptic integral of the second kind",
"score": 2
},
{
"definition": "elementary function",
"score": 1
},
{
"definition": "function",
"score": 1
},
{
"definition": "length of the semi-major axis",
"score": 2
},
{
"definition": "eccentricity",
"score": 2
},
{
"definition": "circumference",
"score": 0
},
{
"definition": "ellipse",
"score": 1
},
{
"definition": "angle",
"score": 1
},
{
"definition": "angular coordinate",
"score": 1
},
{
"definition": "center",
"score": 0
},
{
"definition": "formula",
"score": 0
},
{
"definition": "rotation angle",
"score": 0
}
]
}
]
}
|
| Gold 3 | F(x;k) = u
|
Elliptic integral |
|
|
|
- | - | - | - | x is substituted
|
Full data:
{
"id": 3,
"pid": 53,
"eid": "math.53.6",
"title": "Elliptic integral",
"formulae": [
{
"id": "FORMULA_04e9de23897a3b23dee1a9b7312ad99e",
"formula": "F(x;k) = u",
"semanticFormula": "\\incellintFk@{\\asin@{\\Jacobiellsnk@@{u}{k}}}{k} = u",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "EllipticF[ArcSin[JacobiSN[u, (k)^2]], (k)^2] == u",
"translationInformation": {
"subEquations": [
"EllipticF[ArcSin[JacobiSN[u, (k)^2]], (k)^2] = u"
],
"freeVariables": [
"k",
"u"
],
"constraints": [],
"tokenTranslations": {}
}
},
"Maple": {
"translation": "EllipticF(JacobiSN(u, k), k) = u",
"translationInformation": {
"subEquations": [
"EllipticF(JacobiSN(u, k), k) = u"
],
"freeVariables": [
"k",
"u"
],
"constraints": [],
"tokenTranslations": {}
}
}
},
"positions": [
{
"section": 2,
"sentence": 6,
"word": 5
}
],
"includes": [
"u",
"F",
"x",
"k"
],
"isPartOf": [
"F(\\varphi,k) = F\\left(\\varphi \\,|\\, k^2\\right) = F(\\sin \\varphi ; k) = \\int_0^\\varphi \\frac {\\mathrm{d}\\theta}{\\sqrt{1 - k^2 \\sin^2 \\theta}}",
"F(x ; k) = \\int_{0}^{x} \\frac{\\mathrm{d}t}{\\sqrt{\\left(1 - t^2\\right)\\left(1 - k^2 t^2\\right)}}",
"E(\\varphi,k) = E\\left(\\varphi \\,|\\,k^2\\right) = E(\\sin\\varphi;k) = \\int_0^\\varphi \\sqrt{1-k^2 \\sin^2\\theta}\\,\\mathrm{d}\\theta",
"E(x;k) = \\int_0^x \\frac{\\sqrt{1-k^2 t^2} }{\\sqrt{1-t^2}}\\,\\mathrm{d}t"
],
"definiens": [
{
"definition": "inverse to the elliptic integral",
"score": 1
},
{
"definition": "Jacobian elliptic function",
"score": 2
},
{
"definition": "Legendre",
"score": 1
},
{
"definition": "normal form",
"score": 1
},
{
"definition": "trigonometric form",
"score": 1
},
{
"definition": "incomplete elliptic integral of the second kind",
"score": 0
},
{
"definition": "incomplete elliptic integral of the first kind",
"score": 2
}
]
}
]
}
|
| Gold 4 | \frac{1}{\Gamma(z)} = \frac{i}{2\pi}\int_C (-t)^{-z}e^{-t}\,dt
|
Gamma function |
|
- | - |
|
- | - | - | Contour integrals cannot be translated. | Full data:
{
"id": 4,
"pid": 54,
"eid": "math.54.195",
"title": "Gamma function",
"formulae": [
{
"id": "FORMULA_19a0f00da77cc439ad679c579a295bd2",
"formula": "\\frac{1}{\\Gamma(z)} = \\frac{i}{2\\pi}\\int_C (-t)^{-z}e^{-t}\\,dt",
"semanticFormula": "\\frac{1}{\\EulerGamma@{z}} = \\frac{\\iunit}{2 \\cpi} \\int_C(- t)^{-z} \\expe^{-t} \\diff{t}",
"confidence": 0.8809245132365588,
"translations": {},
"positions": [
{
"section": 11,
"sentence": 10,
"word": 9
}
],
"includes": [
"C",
"\\Gamma",
"\\frac {1}{\\Gamma (z)}",
"z",
"1",
"\\Gamma(r)",
"t",
"\\pi",
"\\Gamma (z)",
"\\Gamma(z)",
"\\Pi\\left(z\\right)",
"\\Gamma\\left(z\\right)",
"e^{-x}"
],
"isPartOf": [],
"definiens": [
{
"definition": "related expression",
"score": 0
},
{
"definition": "integer",
"score": 0
},
{
"definition": "reflection formula",
"score": 1
},
{
"definition": "end",
"score": 0
},
{
"definition": "Hankel contour",
"score": 2
},
{
"definition": "Riemann sphere",
"score": 1
},
{
"definition": "Hankel 's formula for the gamma function",
"score": 2
},
{
"definition": "gamma function",
"score": 2
},
{
"definition": "reciprocal gamma function",
"score": 2
}
]
}
]
}
|
| Gold 5 | 2^{4} = 2 \times2 \times 2 \times 2 = 16
|
Logarithm |
|
|
- | - | - | - | - | - | Full data:
{
"id": 5,
"pid": 55,
"eid": "",
"title": "Logarithm",
"formulae": [
{
"id": "FORMULA_579837194f2124b255d579031524a91c",
"formula": "2^{4} = 2 \\times2 \\times 2 \\times 2 = 16",
"semanticFormula": "2^{4} = 2 \\times2 \\times 2 \\times 2 = 16",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "(2)^(4) == 2 * 2 * 2 * 2 == 16",
"translationInformation": {
"subEquations": [
"(2)^(4) = 2 * 2 * 2 * 2",
"2 * 2 * 2 * 2 = 16"
],
"freeVariables": [],
"constraints": [],
"tokenTranslations": {
"\\times": "was translated to: *"
}
}
},
"Maple": {
"translation": "(2)^(4) = 2 * 2 * 2 * 2 = 16",
"translationInformation": {
"subEquations": [
"(2)^(4) = 2 * 2 * 2 * 2",
"2 * 2 * 2 * 2 = 16"
],
"freeVariables": [],
"constraints": [],
"tokenTranslations": {
"\\times": "was translated to: *"
}
}
}
},
"positions": [
{
"section": 4,
"sentence": 0,
"word": 3
}
],
"includes": [
"2",
"^{4}"
],
"isPartOf": [],
"definiens": [
{
"definition": "example",
"score": 2
}
]
}
]
}
|
| Gold 6 | \psi(x) := \sum_{n=1}^\infty e^{-n^2 \pi x}
|
Riemann zeta function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 6,
"pid": 56,
"eid": "math.56.40",
"title": "Riemann zeta function",
"formulae": [
{
"id": "FORMULA_bd88ec58aa42c7a59bc2f4ff458a54cf",
"formula": "\\psi(x) := \\sum_{n=1}^\\infty e^{-n^2 \\pi x}",
"semanticFormula": "\\psi(x) : = \\sum_{n=1}^\\infty \\expe^{- n^2 \\cpi x}",
"confidence": 0.9073333333333333,
"translations": {
"Mathematica": {
"translation": "\\[Psi][x_] := Sum[Exp[-(n)^(2)*Pi*x], {n, 1, Infinity}]"
},
"Maple": {
"translation": "psi := (x) -> sum(exp(-(n)^(2)*Pi*x), n=1..infinity)"
}
},
"positions": [
{
"section": 4,
"sentence": 7,
"word": 23
}
],
"includes": [
"1",
"n",
"2",
"x",
"\\psi"
],
"isPartOf": [],
"definiens": [
{
"definition": "analytic continuation",
"score": 0
},
{
"definition": "absolute convergence",
"score": 0
},
{
"definition": "convenience",
"score": 0
},
{
"definition": "inversion",
"score": 0
},
{
"definition": "process",
"score": 0
},
{
"definition": "stricter requirement",
"score": 0
},
{
"definition": "series",
"score": 1
},
{
"definition": "definition",
"score": 2
}
]
}
]
}
|
| Gold 7 | \operatorname{li}(x) = \lim_{\varepsilon \to 0+} \left( \int_0^{1-\varepsilon} \frac{dt}{\ln t} + \int_{1+\varepsilon}^x \frac{dt}{\ln t} \right)
|
Logarithmic integral function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 7,
"pid": 57,
"eid": "math.57.2",
"title": "Logarithmic integral function",
"formulae": [
{
"id": "FORMULA_36fb8f8330168b8f8acda0dc36851474",
"formula": "\\operatorname{li}(x) = \\lim_{\\varepsilon \\to 0+} \\left( \\int_0^{1-\\varepsilon} \\frac{dt}{\\ln t} + \\int_{1+\\varepsilon}^x \\frac{dt}{\\ln t} \\right)",
"semanticFormula": "\\logint@{x} = \\lim_{\\varepsilon \\to 0+}(\\int_0^{1-\\varepsilon} \\frac{\\diff{t}}{\\ln t} + \\int_{1+\\varepsilon}^x \\frac{\\diff{t}}{\\ln t})",
"confidence": 0.8728566391293461,
"translations": {
"Mathematica": {
"translation": "LogIntegral[x] == Limit[Integrate[Divide[1,Log[t]], {t, 0, 1 - \\[CurlyEpsilon]}, GenerateConditions->None]+ Integrate[Divide[1,Log[t]], {t, 1 + \\[CurlyEpsilon], x}, GenerateConditions->None], \\[CurlyEpsilon] -> 0, Direction -> \"FromAbove\", GenerateConditions->None]",
"translationInformation": {
"subEquations": [
"LogIntegral[x] = Limit[Integrate[Divide[1,Log[t]], {t, 0, 1 - \\[CurlyEpsilon]}, GenerateConditions->None]+ Integrate[Divide[1,Log[t]], {t, 1 + \\[CurlyEpsilon], x}, GenerateConditions->None], \\[CurlyEpsilon] -> 0, Direction -> \"FromAbove\", GenerateConditions->None]"
],
"freeVariables": [
"x"
],
"constraints": [],
"tokenTranslations": {
"\\logint": "Logarithmic integral; Example: \\logint@{x}\nWill be translated to: LogIntegral[$0]\nConstraints: x > 1\nMathematica uses other branch cuts: (-\\inf, 1)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/6.2#E8\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/LogIntegral.html",
"\\ln": "Natural logarithm; Example: \\ln@@{z}\nWill be translated to: Log[$0]\nConstraints: z != 0\nBranch Cuts: (-\\infty, 0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/4.2#E2\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/Log.html"
}
}
},
"Maple": {
"translation": "Li(x) = limit(int((1)\/(ln(t)), t = 0..1 - varepsilon)+ int((1)\/(ln(t)), t = 1 + varepsilon..x), varepsilon = 0, right)",
"translationInformation": {
"subEquations": [
"Li(x) = limit(int((1)\/(ln(t)), t = 0..1 - varepsilon)+ int((1)\/(ln(t)), t = 1 + varepsilon..x), varepsilon = 0, right)"
],
"freeVariables": [
"x"
],
"constraints": [],
"tokenTranslations": {
"\\logint": "Logarithmic integral; Example: \\logint@{x}\nWill be translated to: Li($0)\nConstraints: x > 1\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/6.2#E8\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=Li",
"\\ln": "Natural logarithm; Example: \\ln@@{z}\nWill be translated to: ln($0)\nConstraints: z != 0\nBranch Cuts: (-\\infty, 0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/4.2#E2\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=ln"
}
}
}
},
"positions": [
{
"section": 1,
"sentence": 2,
"word": 22
}
],
"includes": [
"x",
"x)"
],
"isPartOf": [],
"definiens": [
{
"definition": "Cauchy principal value",
"score": 2
},
{
"definition": "function",
"score": 1
},
{
"definition": "singularity",
"score": 1
},
{
"definition": "special function",
"score": 1
},
{
"definition": "integral representation",
"score": 1
},
{
"definition": "integral logarithm li",
"score": 2
},
{
"definition": "logarithmic integral function",
"score": 2
},
{
"definition": "logarithmic integral",
"score": 2
},
{
"definition": "function li",
"score": 1
}
]
}
]
}
|
| Gold 8 | w_{i} = \frac{1}{p'_{n}(x_{i})}\int_{a}^{b}\omega(x)\frac{p_{n}(x)}{x-x_{i}}dx
|
Gaussian quadrature |
|
|
- | - |
|
- | - | - | Full data:
{
"id": 8,
"pid": 58,
"eid": "math.58.61",
"title": "Gaussian quadrature",
"formulae": [
{
"id": "FORMULA_8c49145544fca24efb8de07eb1275c09",
"formula": "w_{i} = \\frac{1}{p'_{n}(x_{i})}\\int_{a}^{b}\\omega(x)\\frac{p_{n}(x)}{x-x_{i}}dx",
"semanticFormula": "w_{i} = \\frac{1}{p'_{n}(x_{i})} \\int_{a}^{b} \\omega(x) \\frac{p_{n}(x)}{x-x_{i}} \\diff{x}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "Subscript[w, i] = Divide[1, Subscript[p\\[Prime], n][Subscript[x, i]]]*Integrate[\\[Omega][x]*Divide[Subscript[p,n][x], x-Subscript[x,i]], {x, a, b}]"
}
},
"positions": [
{
"section": 5,
"sentence": 4,
"word": 24
}
],
"includes": [
"a",
"b",
"w_{i}",
"p_n(x)",
"p_{k}(x)",
"p_{n}",
"x_{i}",
"\\omega(x)",
"p_{n}(x)",
"\\omega",
"x_i",
"a_{n}",
"P_{n}",
"w_i",
"r(x_{i})",
"i",
"n",
"x",
"P_{n}(x)",
"\\frac{p_{n}(x)}{x-x_{i}}",
"p'_{n}(x_{i})",
"p_{n}(x_{i})",
"x_{j}",
"p_r",
"p_s",
"\\mathbf{e}_n",
"x_j",
"1"
],
"isPartOf": [],
"definiens": [
{
"definition": "yield",
"score": 0
},
{
"definition": "integral expression for the weight",
"score": 2
},
{
"definition": "integrand",
"score": 1
},
{
"definition": "L'H\u00f4pital 's rule",
"score": 0
},
{
"definition": "limit",
"score": 0
},
{
"definition": "polynomial of degree",
"score": 0
}
]
}
]
}
|
| Gold 9 | \begin{align}x & =ue^u, \\[5pt]\frac{dx}{du} & =(u+1)e^u.\end{align}
|
Lambert W function |
|
|
|
- | - | - | - | - | Full data:
{
"id": 9,
"pid": 59,
"eid": "math.59.52",
"title": "Lambert W function",
"formulae": [
{
"id": "FORMULA_fe13643d8449f601f150fd50c0751cf2",
"formula": "\\begin{align}x & =ue^u, \\\\[5pt]\\frac{dx}{du} & =(u+1)e^u.\\end{align}",
"semanticFormula": "\\begin{align}x & =\\LambertW@{x}\\expe^{\\LambertW@{x}}, \\\\ \\deriv{x}{\\LambertW@{x}} &=(\\LambertW@{x} + 1) \\expe^{\\LambertW@{x}} .\\end{align}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "x == ProductLog[x]*(E)^(ProductLog[x])\nD[x,ProductLog[x]] = (ProductLog[x] + 1)*Exp[ProductLog[x]]",
"translationInformation": {
"subEquations": [
"x = ProductLog[x]*(E)^(ProductLog[x])",
"D[x,ProductLog[x]] = (ProductLog[x] + 1)*Exp[ProductLog[x]]"
],
"freeVariables": [
"u",
"x"
],
"constraints": [],
"tokenTranslations": {
"\\expe": "Recognizes e with power as the exponential function. It was translated as a function."
}
}
},
"Maple": {
"translation": "x = LambertW(x)*exp(u); diff(x, [LambertW(x)$1]) = (LambertW(x) + 1)*exp(LambertW(x))",
"translationInformation": {
"subEquations": [
"x = LambertW(x)*exp(u)",
"diff(x, [LambertW(x)$1]) = (LambertW(x) + 1)*exp(LambertW(x))"
],
"freeVariables": [
"u",
"x"
],
"constraints": [],
"tokenTranslations": {
"\\expe": "Recognizes e with power as the exponential function. It was translated as a function."
}
}
}
},
"positions": [
{
"section": 12,
"sentence": 1,
"word": 14
}
],
"includes": [
"e^{w}",
"x"
],
"isPartOf": [],
"definiens": [
{
"definition": "substitution",
"score": 2
},
{
"definition": "third identity",
"score": 0
},
{
"definition": "second identity",
"score": 1
}
]
}
]
}
|
| Gold 10 | \frac{1}{\left| \mathbf{x}-\mathbf{x}' \right|} = \frac{1}{\sqrt{r^2+{r'}^2-2r{r'}\cos\gamma}} = \sum_{\ell=0}^\infty \frac{{r'}^\ell}{r^{\ell+1}} P_\ell(\cos \gamma)
|
Legendre polynomials |
|
|
- | - |
|
- | - | - | Full data:
{
"id": 10,
"pid": 60,
"eid": "math.60.57",
"title": "Legendre polynomials",
"formulae": [
{
"id": "FORMULA_8646bd0d06e9454aaa39dfc506fe54f7",
"formula": "\\frac{1}{\\left| \\mathbf{x}-\\mathbf{x}' \\right|} = \\frac{1}{\\sqrt{r^2+{r'}^2-2r{r'}\\cos\\gamma}} = \\sum_{\\ell=0}^\\infty \\frac{{r'}^\\ell}{r^{\\ell+1}} P_\\ell(\\cos \\gamma)",
"semanticFormula": "\\frac{1}{|\\mathbf{x} - \\mathbf{x} '|} = \\frac{1}{\\sqrt{r^2+{r'}^2-2r{r'}\\cos\\gamma}} = \\sum_{\\ell=0}^\\infty \\frac{{r'}^\\ell}{r^{\\ell+1}} \\LegendrepolyP{\\ell}@{\\cos \\gamma}",
"confidence": 0.808438593520797,
"translations": {
"Mathematica": "Divide[1, Abs[x - x\\[Prime]]] == Divide[1, Sqrt[r^2+(r\\[Prime])^(2)-2*r*r\\[Prime] Cos[\\[Gamma]]]] == Sum[Divide[(r\\[Prime])^(\\[ScriptL]), r^(\\[ScriptL]+1)]*LegendreP[\\[ScriptL], Cos[\\[Gamma]]], {\\[ScriptL], 0, Infinity}]"
},
"positions": [
{
"section": 6,
"sentence": 0,
"word": 21
}
],
"includes": [
"P_n(x)",
"P_n",
"P_n(\\cos\\theta)",
"P_{n}(x)",
"P_m",
"r",
"r{'}",
"\\mathbf{x}",
"\\mathbf{x}{'}",
"\\gamma",
"P"
],
"isPartOf": [],
"definiens": [
{
"definition": "expansion",
"score": 2
},
{
"definition": "Adrien-Marie Legendre as the coefficient",
"score": 0
},
{
"definition": "angle",
"score": 1
},
{
"definition": "Legendre polynomial",
"score": 2
},
{
"definition": "length of the vector",
"score": 1
},
{
"definition": "vector",
"score": 1
},
{
"definition": "polynomial",
"score": 1
}
]
}
]
}
|
| Gold 11 | \operatorname{erf}^{(k)}(z) = \frac{2 (-1)^{k-1}}{\sqrt{\pi}} \mathit{H}_{k-1}(z) e^{-z^2} = \frac{2}{\sqrt{\pi}} \frac{d^{k-1}}{dz^{k-1}} \left(e^{-z^2}\right),\qquad k=1, 2, \dots
|
Error function |
|
|
- | - |
|
- | - | was not detected as k-th derivative but as power. | Full data:
{
"id": 11,
"pid": 61,
"eid": "math.61.27",
"title": "Error function",
"formulae": [
{
"id": "FORMULA_523ec091d0929f0fa69ae7e0d563a72b",
"formula": "\\operatorname{erf}^{(k)}(z) = \\frac{2 (-1)^{k-1}}{\\sqrt{\\pi}} \\mathit{H}_{k-1}(z) e^{-z^2} = \\frac{2}{\\sqrt{\\pi}} \\frac{d^{k-1}}{dz^{k-1}} \\left(e^{-z^2}\\right),\\qquad k=1, 2, \\dots",
"semanticFormula": "\\erf@@{(z)}^{(k)} = \\frac{2 (-1)^{k-1}}{\\sqrt{\\cpi}} \\HermitepolyH{k-1}@{z} \\expe^{-z^2} = \\frac{2}{\\sqrt{\\cpi}} \\deriv [{k-1}]{ }{z}(\\expe^{-z^2}) , \\qquad k = 1 , 2 , \\dots",
"confidence": 0.82607945540953,
"translations": {
"Mathematica": {
"translation": "D[Erf[z], {z, k}] == Divide[2*(- 1)^(k - 1),Sqrt[Pi]]*HermiteH[k - 1, z]*Exp[- (z)^(2)] == Divide[2,Sqrt[Pi]]*D[Exp[- (z)^(2)], {z, k - 1}]",
"translationInformation": {
"subEquations": [
"D[Erf[z], {z, k}] = Divide[2*(- 1)^(k - 1),Sqrt[Pi]]*HermiteH[k - 1, z]*Exp[- (z)^(2)]",
"Divide[2*(- 1)^(k - 1),Sqrt[Pi]]*HermiteH[k - 1, z]*Exp[- (z)^(2)] = Divide[2,Sqrt[Pi]]*D[Exp[- (z)^(2)], {z, k - 1}]"
],
"freeVariables": [
"k",
"z"
],
"constraints": [
"k == 1 , 2 , \\[Ellipsis]"
],
"tokenTranslations": {
"\\deriv1": "Derivative; Example: \\deriv[n]{f}{x}\nWill be translated to: D[$1, {$2, $0}]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/1.4#E4\nMathematica: https:\/\/",
"\\cpi": "Pi was translated to: Pi",
"\\HermitepolyH": "Hermite polynomial; Example: \\HermitepolyH{n}@{x}\nWill be translated to: HermiteH[$0, $1]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/18.3#T1.t1.r13\nMathematica: https:\/\/",
"\\expe": "Recognizes e with power as the exponential function. It was translated as a function.",
"\\erf": "Error function; Example: \\erf@@{z}\nWill be translated to: Erf[$0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/7.2#E1\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/Erf.html"
}
}
},
"Maple": {
"translation": "diff(erf(z), [z$k]) = (2*(- 1)^(k - 1))\/(sqrt(Pi))*HermiteH(k - 1, z)*exp(- (z)^(2)) = (2)\/(sqrt(Pi))*diff(exp(- (z)^(2)), [z$(k - 1)])",
"translationInformation": {
"subEquations": [
"diff(erf(z), [z$k]) = (2*(- 1)^(k - 1))\/(sqrt(Pi))*HermiteH(k - 1, z)*exp(- (z)^(2))",
"(2*(- 1)^(k - 1))\/(sqrt(Pi))*HermiteH(k - 1, z)*exp(- (z)^(2)) = (2)\/(sqrt(Pi))*diff(exp(- (z)^(2)), [z$(k - 1)])"
],
"freeVariables": [
"k",
"z"
],
"constraints": [
"k = 1 , 2 , .."
],
"tokenTranslations": {
"\\deriv1": "Derivative; Example: \\deriv[n]{f}{x}\nWill be translated to: diff($1, [$2$($0)])\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/1.4#E4\nMaple: https:\/\/www.maplesoft.com\/support\/help\/Maple\/view.aspx?path=diff",
"\\cpi": "Pi was translated to: Pi",
"\\HermitepolyH": "Hermite polynomial; Example: \\HermitepolyH{n}@{x}\nWill be translated to: HermiteH($0, $1)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/18.3#T1.t1.r13\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=HermiteH",
"\\expe": "Recognizes e with power as the exponential function. It was translated as a function.",
"\\erf": "Error function; Example: \\erf@@{z}\nWill be translated to: erf($0)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/7.2#E1\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=erf"
}
}
}
},
"positions": [
{
"section": 5,
"sentence": 4,
"word": 6
}
],
"includes": [
"erf",
"e^{-t^2}",
"-1",
"z",
"z)",
"e",
"\\mathit{H}",
"z^{\\bar{n}}"
],
"isPartOf": [],
"definiens": [
{
"definition": "Higher order derivative",
"score": 2
},
{
"definition": "physicists ' Hermite polynomial",
"score": 1
},
{
"definition": "name error function",
"score": 1
},
{
"definition": "erfc",
"score": 1
},
{
"definition": "error function",
"score": 2
},
{
"definition": "erf",
"score": 1
}
]
}
]
}
|
| Gold 12 | x_k = \cos\left(\frac{\pi(k+1/2)}{n}\right),\quad k=0,\ldots,n-1
|
Chebyshev polynomials |
|
|
- | - | - | - | - | - | Full data:
{
"id": 12,
"pid": 62,
"eid": "math.62.44",
"title": "Chebyshev polynomials",
"formulae": [
{
"id": "FORMULA_d9eb68704833b0f525c4ca81a749d9ca",
"formula": "x_k = \\cos\\left(\\frac{\\pi(k+1\/2)}{n}\\right),\\quad k=0,\\ldots,n-1",
"semanticFormula": "x_k = \\cos(\\frac{\\cpi(k + 1 \/ 2)}{n}) , \\quad k = 0 , \\ldots , n - 1",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "Subscript[x, k] == Cos[Divide[Pi*(k + 1\/2),n]]",
"translationInformation": {
"subEquations": [
"Subscript[x, k] = Cos[Divide[Pi*(k + 1\/2),n]]"
],
"freeVariables": [
"Subscript[x, k]",
"k",
"n"
],
"constraints": [
"k == 0 , \\[Ellipsis], n - 1"
],
"tokenTranslations": {
"\\cos": "Cosine; Example: \\cos@@{z}\nWill be translated to: Cos[$0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/4.14#E2\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/Cos.html",
"\\cpi": "Pi was translated to: Pi"
}
}
},
"Maple": {
"translation": "x[k] = cos((Pi*(k + 1\/2))\/(n))",
"translationInformation": {
"subEquations": [
"x[k] = cos((Pi*(k + 1\/2))\/(n))"
],
"freeVariables": [
"k",
"n",
"x[k]"
],
"constraints": [
"k = 0 , .. , n - 1"
],
"tokenTranslations": {
"\\cos": "Cosine; Example: \\cos@@{z}\nWill be translated to: cos($0)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/4.14#E2\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=cos",
"\\cpi": "Pi was translated to: Pi"
}
}
}
},
"positions": [
{
"section": 8,
"sentence": 2,
"word": 18
}
],
"includes": [
"n",
"x",
"n x",
"-1",
"k = 0",
"x_{k}"
],
"isPartOf": [],
"definiens": [
{
"definition": "root",
"score": 2
},
{
"definition": "one",
"score": 0
},
{
"definition": "trigonometric definition",
"score": 0
},
{
"definition": "fact",
"score": 0
},
{
"definition": "different simple root",
"score": 1
},
{
"definition": "Chebyshev polynomial of the first kind",
"score": 1
},
{
"definition": "Chebyshev polynomial",
"score": 1
}
]
}
]
}
|
| Gold 13 | E(x, y; u) := \sum_{n=0}^\infty u^n \, \psi_n (x) \, \psi_n (y) = \frac{1}{\sqrt{\pi (1 - u^2)}} \, \exp\left(-\frac{1 - u}{1 + u} \, \frac{(x + y)^2}{4} - \frac{1 + u}{1 - u} \, \frac{(x - y)^2}{4}\right)
|
Hermite polynomials |
|
|
- | - | - | - | - | - | Full data:
{
"id": 13,
"pid": 63,
"eid": "math.63.109",
"title": "Hermite polynomials",
"formulae": [
{
"id": "FORMULA_249043719eb4dd70350b460363255e11",
"formula": "E(x, y; u) := \\sum_{n=0}^\\infty u^n \\, \\psi_n (x) \\, \\psi_n (y) = \\frac{1}{\\sqrt{\\pi (1 - u^2)}} \\, \\exp\\left(-\\frac{1 - u}{1 + u} \\, \\frac{(x + y)^2}{4} - \\frac{1 + u}{1 - u} \\, \\frac{(x - y)^2}{4}\\right)",
"semanticFormula": "E(x , y ; u) : = \\sum_{n=0}^\\infty u^n \\psi_n(x) \\psi_n(y) = \\frac{1}{\\sqrt{\\cpi(1 - u^2)}} \\exp(- \\frac{1 - u}{1 + u} \\frac{(x + y)^2}{4} - \\frac{1 + u}{1 - u} \\frac{(x - y)^2}{4})",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "\\[CapitalEpsilon][x_, y_, u_] := Sum[(u)^(n)* Subscript[\\[Psi], n][x]* Subscript[\\[Psi], n][y], {n, 0, Infinity}, GenerateConditions->None] == Divide[1,Sqrt[Pi*(1 - (u)^(2))]]*Exp[-Divide[1 - u,1 + u]*Divide[(x + y)^(2),4]-Divide[1 + u,1 - u]*Divide[(x - y)^(2),4]]"
},
"Maple": {
"translation": "Epsilon := (x, y, u) -> sum((u)^(n)* psi[n](x)* psi[n](y), n = 0..infinity) = (1)\/(sqrt(Pi*(1 - (u)^(2))))*exp(-(1 - u)\/(1 + u)*((x + y)^(2))\/(4)-(1 + u)\/(1 - u)*((x - y)^(2))\/(4))"
}
},
"positions": [
{
"section": 25,
"sentence": 2,
"word": 16
}
],
"includes": [
"u",
"\\psi_{n}",
"H_{n}(x)",
"\\psi_{n}(x)",
"x^{n}",
"n",
"x",
"H_{n}",
"He_{n}(x)",
"He_{n}",
"D_{n}(z)",
"E(x,y;u)",
"H_{n}(y)"
],
"isPartOf": [],
"definiens": [
{
"definition": "distributional identity",
"score": 1
},
{
"definition": "separable kernel",
"score": 1
},
{
"definition": "Mehler 's formula",
"score": 2
},
{
"definition": "Hermite polynomial",
"score": 1
},
{
"definition": "Hermite function",
"score": 2
},
{
"definition": "Hermite",
"score": 1
},
{
"definition": "bivariate Gaussian probability density",
"score": 1
},
{
"definition": "Gaussian probability density",
"score": 1
},
{
"definition": "Gaussian probability",
"score": 1
}
]
}
]
}
|
| Gold 14 | x = \pm 1
|
Legendre function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 14,
"pid": 64,
"eid": "math.64.8",
"title": "Legendre function",
"formulae": [
{
"id": "FORMULA_06f9b7b1d3f141742ad1c582b55056ba",
"formula": "x = \\pm 1",
"semanticFormula": "x = \\pm 1",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "x == \\[PlusMinus]1",
"translationInformation": {
"subEquations": [
"x = + 1",
"x = - 1"
],
"freeVariables": [
"x"
],
"constraints": [],
"tokenTranslations": {
"\\pm": "was translated to: \\[PlusMinus]"
}
}
},
"Maple": {
"translation": "x = &+- 1",
"translationInformation": {
"subEquations": [
"x = + 1",
"x = - 1"
],
"freeVariables": [
"x"
],
"constraints": [],
"tokenTranslations": {
"\\pm": "was translated to: &+-"
}
}
}
},
"positions": [
{
"section": 3,
"sentence": 1,
"word": 11
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "value",
"score": 2
}
]
}
]
}
|
| Gold 15 | E_n=2^nE_n(\tfrac{1}{2})
|
Bernoulli polynomials |
|
|
- |
|
- | - | - | Both E where detected as Euler's number but the second is Euler polynomial. | Full data:
{
"id": 15,
"pid": 65,
"eid": "math.65.27",
"title": "Bernoulli polynomials",
"formulae": [
{
"id": "FORMULA_a7fcf738c676932d58f39ff9f7df22ae",
"formula": "E_n=2^nE_n(\\tfrac{1}{2})",
"semanticFormula": "\\EulernumberE{n} = 2^n\\EulerpolyE{n}@{\\tfrac{1}{2}}",
"confidence": 0.8953028732079359,
"translations": {
"Mathematica": {
"translation": "EulerE[n] == (2)^(n)* EulerE[n, Divide[1,2]]"
},
"Maple": {
"translation": "euler(n) = (2)^(n)* euler(n, (1)\/(2))"
}
},
"positions": [
{
"section": 8,
"sentence": 4,
"word": 6
}
],
"includes": [
"B_{n}",
"n",
"E_{k}"
],
"isPartOf": [],
"definiens": [
{
"definition": "Euler number",
"score": 2
}
]
}
]
}
|
| Gold 16 | \operatorname{Si}(ix) = i\operatorname{Shi}(x)
|
Trigonometric integral |
|
|
- | - | - | - | - | Integral was not tagged as a noun by CoreNLP. Hence, the macro for hyperbolic sine function was retrieved too late and not considered for replacements. | Full data:
{
"id": 16,
"pid": 66,
"eid": "math.66.8",
"title": "Trigonometric integral",
"formulae": [
{
"id": "FORMULA_0feb8031b89a9707b164163ec50265f0",
"formula": "\\operatorname{Si}(ix) = i\\operatorname{Shi}(x)",
"semanticFormula": "\\sinint@{\\iunit x} = \\iunit \\sinhint@{x}",
"confidence": 0.8811682126384021,
"translations": {
"Mathematica": {
"translation": "SinIntegral[I*x] == I*SinhIntegral[x]",
"translationInformation": {
"subEquations": [
"SinIntegral[I*x] == I*SinhIntegral[x]"
],
"freeVariables": [
"x"
],
"constraints": [],
"tokenTranslations": {
"Shi": "Was interpreted as a function call because of a leading \\operatorname.",
"\\iunit": "Imaginary unit was translated to: I",
"\\sinint": "Sine integral; Example: \\sinint@{z}\nWill be translated to: SinIntegral[$0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/6.2#E9\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/SinIntegral.html"
}
}
},
"Maple": {
"translation": "Si(I*x) = I*Shi(x)",
"translationInformation": {
"subEquations": [
"Si(I*x) = I*Shi(x)"
],
"freeVariables": [
"x"
],
"constraints": [],
"tokenTranslations": {
"Shi": "Was interpreted as a function call because of a leading \\operatorname.",
"\\iunit": "Imaginary unit was translated to: I",
"\\sinint": "Sine integral; Example: \\sinint@{z}\nWill be translated to: Si($0)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/6.2#E9\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=Si"
}
}
}
},
"positions": [
{
"section": 3,
"sentence": 1,
"word": 9
}
],
"includes": [
"Si",
"Si(x)",
"x"
],
"isPartOf": [],
"definiens": [
{
"definition": "ordinary sine",
"score": 1
},
{
"definition": "Trigonometric integral",
"score": 2
},
{
"definition": "hyperbolic sine integral",
"score": 2
}
]
}
]
}
|
| Gold 17 | f(z)=\frac{1}{\Beta(x,y)}
|
Beta function |
|
|
|
- | - |
|
- | The original formula contained f(z) but should have been f(x,z). This was fixed in the Wikipedia article after we generated the dataset.
|
Full data:
{
"id": 17,
"pid": 67,
"eid": "math.67.29",
"title": "Beta function",
"formulae": [
{
"id": "FORMULA_5f59825d73d63a9990498edca7222261",
"formula": "f(z)=\\frac{1}{\\Beta(x,y)}",
"semanticFormula": "f(x, y) = \\frac{1}{\\EulerBeta@{x}{y}}",
"confidence": 0.8953028732079359,
"translations": {
"Mathematica": {
"translation": "f[x_, y_] := Divide[1,Beta[x, y]]"
},
"Maple": {
"translation": "f := (x,y) -> (1)\/(Beta(x, y))"
}
},
"positions": [
{
"section": 6,
"sentence": 0,
"word": 12
}
],
"includes": [
"x, y",
"\\Beta",
"y",
"x"
],
"isPartOf": [],
"definiens": [
{
"definition": "function about the form",
"score": 0
},
{
"definition": "reciprocal beta function",
"score": 2
},
{
"definition": "definite integral of trigonometric function",
"score": 1
},
{
"definition": "integral representation",
"score": 0
},
{
"definition": "product",
"score": 0
},
{
"definition": "power",
"score": 0
},
{
"definition": "multiple-angle",
"score": 0
},
{
"definition": "beta function",
"score": 2
}
]
}
]
}
|
| Gold 18 | \begin{align}\int x^m e^{ix^n}\,dx & =\frac{x^{m+1}}{m+1}\,_1F_1\left(\begin{array}{c} \frac{m+1}{n}\\1+\frac{m+1}{n}\end{array}\mid ix^n\right) \\[6px]& =\frac{1}{n} i^\frac{m+1}{n}\gamma\left(\frac{m+1}{n},-ix^n\right),\end{align}
|
Fresnel integral |
|
|
- |
|
- | - | - | Matrix argument of does not exist in the DLMF. | Full data:
{
"id": 18,
"pid": 68,
"eid": "math.68.51",
"title": "Fresnel integral",
"formulae": [
{
"id": "FORMULA_b7dae135f3b04317078f86b595fe7dae",
"formula": "\\begin{align}\\int x^m e^{ix^n}\\,dx & =\\frac{x^{m+1}}{m+1}\\,_1F_1\\left(\\begin{array}{c} \\frac{m+1}{n}\\\\1+\\frac{m+1}{n}\\end{array}\\mid ix^n\\right) \\\\[6px]& =\\frac{1}{n} i^\\frac{m+1}{n}\\gamma\\left(\\frac{m+1}{n},-ix^n\\right),\\end{align}",
"semanticFormula": "\\begin{align}\\int x^m \\exp(\\iunit x^n) \\diff{x} &= \\frac{x^{m+1}}{m+1}\\genhyperF{1}{1}@{\\frac{m+1}{n}}{1+\\frac{m+1}{n}}{\\iunit x^n}\\\\ &=\\frac{1}{n} \\iunit^{(m+1)\/n} \\incgamma@{\\frac{m+1}{n}}{-\\iunit x^n}\\end{align}",
"confidence": 0.869061849326977,
"translations": {
"Mathematica": {
"translation": "Integrate[(x)^(m)* Exp[I*(x)^(n)], x, GenerateConditions->None] == Divide[(x)^(m + 1),m + 1]*HypergeometricPFQ[{Divide[m + 1,n]}, {1 +Divide[m + 1,n]}, I*(x)^(n)] == Divide[1,n]*(I)^((m + 1)\/n)* Gamma[Divide[m + 1,n], 0, - I*(x)^(n)]"
},
"Maple": {
"translation": "int((x)^(m)* exp(I*(x)^(n)), x) = ((x)^(m + 1))\/(m + 1)*hypergeom([(m + 1)\/(n)], [1 +(m + 1)\/(n)], I*(x)^(n)) = (1)\/(n)*(I)^((m + 1)\/n)* GAMMA((m + 1)\/(n))-GAMMA((m + 1)\/(n), - I*(x)^(n))"
}
},
"positions": [
{
"section": 5,
"sentence": 0,
"word": 14
}
],
"includes": [
"dx",
"x"
],
"isPartOf": [],
"definiens": [
{
"definition": "incomplete gamma function",
"score": 2
},
{
"definition": "confluent hypergeometric function",
"score": 2
},
{
"definition": "Fresnel integral",
"score": 1
},
{
"definition": "imaginary part",
"score": 1
}
]
}
]
}
|
| Gold 19 | T_n(x) = \frac{\Gamma(1/2)\sqrt{1-x^2}}{(-2)^n\,\Gamma(n+1/2)} \ \frac{d^n}{dx^n}\left([1-x^2]^{n-1/2}\right)
|
Classical orthogonal polynomials |
|
|
- | - | - |
|
- | No info about Gamma function. | Full data:
{
"id": 19,
"pid": 69,
"eid": "math.69.117",
"title": "Classical orthogonal polynomials",
"formulae": [
{
"id": "FORMULA_725c6b6b645d425d3b385ac2c002da77",
"formula": "T_n(x) = \\frac{\\Gamma(1\/2)\\sqrt{1-x^2}}{(-2)^n\\,\\Gamma(n+1\/2)} \\ \\frac{d^n}{dx^n}\\left([1-x^2]^{n-1\/2}\\right)",
"semanticFormula": "\\ChebyshevpolyT{n}@{x} = \\frac{\\EulerGamma{1\/2}\\sqrt{1-x^2}}{(-2)^n\\EulerGamma{n+1\/2}} \\deriv [n]{ }{x}([1 - x^2]^{n-1\/2})",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "ChebyshevT[n, x] == Divide[Gamma[1\/2]*Sqrt[1 - (x)^(2)],(- 2)^(n)* Gamma[n + 1\/2]]*D[(1 - (x)^(2))^(n - 1\/2), {x, n}]"
},
"Maple": {
"translation": "ChebyshevT(n, x) = (GAMMA(1\/2)*sqrt(1 - (x)^(2)))\/((- 2)^(n)* GAMMA(n + 1\/2))*diff((1 - (x)^(2))^(n - 1\/2), [x$(n)])"
}
},
"positions": [
{
"section": 18,
"sentence": 3,
"word": 4
}
],
"includes": [
"\\ L_n",
"H_n",
"P_{n}",
"n-r",
"n",
"P_{n}(x)",
"-1\/2",
"e_{n}",
"P_n",
"\\lambda_{n}",
"-1",
"+1",
"U_n"
],
"isPartOf": [],
"definiens": [
{
"definition": "Rodrigues ' formula",
"score": 2
},
{
"definition": "orthogonal polynomial",
"score": 1
},
{
"definition": "Chebyshev polynomials of the second kind",
"score": 1
},
{
"definition": "classical orthogonal polynomial",
"score": 1
},
{
"definition": "Chebyshev polynomial",
"score": 2
},
{
"definition": "Gamma function",
"score": 2
}
]
}
]
}
|
| Gold 20 | {}_1F_0(1;;z) = \sum_{n \geqslant 0} z^n = (1-z)^{-1}
|
Generalized hypergeometric function |
|
|
- |
|
- | - | - | Empty arguments did not match the semantic macros (bug). | Full data:
{
"id": 20,
"pid": 70,
"eid": "math.70.58",
"title": "Generalized hypergeometric function",
"formulae": [
{
"id": "FORMULA_699b5f465d21dd6af7212cd8414f60c6",
"formula": "{}_1F_0(1;;z) = \\sum_{n \\geqslant 0} z^n = (1-z)^{-1}",
"semanticFormula": "\\genhyperF{1}{0}@{1}{}{z} = \\sum_{n \\geqslant 0} z^n = (1-z)^{-1}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "HypergeometricPFQ[{1}, {}, z] == Sum[(z)^(n), {n, 0, Infinity}, GenerateConditions->None] == (1 - z)^(- 1)"
},
"Maple": {
"translation": "hypergeom([1], [], z) = sum((z)^(n), n = 0..infinity) = (1 - z)^(- 1)"
}
},
"positions": [
{
"section": 17,
"sentence": 2,
"word": 0
}
],
"includes": [
"z",
"n",
"z)",
"_{p}F_{q}",
"^{n}"
],
"isPartOf": [],
"definiens": [
{
"definition": "geometric series with ratio",
"score": 2
},
{
"definition": "coefficient",
"score": 0
},
{
"definition": "hypergeometric function",
"score": 2
}
]
}
]
}
|
| Gold 21 | \chi(-1) = 1
|
Dirichlet L-function |
|
|
- | - | - |
|
- | It was translated to DirichletCharacter[1, k, - 1] == 1. The only valid input for k is 1.
|
Full data:
{
"id": 21,
"pid": 71,
"eid": "math.71.1-1",
"title": "Dirichlet L-function",
"formulae": [
{
"id": "FORMULA_dcb9beab8f504cfc907c3165d24e5ad3",
"formula": "\\chi(-1) = 1",
"semanticFormula": "\\Dirichletchar@@{- 1}{k} = 1",
"confidence": 0.746792096089683,
"translations": {
"Mathematica": {
"translation": "DirichletCharacter[1, 1, -1] == 1"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 7
}
],
"includes": [
"\\chi"
],
"isPartOf": [
"a=\\begin{cases}0;&\\mbox{if }\\chi(-1)=1, \\\\ 1;&\\mbox{if }\\chi(-1)=-1,\\end{cases}"
],
"definiens": [
{
"definition": "primitive character",
"score": 2
},
{
"definition": "integer",
"score": 1
},
{
"definition": "only zero",
"score": 0
},
{
"definition": "Gamma function",
"score": 0
},
{
"definition": "symbol",
"score": 0
},
{
"definition": "functional equation",
"score": 0
},
{
"definition": "Gauss sum",
"score": 0
},
{
"definition": "Dirichlet character",
"score": 2
}
]
}
]
}
|
| Gold 22 | \operatorname{Bi}'(z)\sim \frac{z^{\frac{1}{4}}e^{\frac{2}{3}z^{\frac{3}{2}}}}{\sqrt\pi\,}\left[ \sum_{n=0}^{\infty}\frac{1+6n}{1-6n} \dfrac{ \Gamma(n+\frac{5}{6})\Gamma(n+\frac{1}{6})\left(\frac{3}{4}\right)^n}{2\pi n! z^{3n/2}} \right]
|
Airy function |
|
- | - | - |
|
- | - | No translation possible for \sim
|
Full data:
{
"id": 22,
"pid": 72,
"eid": "math.72.15",
"title": "Airy function",
"formulae": [
{
"id": "FORMULA_3b2520d05d324290456841271e8d565b",
"formula": "\\operatorname{Bi}'(z)\\sim \\frac{z^{\\frac{1}{4}}e^{\\frac{2}{3}z^{\\frac{3}{2}}}}{\\sqrt\\pi\\,}\\left[ \\sum_{n=0}^{\\infty}\\frac{1+6n}{1-6n} \\dfrac{ \\Gamma(n+\\frac{5}{6})\\Gamma(n+\\frac{1}{6})\\left(\\frac{3}{4}\\right)^n}{2\\pi n! z^{3n\/2}} \\right]",
"semanticFormula": "\\AiryBi'@{z} \\sim \\frac{z^{\\frac{1}{4}} \\expe^{\\frac{2}{3}z^{\\frac{3}{2}}}}{\\sqrt{\\cpi}} [\\sum_{n=0}^{\\infty} \\frac{1+6n}{1-6n} \\frac{\\EulerGamma@{n + \\frac{5}{6}} \\EulerGamma@{n + \\frac{1}{6}}(\\frac{3}{4})^n{2 \\cpi n! z^{3n\/2}}}]",
"confidence": 0.6525418663370697,
"translations": {},
"positions": [
{
"section": 3,
"sentence": 9,
"word": 9
}
],
"includes": [
"z",
"z)",
"= 0"
],
"isPartOf": [],
"definiens": [
{
"definition": "z",
"score": 0
},
{
"definition": "asymptotic formula for Ai",
"score": 1
},
{
"definition": "Bi",
"score": 1
},
{
"definition": "asymptotic behaviour of the Airy function",
"score": 1
},
{
"definition": "Ai",
"score": 1
},
{
"definition": "cosine",
"score": 2
},
{
"definition": "definition of the Airy function",
"score": 1
},
{
"definition": "Airy function",
"score": 2
},
{
"definition": "Gamma function",
"score": 2
}
]
}
]
}
|
| Gold 23 | F'(y)=1-2yF(y)
|
Dawson function |
|
|
- | - |
|
|
- | No dependency to Dawson. | Full data:
{
"id": 23,
"pid": 73,
"eid": "math.73.41",
"title": "Dawson function",
"formulae": [
{
"id": "FORMULA_f6b555bd8ce626d90119ab5eafdaeff2",
"formula": "F'(y)=1-2yF(y)",
"semanticFormula": "\\DawsonsintF'@{y}=1-2y\\DawsonsintF@{y}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "D[DawsonF[y], {y, 1}] == 1 - 2*y*DawsonF[y]"
},
"Maple": {
"translation": "diff( dawson(y), y$(1) ) = 1 - 2*y*dawson(y)"
}
},
"positions": [
{
"section": 2,
"sentence": 9,
"word": 1
}
],
"includes": [
"y"
],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 0
},
{
"definition": "Dawson function",
"score": 2
}
]
}
]
}
|
| Gold 24 | s\not =1
|
Hurwitz zeta function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 24,
"pid": 74,
"eid": "math.74.0-1",
"title": "Hurwitz zeta function",
"formulae": [
{
"id": "FORMULA_80a3608d4c2aae63f082861007c16c38",
"formula": "s\\not =1",
"semanticFormula": "s \\neq 1",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "s \\[NotEqual] 1"
}
},
"positions": [
{
"section": 0,
"sentence": 2,
"word": 24
}
],
"includes": [
"s",
"1",
"\\not = 1"
],
"isPartOf": [],
"definiens": [
{
"definition": "value",
"score": 2
}
]
}
]
}
|
| Gold 25 | q = e^{i\pi\tau}
|
Theta function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 25,
"pid": 75,
"eid": "math.75.6-1",
"title": "Theta function",
"formulae": [
{
"id": "FORMULA_bfba6c35dbbcd8b89c6a29b1ffd6f517",
"formula": "q = e^{i\\pi\\tau}",
"semanticFormula": "q = \\expe^{\\iunit \\cpi \\tau}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "q == Exp[I*Pi*\\[Tau]]"
},
"Maple": {
"translation": "q = exp(I*Pi*tau)"
}
},
"positions": [
{
"section": 2,
"sentence": 0,
"word": 57
}
],
"includes": [
"\\tau",
"q",
"w = e^{\\pi iz}",
"q = e^{\\pi i\\tau}"
],
"isPartOf": [
"q = e^{\\pi i\\tau}",
"q = e^{2\\pi i\\tau}",
"\\theta_F (z)= \\sum_{m\\in \\Z^n} e^{2\\pi izF(m)}",
"\\hat{\\theta}_F (z) = \\sum_{k=0}^\\infty R_F(k) e^{2\\pi ikz}"
],
"definiens": [
{
"definition": "term of the nome",
"score": 2
},
{
"definition": "nome",
"score": 2
}
]
}
]
}
|
| Gold 26 | \frac{\mathrm{d}}{\mathrm{d}z} \operatorname{dn}(z) = - k^2 \operatorname{sn}(z) \operatorname{cn}(z)
|
Jacobi elliptic functions |
|
|
- | - | - | - | - | - | Full data:
{
"id": 26,
"pid": 76,
"eid": "math.76.155",
"title": "Jacobi elliptic functions",
"formulae": [
{
"id": "FORMULA_b54c03865b3efa9ea9112567cd66f59d",
"formula": "\\frac{\\mathrm{d}}{\\mathrm{d}z} \\operatorname{dn}(z) = - k^2 \\operatorname{sn}(z) \\operatorname{cn}(z)",
"semanticFormula": "\\deriv [1]{ }{z} \\Jacobielldnk@@{(z)}{k} = - k^2 \\Jacobiellsnk@@{(z)}{k} \\Jacobiellcnk@@{(z)}{k}",
"confidence": 0.6954186066124032,
"translations": {
"Mathematica": {
"translation": "D[JacobiDN[z, (k)^2], {z, 1}] == - (k)^(2)* JacobiSN[z, (k)^2]*JacobiCN[z, (k)^2]"
},
"Maple": {
"translation": "diff(JacobiDN(z, k), [z$(1)]) = - (k)^(2)* JacobiSN(z, k)*JacobiCN(z, k)"
}
},
"positions": [
{
"section": 16,
"sentence": 0,
"word": 15
}
],
"includes": [
"k",
"^{2}"
],
"isPartOf": [],
"definiens": [
{
"definition": "derivative",
"score": 2
},
{
"definition": "elliptic function",
"score": 2
},
{
"definition": "basic Jacobi",
"score": 0
},
{
"definition": "sn",
"score": 2
},
{
"definition": "dn",
"score": 2
},
{
"definition": "cn",
"score": 2
}
]
}
]
}
|
| Gold 27 | \int_{-\infty}^\infty \frac {\gamma\left(\frac s 2, z^2 \pi \right)} {(z^2 \pi)^\frac s 2} e^{-2 \pi i k z} \mathrm d z = \frac {\Gamma\left(\frac {1-s} 2, k^2 \pi \right)} {(k^2 \pi)^\frac {1-s} 2}
|
Incomplete gamma function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 27,
"pid": 77,
"eid": "math.77.118",
"title": "Incomplete gamma function",
"formulae": [
{
"id": "FORMULA_c82b4ceebacd2b4a03b2eff406834e61",
"formula": "\\int_{-\\infty}^\\infty \\frac {\\gamma\\left(\\frac s 2, z^2 \\pi \\right)} {(z^2 \\pi)^\\frac s 2} e^{-2 \\pi i k z} \\mathrm d z = \\frac {\\Gamma\\left(\\frac {1-s} 2, k^2 \\pi \\right)} {(k^2 \\pi)^\\frac {1-s} 2}",
"semanticFormula": "\\int_{-\\infty}^\\infty \\frac{\\incgamma@{\\frac s 2}{z^2 \\cpi}}{(z^2 \\cpi)^\\frac s 2} \\expe^{- 2 \\cpi \\iunit k z} \\diff{z} = \\frac{\\incGamma@{\\frac {1-s} 2}{k^2 \\cpi}}{(k^2 \\cpi)^\\frac {1-s} 2}}",
"confidence": 0.8121295595054496,
"translations": {
"Mathematica": {
"translation": "Integrate[Divide[Gamma[Divide[s,2], 0, (z)^(2)* Pi],((z)^(2)* Pi)^(Divide[s,2])]*Exp[- 2*Pi*I*k*z], {z, - Infinity, Infinity}, GenerateConditions->None] == Divide[Gamma[Divide[1 - s,2], (k)^(2)* Pi],((k)^(2)* Pi)^(Divide[1 - s,2])]"
},
"Maple": {
"translation": "int((GAMMA((s)\/(2))-GAMMA((s)\/(2), (z)^(2)* Pi))\/(((z)^(2)* Pi)^((s)\/(2)))*exp(- 2*Pi*I*k*z), z = - infinity..infinity) = (GAMMA((1 - s)\/(2), (k)^(2)* Pi))\/(((k)^(2)* Pi)^((1 - s)\/(2)))"
}
},
"positions": [
{
"section": 25,
"sentence": 1,
"word": 15
}
],
"includes": [
"\\gamma(s, z)",
"\\gamma",
"z^s",
"\\Gamma",
"\\gamma(s,z)",
"k",
"z",
"z=",
"2\\pi",
"\\gamma(u,v)",
"\\gamma(s,x)",
"s",
"z^{s}",
"e^{-x}",
"\\gamma(a,x)"
],
"isPartOf": [],
"definiens": [
{
"definition": "Fourier",
"score": 1
},
{
"definition": "upper incomplete Gamma function",
"score": 2
},
{
"definition": "lower incomplete Gamma function",
"score": 2
}
]
}
]
}
|
| Gold 28 | _{1}(z) =
|
Polylogarithm |
|
|
- | - | - | - | - | Wrong math detection. | Full data:
{
"id": 28,
"pid": 78,
"eid": "math.78.0-1",
"title": "Polylogarithm",
"formulae": [
{
"id": "FORMULA_e939f30d07578c2fb0d8cb5201db3c79",
"formula": "_{1}(z) =",
"semanticFormula": "\\polylog{1}@{z} = -\\ln@{1-z}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "PolyLog[1, z] = -Log[1 - z]"
},
"Maple": {
"translation": "polylog(1, z) = -ln(1 - z)"
}
},
"positions": [
{
"section": 0,
"sentence": 7,
"word": 11
}
],
"includes": [
"_{1}",
"z",
"z) =",
"z)",
"1"
],
"isPartOf": [
"\\operatorname{Li}_{1}(z) = -\\ln(1-z)",
"\\operatorname{Ti}_0(z) = {z \\over 1+z^2}, \\quad \\operatorname{Ti}_1(z) = \\arctan z, \\quad \\operatorname{Ti}_2(z) = \\int_0^z {\\arctan t \\over t} dt, \\quad \\ldots\\quad \\operatorname{Ti}_{n+1}(z) = \\int_0^z \\frac{\\operatorname{Ti}_n(t)}{t} dt"
],
"definiens": [
{
"definition": "natural logarithm",
"score": 2
},
{
"definition": "logarithm",
"score": 2
},
{
"definition": "polylogarithm function",
"score": 2
},
{
"definition": "dilogarithm",
"score": 1
},
{
"definition": "trilogarithm",
"score": 1
}
]
}
]
}
|
| Gold 29 | \int_{-\infty}^\infty \operatorname{sinc}(t) \, e^{-i 2 \pi f t}\,dt = \operatorname{rect}(f)
|
Sinc function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 29,
"pid": 79,
"eid": "math.79.11",
"title": "Sinc function",
"formulae": [
{
"id": "FORMULA_6340f4a043f912a3557e084aaf03792a",
"formula": "\\int_{-\\infty}^\\infty \\operatorname{sinc}(t) \\, e^{-i 2 \\pi f t}\\,dt = \\operatorname{rect}(f)",
"semanticFormula": "\\int_{-\\infty}^\\infty \\operatorname{sinc}(t) \\expe^{- \\iunit 2 \\cpi f t} \\diff{t} = \\operatorname{rect}(f)",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "Integrate[sinc[(t)]*Exp[- I*2*Pi*f*t], {t, - Infinity, Infinity}, GenerateConditions->None] == rect[f]"
},
"Maple": {
"translation": "int(sinc((t))*exp(- I*2*Pi*f*t), t = - infinity..infinity) = rect(f)"
}
},
"positions": [
{
"section": 1,
"sentence": 9,
"word": 16
}
],
"includes": [
"\\pi",
"\\infty",
"sinc"
],
"isPartOf": [],
"definiens": [
{
"definition": "argument",
"score": 0
},
{
"definition": "continuous Fourier",
"score": 2
},
{
"definition": "rectangular function",
"score": 2
},
{
"definition": "sinc",
"score": 2
},
{
"definition": "ordinary frequency",
"score": 1
}
]
}
]
}
|
| Gold 30 | N=1
|
Exponential integral |
|
|
- | - | - | - | - | - | Full data:
{
"id": 30,
"pid": 80,
"eid": "math.80.26",
"title": "Exponential integral",
"formulae": [
{
"id": "FORMULA_a9a738ef9d4e46360dd9b87b39c691bf",
"formula": "N=1",
"semanticFormula": "N=1",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "N == 1"
},
"Maple": {
"translation": "N = 1"
}
},
"positions": [
{
"section": 4,
"sentence": 3,
"word": 30
}
],
"includes": [
"N"
],
"isPartOf": [],
"definiens": [
{
"definition": "large value",
"score": 2
},
{
"definition": "value",
"score": 2
}
]
}
]
}
|
| Gold 31 | \sum_{n=0}^\infty \frac{n!\,\Gamma\left(\alpha + 1\right)}{\Gamma\left(n+\alpha+1\right)}L_n^{(\alpha)}(x)L_n^{(\alpha)}(y)t^n=\frac{1}{(1-t)^{\alpha + 1}}e^{-(x+y)t/(1-t)}\,_0F_1\left(;\alpha + 1;\frac{xyt}{(1-t)^2}\right)
|
Laguerre polynomials |
|
|
- | - | - |
|
- | No infos about the gamma function. | Full data:
{
"id": 31,
"pid": 81,
"eid": "math.81.84",
"title": "Laguerre polynomials",
"formulae": [
{
"id": "FORMULA_f179a85d8102cbedb67cf60b188a68b7",
"formula": "\\sum_{n=0}^\\infty \\frac{n!\\,\\Gamma\\left(\\alpha + 1\\right)}{\\Gamma\\left(n+\\alpha+1\\right)}L_n^{(\\alpha)}(x)L_n^{(\\alpha)}(y)t^n=\\frac{1}{(1-t)^{\\alpha + 1}}e^{-(x+y)t\/(1-t)}\\,_0F_1\\left(;\\alpha + 1;\\frac{xyt}{(1-t)^2}\\right)",
"semanticFormula": "\\sum_{n=0}^\\infty \\frac{n! \\EulerGamma@{\\alpha + 1}}{\\EulerGamma@{n + \\alpha + 1}} \\LaguerrepolyL[\\alpha]{n}@{x} \\LaguerrepolyL[\\alpha]{n}@{x} t^n = \\frac{1}{(1-t)^{\\alpha + 1}} \\expe^{-(x+y)t\/(1-t)} \\genhyperF{0}{1}@{}{\\alpha + 1}{\\frac{xyt}{(1-t)^2}}",
"confidence": 0.8953028732079359,
"translations": {
"Mathematica": {
"translation": "Sum[Divide[(n)!*Gamma[\\[Alpha]+ 1],Gamma[n + \\[Alpha]+ 1]]*LaguerreL[n, \\[Alpha], x]*LaguerreL[n, \\[Alpha], x]*(t)^(n), {n, 0, Infinity}, GenerateConditions->None] == Divide[1,(1 - t)^(\\[Alpha]+ 1)]*Exp[-(x + y)*t\/(1 - t)]*HypergeometricPFQ[{}, {\\[Alpha]+ 1}, Divide[x*y*t,(1 - t)^(2)]]"
},
"Maple": {
"translation": "sum((factorial(n)*GAMMA(alpha + 1))\/(GAMMA(n + alpha + 1))*LaguerreL(n, alpha, x)*LaguerreL(n, alpha, x)*(t)^(n), n = 0..infinity) = (1)\/((1 - t)^(alpha + 1))*exp(-(x + y)*t\/(1 - t))*hypergeom([], [alpha + 1], (x*y*t)\/((1 - t)^(2)))"
}
},
"positions": [
{
"section": 15,
"sentence": 0,
"word": 10
}
],
"includes": [
"\\alpha",
"L_{n}^{(\\alpha)}",
"L_n^{(\\alpha)}(x)",
"n"
],
"isPartOf": [],
"definiens": [
{
"definition": "Hille formula",
"score": 2
},
{
"definition": "Laguerre polynomial",
"score": 2
},
{
"definition": "series on the left converge",
"score": 0
},
{
"definition": "generalized Laguerre polynomial",
"score": 2
},
{
"definition": "confluent hypergeometric function",
"score": 2
}
]
}
]
}
|
| Gold 32 | c_{lm} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!}
|
Associated Legendre polynomials |
|
|
- | - | - | - | - | - | Full data:
{
"id": 32,
"pid": 82,
"eid": "math.82.8",
"title": "Associated Legendre polynomials",
"formulae": [
{
"id": "FORMULA_6f29e15c07089506a70db1b3f54b27a5",
"formula": "c_{lm} = (-1)^m \\frac{(\\ell-m)!}{(\\ell+m)!}",
"semanticFormula": "c_{lm} = (-1)^m \\frac{(\\ell-m)!}{(\\ell+m)!}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "Subscript[c, l, m] == (- 1)^(m)*Divide[(\\[ScriptL]- m)!,(\\[ScriptL]+ m)!]"
},
"Maple": {
"translation": "c[l, m] = (- 1)^(m)*(factorial(ell - m))\/(factorial(ell + m))"
}
},
"positions": [
{
"section": 1,
"sentence": 7,
"word": 26
}
],
"includes": [
"m",
"(-1)^{m}",
"- 1"
],
"isPartOf": [],
"definiens": [
{
"definition": "proportionality constant",
"score": 2
}
]
}
]
}
|
| Gold 33 | \mathrm{Gi}(x) = \frac{1}{\pi} \int_0^\infty \sin\left(\frac{t^3}{3} + xt\right)\, dt
|
Scorer's function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 33,
"pid": 83,
"eid": "math.83.3",
"title": "Scorer's function",
"formulae": [
{
"id": "FORMULA_c8116180276232704ca3e9f67f207565",
"formula": "\\mathrm{Gi}(x) = \\frac{1}{\\pi} \\int_0^\\infty \\sin\\left(\\frac{t^3}{3} + xt\\right)\\, dt",
"semanticFormula": "\\ScorerGi@{x} = \\frac{1}{\\cpi} \\int_0^\\infty \\sin(\\frac{t^3}{3} + xt) \\diff{t}",
"confidence": 0.7929614010341081,
"translations": {
"Mathematica": {
"translation": "ScorerGi[x] == Divide[1,Pi]*Integrate[Sin[Divide[(t)^(3),3]+ x*t], {t, 0, Infinity}, GenerateConditions->None]",
"translationInformation": {
"subEquations": [
"ScorerGi[x] = Divide[1,Pi]*Integrate[Sin[Divide[(t)^(3),3]+ x*t], {t, 0, Infinity}, GenerateConditions->None]"
],
"freeVariables": [
"x"
],
"constraints": [],
"tokenTranslations": {
"\\ScorerGi": "Scorer function Gi; Example: \\ScorerGi@{z}\nWill be translated to: ScorerGi[$0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/9.12#i\nMathematica: https:\/\/",
"\\cpi": "Pi was translated to: Pi",
"\\sin": "Sine; Example: \\sin@@{z}\nWill be translated to: Sin[$0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/4.14#E1\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/Sin.html"
}
}
},
"Maple": {
"translation": "AiryBi(x)*(int(AiryAi(t), t = (x) .. infinity))+AiryAi(x)*(int(AiryBi(t), t = 0 .. (x))) = (1)\/(Pi)*int(sin(((t)^(3))\/(3)+ x*t), t = 0..infinity)",
"translationInformation": {
"subEquations": [
"AiryBi(x)*(int(AiryAi(t), t = (x) .. infinity))+AiryAi(x)*(int(AiryBi(t), t = 0 .. (x))) = (1)\/(Pi)*int(sin(((t)^(3))\/(3)+ x*t), t = 0..infinity)"
],
"freeVariables": [
"x"
],
"constraints": [],
"tokenTranslations": {
"\\ScorerGi": "Scorer function Gi; Example: \\ScorerGi@{z}\nWill be translated to: \nAlternative translations: [AiryBi($0)*(int(AiryAi(t), t = ($0) .. infinity))+AiryAi($0)*(int(AiryBi(t), t = 0 .. ($0)))]Relevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/9.12#i\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=Airy",
"\\cpi": "Pi was translated to: Pi",
"\\sin": "Sine; Example: \\sin@@{z}\nWill be translated to: sin($0)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/4.14#E1\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=sin"
}
}
}
},
"positions": [
{
"section": 0,
"sentence": 1,
"word": 15
}
],
"includes": [
"x",
"x)"
],
"isPartOf": [],
"definiens": [
{
"definition": "Scorer 's function",
"score": 2
},
{
"definition": "special function",
"score": 1
}
]
}
]
}
|
| Gold 34 | \frac{\partial^2}{\partial x^2} V(x;\sigma,\gamma)= \frac{x^2-\gamma^2-\sigma^2}{\sigma^4} \frac{\operatorname{Re}[w(z)]}{\sigma\sqrt{2 \pi}}-\frac{2 x \gamma}{\sigma^4} \frac{\operatorname{Im}[w(z)]}{\sigma\sqrt{2 \pi}}+\frac{\gamma}{\sigma^4}\frac{1}{\pi}
|
Voigt profile |
|
|
- | - | - | - |
|
- | Full data:
{
"id": 34,
"pid": 84,
"eid": "math.84.31",
"title": "Voigt profile",
"formulae": [
{
"id": "FORMULA_e663d20df3cca1ae5dec645d320cd511",
"formula": "\\frac{\\partial^2}{\\partial x^2} V(x;\\sigma,\\gamma)= \\frac{x^2-\\gamma^2-\\sigma^2}{\\sigma^4} \\frac{\\operatorname{Re}[w(z)]}{\\sigma\\sqrt{2 \\pi}}-\\frac{2 x \\gamma}{\\sigma^4} \\frac{\\operatorname{Im}[w(z)]}{\\sigma\\sqrt{2 \\pi}}+\\frac{\\gamma}{\\sigma^4}\\frac{1}{\\pi}",
"semanticFormula": "\\deriv[2]{}{x} V(x ; \\sigma , \\gamma) = \\frac{x^2-\\gamma^2-\\sigma^2}{\\sigma^4} \\frac{\\realpart [\\Faddeevaw@{z}]}{\\sigma \\sqrt{2 \\cpi}} - \\frac{2 x \\gamma}{\\sigma^4} \\frac{\\imagpart [\\Faddeevaw@{z}]}{\\sigma \\sqrt{2 \\cpi}} + \\frac{\\gamma}{\\sigma^4} \\frac{1}{\\cpi}",
"confidence": 0.8620216359266987,
"translations": {
"Mathematica": {
"translation": "D[PDF[VoigtDistribution[\\[Gamma], \\[Sigma]], x], {x, 2}] == Divide[x^2 - \\[Gamma]^2 - \\[Sigma]^2, \\[Sigma]^4] * Divide[ Re[ Exp[-(Divide[x+I*y,\\[Sigma]*Sqrt[2]])^2]*Erfc[-I*(Divide[x+I*y,\\[Sigma]*Sqrt[2]])] ], \\[Sigma]*Sqrt[2*Pi]] - Divide[2*x*y, \\[Sigma]^4] * Divide[Im[Exp[-(Divide[x+I*y,\\[Sigma]*Sqrt[2]])^2]*Erfc[-I*(Divide[x+I*y,\\[Sigma]*Sqrt[2]])]], \\[Sigma]*Sqrt[2*Pi]] + Divide[\\[Gamma],\\[Sigma]^4]*Divide[1,Pi]"
}
},
"positions": [
{
"section": 6,
"sentence": 0,
"word": 20
}
],
"includes": [
"w(z)]",
"z",
"V(x;\\sigma,\\gamma)",
"x",
"w(z)"
],
"isPartOf": [],
"definiens": [
{
"definition": "term of the Faddeeva function",
"score": 2
},
{
"definition": "second derivative profile",
"score": 2
},
{
"definition": "real part of the Faddeeva function",
"score": 2
},
{
"definition": "Faddeeva function",
"score": 2
},
{
"definition": "Voigt function",
"score": 2
},
{
"definition": "Voigt profile",
"score": 2
}
]
}
]
}
|
| Gold 35 | \Phi(z,s,a) = \frac{1}{1-z} \frac{1}{a^{s}} + \sum_{n=1}^{N-1} \frac{(-1)^{n} \mathrm{Li}_{-n}(z)}{n!} \frac{(s)_{n}}{a^{n+s}} +O(a^{-N-s})
|
Lerch zeta function |
|
|
- | - | - | - |
|
Landau notation. | Full data:
{
"id": 35,
"pid": 85,
"eid": "math.85.57",
"title": "Lerch zeta function",
"formulae": [
{
"id": "FORMULA_a0cc62efe3cabac6d8bebe5b8b94b5fa",
"formula": "\\Phi(z,s,a) = \\frac{1}{1-z} \\frac{1}{a^{s}} + \\sum_{n=1}^{N-1} \\frac{(-1)^{n} \\mathrm{Li}_{-n}(z)}{n!} \\frac{(s)_{n}}{a^{n+s}} +O(a^{-N-s})",
"semanticFormula": "\\Phi(z , s , a) = \\frac{1}{1-z} \\frac{1}{a^{s}} + \\sum_{n=1}^{N-1} \\frac{(-1)^{n} \\polylog{-n}@{z}}{n!} \\frac{\\Pochhammersym{s}{n}}{a^{n+s}} + \\bigO{a^{-N-s}}",
"confidence": 0.8662724998444776,
"translations": {
"Mathematica": {
"translation": "\\[CapitalPhi][z, s, a] == Divide[1,1 - z]*Divide[1,(a)^(s)]+ Sum[Divide[(- 1)^(n)* PolyLog[-n, z],(n)!]*Divide[Pochhammer[s, n],(a)^(n + s)], {n, 1, N - 1}, GenerateConditions->None]+ O[a]^(- N - s)"
}
},
"positions": [
{
"section": 6,
"sentence": 1,
"word": 23
}
],
"includes": [
"a",
"\\Phi(z,s,a)",
"z",
"s"
],
"isPartOf": [],
"definiens": [
{
"definition": "asymptotic expansion",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 1
},
{
"definition": "Lerch transcendent",
"score": 2
},
{
"definition": "polylogarithm",
"score": 2
},
{
"definition": "polylogarithm function",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 2
}
]
}
]
}
|
| Gold 36 | M(1,2,z)=(e^z-1)/z,\ \ M(1,3,z)=2!(e^z-1-z)/z^2
|
Confluent hypergeometric function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 36,
"pid": 86,
"eid": "math.86.44",
"title": "Confluent hypergeometric function",
"formulae": [
{
"id": "FORMULA_d83a3ce5244b566d8f71edb7f81afa43",
"formula": "M(1,2,z)=(e^z-1)\/z,\\ \\ M(1,3,z)=2!(e^z-1-z)\/z^2",
"semanticFormula": "\\KummerconfhyperM@{1}{2}{z} = (\\expe^z - 1) \/ z , \\KummerconfhyperM@{1}{3}{z} = 2! (\\expe^z - 1 - z) \/ z^2",
"confidence": 0.912945064646862,
"translations": {
"Mathematica": {
"translation": "Hypergeometric1F1[1, 2, z] == (Exp[z]- 1)\/z\n Hypergeometric1F1[1, 3, z] == (2)!*(Exp[z]- 1 - z)\/(z)^(2)"
},
"Maple": {
"translation": "KummerM(1, 2, z) = (exp(z)- 1)\/z; KummerM(1, 3, z) = factorial(2)*(exp(z)- 1 - z)\/(z)^(2)"
}
},
"positions": [
{
"section": 10,
"sentence": 4,
"word": 0
}
],
"includes": [
"M",
"U(a, b, z)",
"z",
"U(n,c,z)",
"\\Phi(a, b, z)",
"M(n,b,z)",
"M(a, b, z)"
],
"isPartOf": [],
"definiens": [
{
"definition": "etc",
"score": 0
},
{
"definition": "Kummer 's function of the first kind",
"score": 2
},
{
"definition": "confluent hypergeometric function",
"score": 1
},
{
"definition": "hypergeometric function",
"score": 1
}
]
}
]
}
|
| Gold 37 | \sigma = \pm 1
|
Mathieu function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 37,
"pid": 87,
"eid": "math.87.54",
"title": "Mathieu function",
"formulae": [
{
"id": "FORMULA_f694135eafc20195a9d96ca3ce8af674",
"formula": "\\sigma = \\pm 1",
"semanticFormula": "\\sigma = \\pm 1",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "\\[Sigma] == \\[PlusMinus]1",
"translationInformation": {
"subEquations": [
"\\[Sigma] = + 1",
"\\[Sigma] = - 1"
],
"freeVariables": [
"\\[Sigma]"
],
"constraints": [],
"tokenTranslations": {
"\\pm": "was translated to: \\[PlusMinus]"
}
}
},
"Maple": {
"translation": "sigma = &+- 1",
"translationInformation": {
"subEquations": [
"sigma = + 1",
"sigma = - 1"
],
"freeVariables": [
"sigma"
],
"constraints": [],
"tokenTranslations": {
"\\pm": "was translated to: &+-"
}
}
}
},
"positions": [
{
"section": 4,
"sentence": 1,
"word": 27
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "value",
"score": 2
}
]
}
]
}
|
| Gold 38 | \frac{d^2f}{dz^2} + \left(\tilde{a}z^2+\tilde{b}z+\tilde{c}\right)f=0
|
Parabolic cylinder function |
|
|
- | - | - |
|
- | ODE. f does not show the argument z. | Full data:
{
"id": 38,
"pid": 88,
"eid": "math.88.0",
"title": "Parabolic cylinder function",
"formulae": [
{
"id": "FORMULA_bec6388631b20f2af14e375b13e1533f",
"formula": "\\frac{d^2f}{dz^2} + \\left(\\tilde{a}z^2+\\tilde{b}z+\\tilde{c}\\right)f=0",
"semanticFormula": "\\deriv [2]{f}{z} +(\\tilde{a} z^2 + \\tilde{b} z + \\tilde{c}) f = 0",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "D[f[z], {z, 2}] + (a*z^2 + b*z + c)*f[z] == 0"
}
},
"positions": [
{
"section": 0,
"sentence": 0,
"word": 19
}
],
"includes": [
"z"
],
"isPartOf": [],
"definiens": [
{
"definition": "solution to the differential equation",
"score": 2
},
{
"definition": "special function",
"score": 1
},
{
"definition": "mathematics",
"score": 0
},
{
"definition": "parabolic cylinder function",
"score": 1
}
]
}
]
}
|
| Gold 39 | c=\infty
|
Painlevé transcendents |
|
|
- | - | - | - | - | - | Full data:
{
"id": 39,
"pid": 89,
"eid": "math.89.23",
"title": "Painlev\u00e9 transcendents",
"formulae": [
{
"id": "FORMULA_0a306ab913684a1ba3935715d3dd8ad8",
"formula": "c=\\infty",
"semanticFormula": "c=\\infty",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "c == Infinity"
},
"Maple": {
"translation": "c = infinity"
}
},
"positions": [
{
"section": 9,
"sentence": 5,
"word": 23
}
],
"includes": [
"c"
],
"isPartOf": [],
"definiens": [
{
"definition": "central charge of the Virasoro algebra",
"score": 2
},
{
"definition": "combination of conformal block",
"score": 1
},
{
"definition": "Painlev\u00e9 VI equation",
"score": 1
},
{
"definition": "two-dimensional conformal field theory",
"score": 1
}
]
}
]
}
|
| Gold 40 | c = a + 1
|
Hypergeometric function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 40,
"pid": 90,
"eid": "math.90.7",
"title": "Hypergeometric function",
"formulae": [
{
"id": "FORMULA_aaffb0ad8dea17d68491d9fb6ebcfbe3",
"formula": "c = a + 1",
"semanticFormula": "c = a + 1",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "c == a + 1"
},
"Maple": {
"translation": "c := a + 1"
}
},
"positions": [
{
"section": 3,
"sentence": 0,
"word": 20
}
],
"includes": [
"a",
"c"
],
"isPartOf": [],
"definiens": [
{
"definition": "value",
"score": 2
}
]
}
]
}
|
| Gold 41 | \frac{1}{\Gamma(z)}= z e^{\gamma z} \prod_{k=1}^\infty \left\{ \left(1+\frac{z}{k}\right)e^{-z/k} \right\}
|
Barnes G-function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 41,
"pid": 91,
"eid": "math.91.47",
"title": "Barnes G-function",
"formulae": [
{
"id": "FORMULA_6bc0d742c4d25c1abb61158150489676",
"formula": "\\frac{1}{\\Gamma(z)}= z e^{\\gamma z} \\prod_{k=1}^\\infty \\left\\{ \\left(1+\\frac{z}{k}\\right)e^{-z\/k} \\right\\}",
"semanticFormula": "\\frac{1}{\\EulerGamma@{z}} = z \\expe^{\\EulerConstant z} \\prod_{k=1}^\\infty \\{(1 + \\frac{z}{k}) \\expe^{-z\/k} \\}",
"confidence": 0.8614665289982916,
"translations": {
"Mathematica": {
"translation": "Divide[1,Gamma[z]] == z*Exp[EulerGamma*z]*Product[(1 +Divide[z,k])*Exp[- z\/k], {k, 1, Infinity}, GenerateConditions->None]",
"translationInformation": {
"subEquations": [
"Divide[1,Gamma[z]] = z*Exp[EulerGamma*z]*Product[(1 +Divide[z,k])*Exp[- z\/k], {k, 1, Infinity}, GenerateConditions->None]"
],
"freeVariables": [
"z"
],
"constraints": [],
"tokenTranslations": {
"\\expe": "Recognizes e with power as the exponential function. It was translated as a function.",
"\\EulerConstant": "Euler-Mascheroni constant was translated to: EulerGamma",
"\\EulerGamma": "Euler Gamma function; Example: \\EulerGamma@{z}\nWill be translated to: Gamma[$0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/5.2#E1\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/Gamma.html"
}
}
},
"Maple": {
"translation": "(1)\/(GAMMA(z)) = z*exp(gamma*z)*product((1 +(z)\/(k))*exp(- z\/k), k = 1..infinity)",
"translationInformation": {
"subEquations": [
"(1)\/(GAMMA(z)) = z*exp(gamma*z)*product((1 +(z)\/(k))*exp(- z\/k), k = 1..infinity)"
],
"freeVariables": [
"z"
],
"constraints": [],
"tokenTranslations": {
"\\expe": "Recognizes e with power as the exponential function. It was translated as a function.",
"\\EulerConstant": "Euler-Mascheroni constant was translated to: gamma",
"\\EulerGamma": "Euler Gamma function; Example: \\EulerGamma@{z}\nWill be translated to: GAMMA($0)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/5.2#E1\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=GAMMA"
}
}
}
},
"positions": [
{
"section": 8,
"sentence": 0,
"word": 55
}
],
"includes": [
"\\,\\Gamma(x)",
"\\, \\gamma",
"z",
"\\,\\gamma"
],
"isPartOf": [],
"definiens": [
{
"definition": "Euler",
"score": 1
},
{
"definition": "Mascheroni",
"score": 1
},
{
"definition": "gamma function",
"score": 2
}
]
}
]
}
|
| Gold 42 | 192/24 = 8 = 2 \times 4
|
Heun function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 42,
"pid": 92,
"eid": "math.92.1-1",
"title": "Heun function",
"formulae": [
{
"id": "FORMULA_8c78ef87048e61947a6d7d4b5e06aa63",
"formula": "192\/24 = 8 = 2 \\times 4",
"semanticFormula": "192\/24 = 8 = 2 \\times 4",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "192\/24 == 8 == 2 * 4",
"translationInformation": {
"subEquations": [
"192\/24 = 8",
"8 = 2 * 4"
],
"freeVariables": [],
"constraints": [],
"tokenTranslations": {
"\\times": "was translated to: *"
}
},
"numericResults": {
"overallResult": "SUCCESS",
"numberOfTests": 2,
"numberOfFailedTests": 0,
"numberOfSuccessfulTests": 2,
"numberOfSkippedTests": 0,
"numberOfErrorTests": 0,
"wasAborted": false,
"crashed": false,
"testCalculationsGroups": [
{
"lhs": "192\/24",
"rhs": "8",
"testExpression": "(192\/24)-(8)",
"activeConstraints": [],
"testCalculations": [
{
"result": "SUCCESS",
"resultExpression": "0.",
"testValues": {}
}
]
},
{
"lhs": "8",
"rhs": "2 * 4",
"testExpression": "(8)-(2 * 4)",
"activeConstraints": [],
"testCalculations": [
{
"result": "SUCCESS",
"resultExpression": "0.",
"testValues": {}
}
]
}
]
},
"symbolicResults": {
"overallResult": "SUCCESS",
"numberOfTests": 2,
"numberOfFailedTests": 0,
"numberOfSuccessfulTests": 2,
"numberOfSkippedTests": 0,
"numberOfErrorTests": 0,
"crashed": false,
"testCalculationsGroup": [
{
"lhs": "192\/24",
"rhs": "8",
"testExpression": "(192\/24)-(8)",
"testCalculations": [
{
"result": "SUCCESS",
"testTitle": "Simple",
"testExpression": "FullSimplify[(192\/24)-(8)]",
"resultExpression": "0",
"wasAborted": false,
"conditionallySuccessful": false
}
]
},
{
"lhs": "8",
"rhs": "2 * 4",
"testExpression": "(8)-(2 * 4)",
"testCalculations": [
{
"result": "SUCCESS",
"testTitle": "Simple",
"testExpression": "FullSimplify[(8)-(2 * 4)]",
"resultExpression": "0",
"wasAborted": false,
"conditionallySuccessful": false
}
]
}
]
}
},
"SymPy": {
"translation": "192\/24 == 8 == 2 * 4",
"translationInformation": {
"subEquations": [
"192\/24 = 8",
"8 = 2 * 4"
],
"freeVariables": [],
"constraints": [],
"tokenTranslations": {
"\\times": "was translated to: *"
}
}
},
"Maple": {
"translation": "192\/24 = 8 = 2 * 4",
"translationInformation": {
"subEquations": [
"192\/24 = 8",
"8 = 2 * 4"
],
"freeVariables": [],
"constraints": [],
"tokenTranslations": {
"\\times": "was translated to: *"
}
},
"numericResults": {
"overallResult": "SUCCESS",
"numberOfTests": 2,
"numberOfFailedTests": 0,
"numberOfSuccessfulTests": 2,
"numberOfSkippedTests": 0,
"numberOfErrorTests": 0,
"wasAborted": false,
"crashed": false,
"testCalculationsGroups": [
{
"lhs": "192\/24",
"rhs": "8",
"testExpression": "evalf((192\/24)-(8))",
"activeConstraints": [],
"testCalculations": [
{
"result": "SUCCESS",
"resultExpression": "0.",
"testValues": {}
}
]
},
{
"lhs": "8",
"rhs": "2 * 4",
"testExpression": "evalf((8)-(2 * 4))",
"activeConstraints": [],
"testCalculations": [
{
"result": "SUCCESS",
"resultExpression": "0.",
"testValues": {}
}
]
}
]
},
"symbolicResults": {
"overallResult": "SUCCESS",
"numberOfTests": 2,
"numberOfFailedTests": 0,
"numberOfSuccessfulTests": 2,
"numberOfSkippedTests": 0,
"numberOfErrorTests": 0,
"crashed": false,
"testCalculationsGroup": [
{
"lhs": "192\/24",
"rhs": "8",
"testExpression": "(192\/24)-(8)",
"testCalculations": [
{
"result": "SUCCESS",
"testTitle": "Simple",
"testExpression": "simplify((192\/24)-(8))",
"resultExpression": "0",
"wasAborted": false,
"conditionallySuccessful": false
}
]
},
{
"lhs": "8",
"rhs": "2 * 4",
"testExpression": "(8)-(2 * 4)",
"testCalculations": [
{
"result": "SUCCESS",
"testTitle": "Simple",
"testExpression": "simplify((8)-(2 * 4))",
"resultExpression": "0",
"wasAborted": false,
"conditionallySuccessful": false
}
]
}
]
}
}
},
"positions": [
{
"section": 3,
"sentence": 1,
"word": 25
}
],
"includes": [],
"isPartOf": [],
"definiens": []
}
]
}
|
| Gold 43 | =2
|
Gegenbauer polynomials |
|
|
- | - | - | - | - | Wrong math detection. | Full data:
{
"id": 43,
"pid": 93,
"eid": "math.93.0-1",
"title": "Gegenbauer polynomials",
"formulae": [
{
"id": "FORMULA_34d9d355f0c0e28d91465c3b575fb0a1",
"formula": "=2",
"semanticFormula": "\\alpha = 2",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "\\[Alpha] = 2"
},
"Maple": {
"translation": "alpha = 2"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 17
}
],
"includes": [],
"isPartOf": [
"\\begin{align}C_0^\\alpha(x) & = 1 \\\\C_1^\\alpha(x) & = 2 \\alpha x \\\\C_n^\\alpha(x) & = \\frac{1}{n}[2x(n+\\alpha-1)C_{n-1}^\\alpha(x) - (n+2\\alpha-2)C_{n-2}^\\alpha(x)].\\end{align}"
],
"definiens": [
{
"definition": "value",
"score": 2
}
]
}
]
}
|
| Gold 44 | \lim_{q\to 1}\;_{j}\phi_k \left[\begin{matrix} q^{a_1} & q^{a_2} & \ldots & q^{a_j} \\ q^{b_1} & q^{b_2} & \ldots & q^{b_k} \end{matrix} ; q,(q-1)^{1+k-j} z \right]=\;_{j}F_k \left[\begin{matrix} a_1 & a_2 & \ldots & a_j \\ b_1 & b_2 & \ldots & b_k \end{matrix} ;z \right]
|
Basic hypergeometric series |
|
- | - |
|
- | - | - | Indef length of arguments are not translatable. | Full data:
{
"id": 44,
"pid": 94,
"eid": "math.94.4",
"title": "Basic hypergeometric series",
"formulae": [
{
"id": "FORMULA_33e3b57bb75d5ea3b5b8ddcceef38430",
"formula": "\\lim_{q\\to 1}\\;_{j}\\phi_k \\left[\\begin{matrix} q^{a_1} & q^{a_2} & \\ldots & q^{a_j} \\\\ q^{b_1} & q^{b_2} & \\ldots & q^{b_k} \\end{matrix} ; q,(q-1)^{1+k-j} z \\right]=\\;_{j}F_k \\left[\\begin{matrix} a_1 & a_2 & \\ldots & a_j \\\\ b_1 & b_2 & \\ldots & b_k \\end{matrix} ;z \\right]",
"semanticFormula": "\\lim_{q\\to 1} \\qgenhyperphi{j}{k}@{q^{a_1} , q^{a_2} , \\ldots , q^{a_j}}{q^{b_1} , q^{b_2} , \\ldots , q^{b_k}}{q}{(q - 1)^{1+k-j} z} = \\genhyperF{j}{k}@{a_1 , a_2 , \\ldots , a_j}{b_1 , b_2 , \\ldots , b_k}{z}",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 1,
"sentence": 5,
"word": 13
}
],
"includes": [
"q^{n}",
"q",
"b",
"a",
"z"
],
"isPartOf": [],
"definiens": [
{
"definition": "q-analog of the hypergeometric series",
"score": 2
},
{
"definition": "unilateral basic hypergeometric series",
"score": 2
},
{
"definition": "basic hypergeometric series",
"score": 2
}
]
}
]
}
|
| Gold 45 | \frac{d^2w}{dz^2}+\left(-\frac{1}{4}+\frac{\kappa}{z}+\frac{1/4-\mu^2}{z^2}\right)w=0
|
Whittaker function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 45,
"pid": 95,
"eid": "math.95.0",
"title": "Whittaker function",
"formulae": [
{
"id": "FORMULA_16ec3a3583ee2b4621d316bf839c1725",
"formula": "\\frac{d^2w}{dz^2}+\\left(-\\frac{1}{4}+\\frac{\\kappa}{z}+\\frac{1\/4-\\mu^2}{z^2}\\right)w=0",
"semanticFormula": "\\deriv [2]{w}{z} +(- \\frac{1}{4} + \\frac{\\kappa}{z} + \\frac{1\/4-\\mu^2}{z^2}) w = 0",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "D[w, {z, 2}]+(-Divide[1,4]+Divide[\\[Kappa],z]+Divide[1\/4 - \\[Mu]^(2),(z)^(2)])*w == 0"
},
"Maple": {
"translation": "diff(w, [z$(2)])+(-(1)\/(4)+(kappa)\/(z)+(1\/4 - (mu)^(2))\/((z)^(2)))*w = 0"
}
},
"positions": [
{
"section": 0,
"sentence": 2,
"word": 4
}
],
"includes": [
"\\mu",
"\\kappa",
"z"
],
"isPartOf": [],
"definiens": [
{
"definition": "Whittaker 's equation",
"score": 2
},
{
"definition": "Whittaker function",
"score": 1
}
]
}
]
}
|
| Gold 46 | e_1=\tfrac12,\qquad e_2=0,\qquad e_3=-\tfrac12
|
Lemniscatic elliptic function |
|
|
- | - | - | - | - | Multi-equation problem (bug). | Full data:
{
"id": 46,
"pid": 96,
"eid": "math.96.1",
"title": "Lemniscatic elliptic function",
"formulae": [
{
"id": "FORMULA_24137d79f0a282f42fdf9ea93576e998",
"formula": "e_1=\\tfrac12,\\qquad e_2=0,\\qquad e_3=-\\tfrac12",
"semanticFormula": "e_1=\\tfrac12,\\qquad e_2=0,\\qquad e_3=-\\tfrac12",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "Subscript[e, 1] == Divide[1,2]\n Subscript[e, 2] = 0\n Subscript[e, 3] = -Divide[1,2]"
},
"Maple": {
"translation": "e[1] := (1)\/(2); e[2] := 0; e[3] := -(1)\/(2)"
}
},
"positions": [
{
"section": 0,
"sentence": 5,
"word": 11
}
],
"includes": [
"e_{1}",
"e_{2}",
"e_{3}"
],
"isPartOf": [],
"definiens": [
{
"definition": "constant",
"score": 2
}
]
}
]
}
|
| Gold 47 | \gamma> 0,n-p=m-q> 0
|
Meijer G-function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 47,
"pid": 98,
"eid": "math.98.53-1",
"title": "Meijer G-function",
"formulae": [
{
"id": "FORMULA_028eb01ef675c90ea0f74fcdd93fc78c",
"formula": "\\gamma> 0,n-p=m-q> 0",
"semanticFormula": "\\gamma> 0,n-p=m-q> 0",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "\\[Gamma] > 0\n n - p == m - q > 0"
},
"Maple": {
"translation": "gamma > 0; n - p = m - q > 0"
}
},
"positions": [
{
"section": 12,
"sentence": 0,
"word": 17
}
],
"includes": [
"m",
"q",
"p=q> 0",
"n",
"p=q",
"\\gamma>"
],
"isPartOf": [],
"definiens": [
{
"definition": "constraint",
"score": 2
}
]
}
]
}
|
| Gold 48 | \begin{pmatrix} j \\ m \quad m'\end{pmatrix}:= \sqrt{2 j + 1}\begin{pmatrix} j & 0 & j \\ m & 0 & m'\end{pmatrix}= (-1)^{j - m'} \delta_{m, -m'}
|
3-j symbol |
|
|
- | - | - | - | - | LCT does not support matrix translations yet. | Full data:
{
"id": 48,
"pid": 99,
"eid": "math.99.30",
"title": "3-j symbol",
"formulae": [
{
"id": "FORMULA_3f987b881a59a03904ff9a79476faae0",
"formula": "\\begin{pmatrix} j \\\\ m \\quad m'\\end{pmatrix}:= \\sqrt{2 j + 1}\\begin{pmatrix} j & 0 & j \\\\ m & 0 & m'\\end{pmatrix}= (-1)^{j - m'} \\delta_{m, -m'}",
"semanticFormula": "\\begin{pmatrix} j \\\\ m \\quad m'\\end{pmatrix}:= \\sqrt{2 j + 1}\\begin{pmatrix} j & 0 & j \\\\ m & 0 & m'\\end{pmatrix}= (-1)^{j - m'} \\delta_{m, -m'}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "Wigner[j_, m_, m\\[Prime]_] := Sqrt[2*j+1] * {{j, 0, j}, {m, 0, m\\[Prime]}} = (-1)^(j-m\\[Prime])*Subscript[\\[Delta], m, -m\\[Prime]]"
}
},
"positions": [
{
"section": 10,
"sentence": 0,
"word": 23
}
],
"includes": [
"j",
"m"
],
"isPartOf": [],
"definiens": [
{
"definition": "Wigner 1-jm symbol",
"score": 2
},
{
"definition": "metric tensor in angular-momentum theory",
"score": 2
},
{
"definition": "quantity",
"score": 0
}
]
}
]
}
|
| Gold 49 | \begin{Bmatrix} i & j & \ell\\ k & m & n \end{Bmatrix}= (\Phi_{i,j}^{k,m})_{\ell,n}
|
6-j symbol |
|
- | - | - | - |
|
- | A matrix cannot be defined as a function in Mathematica. | Full data:
{
"id": 49,
"pid": 100,
"eid": "math.100.14",
"title": "6-j symbol",
"formulae": [
{
"id": "FORMULA_21d6ec52b25bb130bf068c4857bbcb93",
"formula": "\\begin{Bmatrix} i & j & \\ell\\\\ k & m & n \\end{Bmatrix}= (\\Phi_{i,j}^{k,m})_{\\ell,n}",
"semanticFormula": "\\Wignersixjsym{i}{j}{\\ell}{k}{m}{n} = (\\Phi_{i,j}^{k,m})_{\\ell,n}",
"confidence": 0.8624533614429312,
"translations": {},
"positions": [
{
"section": 5,
"sentence": 4,
"word": 23
}
],
"includes": [
"j"
],
"isPartOf": [],
"definiens": [
{
"definition": "6j symbol",
"score": 2
},
{
"definition": "associativity isomorphism",
"score": 2
},
{
"definition": "symbol",
"score": 1
},
{
"definition": "vector space isomorphism",
"score": 2
},
{
"definition": "Wigner",
"score": 1
},
{
"definition": "Wigner 's 6 - j symbol",
"score": 2
}
]
}
]
}
|
| Gold 50 | \sum_{j_7 j_8} (2j_7+1)(2j_8+1) \begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6\\ j_7 & j_8 & j_9 \end{Bmatrix} \begin{Bmatrix} j_1 & j_2 & j_3'\\ j_4 & j_5 & j_6'\\ j_7 & j_8 & j_9 \end{Bmatrix} = \frac{\delta_{j_3j_3'}\delta_{j_6j_6'} \begin{Bmatrix} j_1 & j_2 & j_3 \end{Bmatrix} \begin{Bmatrix} j_4 & j_5 & j_6\end{Bmatrix} \begin{Bmatrix} j_3 & j_6 & j_9 \end{Bmatrix}} {(2j_3+1)(2j_6+1)}
|
9-j symbol |
|
- | - | - | - | - | - | Mistakenly interpreted as Wigner 6-j rather than 9-j. | Full data:
{
"id": 50,
"pid": 101,
"eid": "math.101.32",
"title": "9-j symbol",
"formulae": [
{
"id": "FORMULA_08d08037d9e64d85aa3645470ce645af",
"formula": "\\sum_{j_7 j_8} (2j_7+1)(2j_8+1) \\begin{Bmatrix} j_1 & j_2 & j_3\\\\ j_4 & j_5 & j_6\\\\ j_7 & j_8 & j_9 \\end{Bmatrix} \\begin{Bmatrix} j_1 & j_2 & j_3'\\\\ j_4 & j_5 & j_6'\\\\ j_7 & j_8 & j_9 \\end{Bmatrix} = \\frac{\\delta_{j_3j_3'}\\delta_{j_6j_6'} \\begin{Bmatrix} j_1 & j_2 & j_3 \\end{Bmatrix} \\begin{Bmatrix} j_4 & j_5 & j_6\\end{Bmatrix} \\begin{Bmatrix} j_3 & j_6 & j_9 \\end{Bmatrix}} {(2j_3+1)(2j_6+1)}",
"semanticFormula": "\\sum_{j_7 j_8} (2j_7+1)(2j_8+1) \\Wignerninejsym{j_1}{j_2}{j_3}{j_4}{j_5}{j_6}{j_7}{j_8}{j_9} \\Wignerninejsym{j_1}{j_2}{j_3'}{j_4}{j_5}{j_6'}{j_7}{j_8}{j_9} = \\frac{\\delta_{j_3j_3'}\\delta_{j_6j_6'} \\begin{Bmatrix} j_1 & j_2 & j_3 \\end{Bmatrix} \\begin{Bmatrix} j_4 & j_5 & j_6\\end{Bmatrix} \\begin{Bmatrix} j_3 & j_6 & j_9 \\end{Bmatrix}}{(2j_3+1)(2j_6+1)}",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 5,
"sentence": 0,
"word": 10
}
],
"includes": [
"j",
"_{4}"
],
"isPartOf": [],
"definiens": [
{
"definition": "orthogonality relation",
"score": 1
},
{
"definition": "triangular delta",
"score": 2
},
{
"definition": "symbol",
"score": 1
},
{
"definition": "Wigner 's 9 - j symbol",
"score": 2
}
]
}
]
}
|
| Gold 51 | \mathcal{K}_k(x; n,q) = \sum_{j=0}^{k}(-q)^j (q-1)^{k-j} \binom {n-j}{k-j} \binom{x}{j}
|
Kravchuk polynomials |
|
|
- | - | - |
|
- | Krawtchouk vs Kravchuk (synonym problem) | Full data:
{
"id": 51,
"pid": 102,
"eid": "math.102.5",
"title": "Kravchuk polynomials",
"formulae": [
{
"id": "FORMULA_6b7eb62a3e02e45fb1365dd2f07a5bbc",
"formula": "\\mathcal{K}_k(x; n,q) = \\sum_{j=0}^{k}(-q)^j (q-1)^{k-j} \\binom {n-j}{k-j} \\binom{x}{j}",
"semanticFormula": "\\KrawtchoukpolyK{k}@{x}{n}{q} = \\sum_{j=0}^{k}(-q)^j (q-1)^{k-j} \\binom {n-j}{k-j} \\binom{x}{j}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "K[k_, x_, n_, q_] := Sum[(- q)^(j)*(q - 1)^(k - j)*Binomial[n - j,k - j]*Binomial[x,j], {j, 0, k}, GenerateConditions->None]"
}
},
"positions": [
{
"section": 2,
"sentence": 0,
"word": 9
}
],
"includes": [
"q",
"n"
],
"isPartOf": [],
"definiens": [
{
"definition": "following alternative expression",
"score": 0
},
{
"definition": "Kravchuk polynomial",
"score": 2
}
]
}
]
} |
| Gold 52 | g_1(x) = \sum_{k \geq 1} \frac{\sin(k \pi / 4)}{k! (8x)^k} \prod_{l = 1}^k (2l - 1)^2
|
Kelvin functions |
|
|
|
- | - | - | - | - | Full data:
{
"id": 52,
"pid": 103,
"eid": "math.103.8",
"title": "Kelvin functions",
"formulae": [
{
"id": "FORMULA_07453e6baf8f216467f9b664de795bfc",
"formula": "g_1(x) = \\sum_{k \\geq 1} \\frac{\\sin(k \\pi \/ 4)}{k! (8x)^k} \\prod_{l = 1}^k (2l - 1)^2",
"semanticFormula": "g_1(x) = \\sum_{k \\geq 1} \\frac{\\sin(k \\cpi \/ 4)}{k! (8x)^k} \\prod_{l = 1}^k(2 l - 1)^2",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "Subscript[g, 1][x_] := Sum[Divide[Sin[k*Pi\/4],(k)!*(8*x)^(k)]*Product[(2*l - 1)^(2), {l, 1, k}, GenerateConditions->None], {k, 1, Infinity}, GenerateConditions->None]"
},
"Maple": {
"translation": "g[1] := (x) -> sum((sin(k*Pi\/4))\/(factorial(k)*(8*x)^(k))*product((2*l - 1)^(2), l = 1..k), k = 1..infinity)"
}
},
"positions": [
{
"section": 1,
"sentence": 1,
"word": 28
}
],
"includes": [
"x",
"x)",
"g_1(x)"
],
"isPartOf": [],
"definiens": [
{
"definition": "series expansion",
"score": 1
},
{
"definition": "special case",
"score": 0
},
{
"definition": "asymptotic series",
"score": 1
},
{
"definition": "definition",
"score": 2
}
]
}
]
} |
| Gold 53 | S_{\mu,\nu}(z) = s_{\mu,\nu}(z) + 2^{\mu-1} \Gamma\left(\frac{\mu + \nu + 1}{2}\right) \Gamma\left(\frac{\mu - \nu + 1}{2}\right)\left(\sin \left[(\mu - \nu)\frac{\pi}{2}\right] J_\nu(z) - \cos \left[(\mu - \nu)\frac{\pi}{2}\right] Y_\nu(z)\right)
|
Lommel function |
|
|
- | - | - |
|
- | No information about gamma function | Full data:
{
"id": 53,
"pid": 104,
"eid": "math.104.2",
"title": "Lommel function",
"formulae": [
{
"id": "FORMULA_03f5cb50caaedb9f0a4ada231fd61c58",
"formula": "S_{\\mu,\\nu}(z) = s_{\\mu,\\nu}(z) + 2^{\\mu-1} \\Gamma\\left(\\frac{\\mu + \\nu + 1}{2}\\right) \\Gamma\\left(\\frac{\\mu - \\nu + 1}{2}\\right)\\left(\\sin \\left[(\\mu - \\nu)\\frac{\\pi}{2}\\right] J_\\nu(z) - \\cos \\left[(\\mu - \\nu)\\frac{\\pi}{2}\\right] Y_\\nu(z)\\right)",
"semanticFormula": "\\LommelS{\\mu}{\\nu}@{z} = \\Lommels{\\mu}{\\nu}@{z} + 2^{\\mu-1} \\EulerGamma@{\\frac{\\mu + \\nu + 1}{2}} \\EulerGamma@{\\frac{\\mu - \\nu + 1}{2}}(\\sin [(\\mu - \\nu) \\frac{\\cpi}{2}] \\BesselJ{\\nu}@{z} - \\cos [(\\mu - \\nu) \\frac{\\cpi}{2}] \\BesselY{\\nu}@{z})",
"confidence": 0.8775479393290169,
"translations": {
"Mathematica": {
"translation": "S[\\[Mu]_, \\[Nu]_, z_] := Divide[Pi,2]*(BesselY[\\[Nu], z]*Integrate[(x)^\\[Mu]* BesselJ[\\[Nu], x], {x, 0, z}, GenerateConditions->None]- BesselJ[\\[Nu], z]*Integrate[(x)^\\[Mu]* BesselY[\\[Nu], x], {x, 0, z}, GenerateConditions->None]) + (2)^(\\[Mu]- 1)* Gamma[Divide[\\[Mu]+ \\[Nu]+ 1,2]]*Gamma[Divide[\\[Mu]- \\[Nu]+ 1,2]]*(Sin[((\\[Mu]- \\[Nu])*Divide[Pi,2])*]*BesselJ[\\[Nu], z]- Cos[((\\[Mu]- \\[Nu])*Divide[Pi,2])*]*BesselY[\\[Nu], z])"
},
"Maple": {
"translation": "LommelS1(mu, nu, z) = (Pi)\/(2)*(BesselY(nu, z)*int((x)^(mu)* BesselJ(nu, x), x = 0..z)- BesselJ(nu, z)*int((x)^(mu)* BesselY(nu, x), x = 0..z))"
}
},
"positions": [
{
"section": 0,
"sentence": 1,
"word": 18
}
],
"includes": [
"s_{\\mu,\\nu}(z)",
"S_{\\mu,\\nu}(z)",
"J_{\\nu}(z)",
"Y_{\\nu}(z)"
],
"isPartOf": [],
"definiens": [
{
"definition": "Lommel function",
"score": 2
},
{
"definition": "Bessel function of the first kind",
"score": 2
},
{
"definition": "Bessel function of the second kind",
"score": 2
}
]
}
]
} |
| Gold 54 | \mathbf{H}_{\alpha}(z) = \frac{z^{\alpha+1}}{2^{\alpha}\sqrt{\pi} \Gamma \left (\alpha+\tfrac{3}{2} \right )} {}_1F_2 \left (1,\tfrac{3}{2}, \alpha+\tfrac{3}{2},-\tfrac{z^2}{4} \right )
|
Struve function |
|
|
- |
|
- | - | - | Arguments of are split by commas. That is wrong notation. Hence, our semantic patterns did not match. | Full data:
{
"id": 54,
"pid": 105,
"eid": "math.105.18",
"title": "Struve function",
"formulae": [
{
"id": "FORMULA_6dc2da7f595d2f199fbc15768167f006",
"formula": "\\mathbf{H}_{\\alpha}(z) = \\frac{z^{\\alpha+1}}{2^{\\alpha}\\sqrt{\\pi} \\Gamma \\left (\\alpha+\\tfrac{3}{2} \\right )} {}_1F_2 \\left (1,\\tfrac{3}{2}, \\alpha+\\tfrac{3}{2},-\\tfrac{z^2}{4} \\right )",
"semanticFormula": "\\StruveH{\\alpha}@{z} = \\frac{z^{\\alpha+1}}{2^{\\alpha} \\sqrt{\\cpi} \\EulerGamma@{\\alpha + \\tfrac{3}{2}}} \\genhyperF{1}{2}@{1}{\\tfrac{3}{2}, \\alpha + \\tfrac{3}{2}}{- \\tfrac{z^2}{4}}",
"confidence": 0.8740850655136605,
"translations": {
"Mathematica": {
"translation": "StruveH[\\[Alpha], z] == Divide[(z)^(\\[Alpha]+ 1),(2)^\\[Alpha]*Sqrt[Pi]*Gamma[\\[Alpha]+Divide[3,2]]]*HypergeometricPFQ[{1}, {Divide[3,2], \\[Alpha]+Divide[3,2]}, -Divide[(z)^(2),4]]"
},
"Maple": {
"translation": "StruveH(alpha, z) = ((z)^(alpha + 1))\/((2)^(alpha)*sqrt(Pi)*GAMMA(alpha +(3)\/(2)))*hypergeom([1], [(3)\/(2), alpha +(3)\/(2)], -((z)^(2))\/(4))"
}
},
"positions": [
{
"section": 6,
"sentence": 2,
"word": 31
}
],
"includes": [
"_{1}F_{2}",
"\\mathbf{K}_\\alpha(x)",
"\\alpha",
"\\Gamma(z)",
"\\mathbf{H}_{\\alpha}(x)",
"\\mathbf{L}_{\\alpha}(x)",
"\\mathbf{H}_{\\alpha}(z)",
"Y_{\\alpha}(x)",
"\\mathbf{M}_\\alpha(x)"
],
"isPartOf": [],
"definiens": [
{
"definition": "hypergeometric function",
"score": 2
},
{
"definition": "Struve",
"score": 2
},
{
"definition": "Struve function",
"score": 2
},
{
"definition": "gamma function",
"score": 2
}
]
}
]
} |
| Gold 55 | f(t+p) = f(t)
|
Hill differential equation |
|
|
- | - | - | - | - | - | Full data:
{
"id": 55,
"pid": 106,
"eid": "math.106.7",
"title": "Hill differential equation",
"formulae": [
{
"id": "FORMULA_3a6745862e8f6ef2b93c343ad82b40c0",
"formula": "f(t+p) = f(t)",
"semanticFormula": "f(t+p) = f(t)",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "f[t + p] == f[t]"
},
"Maple": {
"translation": "f(t + p) = f(t)"
}
},
"positions": [
{
"section": 0,
"sentence": 1,
"word": 21
}
],
"includes": [
"f(t)",
"t",
"p",
"f(t+\\pi)=f(t)"
],
"isPartOf": [
"f(t+\\pi)=f(t)"
],
"definiens": [
{
"definition": "function",
"score": 2
},
{
"definition": "periodic function by minimal period",
"score": 2
}
]
}
]
} |
| Gold 56 | \mathbf{J}_\nu(z)=\cos\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k}}{4^k\Gamma\left(k+\frac{\nu}{2}+1\right)\Gamma\left(k-\frac{\nu}{2}+1\right)}+\sin\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k+1}}{2^{2k+1}\Gamma\left(k+\frac{\nu}{2}+\frac{3}{2}\right)\Gamma\left(k-\frac{\nu}{2}+\frac{3}{2}\right)}
|
Anger function |
|
|
- | - | - |
|
- | No information about gamma function. | Full data:
{
"id": 56,
"pid": 108,
"eid": "math.108.3",
"title": "Anger function",
"formulae": [
{
"id": "FORMULA_014efde25f995ccd08168a36ec7ef86d",
"formula": "\\mathbf{J}_\\nu(z)=\\cos\\frac{\\pi\\nu}{2}\\sum_{k=0}^\\infty\\frac{(-1)^kz^{2k}}{4^k\\Gamma\\left(k+\\frac{\\nu}{2}+1\\right)\\Gamma\\left(k-\\frac{\\nu}{2}+1\\right)}+\\sin\\frac{\\pi\\nu}{2}\\sum_{k=0}^\\infty\\frac{(-1)^kz^{2k+1}}{2^{2k+1}\\Gamma\\left(k+\\frac{\\nu}{2}+\\frac{3}{2}\\right)\\Gamma\\left(k-\\frac{\\nu}{2}+\\frac{3}{2}\\right)}",
"semanticFormula": "\\AngerJ{\\nu}@{z} = \\cos \\frac{\\cpi\\nu}{2} \\sum_{k=0}^\\infty \\frac{(-1)^k z^{2k}}{4^k\\EulerGamma@{k+\\frac{\\nu}{2}+1}\\EulerGamma@{k-\\frac{\\nu}{2}+1}}+\\sin\\frac{\\cpi\\nu}{2}\\sum_{k=0}^\\infty\\frac{(-1)^k z^{2k+1}}{2^{2k+1}\\EulerGamma@{k+\\frac{\\nu}{2}+\\frac{3}{2}}\\EulerGamma@{k-\\frac{\\nu}{2}+\\frac{3}{2}}}",
"confidence": 0.8648813564530858,
"translations": {
"Mathematica": {
"translation": "AngerJ[\\[Nu], z] == Cos[Divide[Pi*\\[Nu],2]]*Sum[Divide[(- 1)^(k)* (z)^(2*k),(4)^(k)* Gamma[k +Divide[\\[Nu],2]+ 1]*Gamma[k -Divide[\\[Nu],2]+ 1]], {k, 0, Infinity}, GenerateConditions->None]+ Sin[Divide[Pi*\\[Nu],2]]*Sum[Divide[(- 1)^(k)* (z)^(2*k + 1),(2)^(2*k + 1)* Gamma[k +Divide[\\[Nu],2]+Divide[3,2]]*Gamma[k -Divide[\\[Nu],2]+Divide[3,2]]], {k, 0, Infinity}, GenerateConditions->None]"
},
"Maple": {
"translation": "AngerJ(nu, z) = cos((Pi*nu)\/(2))*sum(((- 1)^(k)* (z)^(2*k))\/((4)^(k)* GAMMA(k +(nu)\/(2)+ 1)*GAMMA(k -(nu)\/(2)+ 1)), k = 0..infinity)+ sin((Pi*nu)\/(2))*sum(((- 1)^(k)* (z)^(2*k + 1))\/((2)^(2*k + 1)* GAMMA(k +(nu)\/(2)+(3)\/(2))*GAMMA(k -(nu)\/(2)+(3)\/(2))), k = 0..infinity)"
}
},
"positions": [
{
"section": 2,
"sentence": 0,
"word": 8
}
],
"includes": [
"J_{\\nu}",
"\\mathbf{J}_{\\nu}",
"\\nu"
],
"isPartOf": [],
"definiens": [
{
"definition": "power series expansion",
"score": 2
},
{
"definition": "Anger function",
"score": 2
},
{
"definition": "Gamma function",
"score": 2
}
]
}
]
} |
| Gold 57 | (\operatorname{Ec})^'_{2K} = (\operatorname{Ec})^'_0 = 0, \;\; (\operatorname{Es})^'_{2K} = (\operatorname{Es})^'_0 = 0
|
Lamé function |
|
- | - | - | - | - | - | No translation possible. | Full data:
{
"id": 57,
"pid": 109,
"eid": "math.109.27",
"title": "Lam\u00e9 function",
"formulae": [
{
"id": "FORMULA_7d20395e75eeb74df48a681897d9d727",
"formula": "(\\operatorname{Ec})^'_{2K} = (\\operatorname{Ec})^'_0 = 0, \\;\\; (\\operatorname{Es})^'_{2K} = (\\operatorname{Es})^'_0 = 0",
"semanticFormula": "(\\operatorname{Ec})_{2K}^' =(\\operatorname{Ec})_0^' = 0 ,(\\operatorname{Es})_{2K}^' =(\\operatorname{Es})_0^' = 0",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 2,
"sentence": 3,
"word": 31
}
],
"includes": [
"\\operatorname{Ec}",
"\\operatorname{Es}"
],
"isPartOf": [],
"definiens": [
{
"definition": "boundary condition",
"score": 2
},
{
"definition": "ellipsoidal wave",
"score": 2
}
]
}
]
} |
| Gold 58 | \int_{-\infty}^{+\infty} e^{-x^2} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)
|
Gauss–Hermite quadrature |
|
- | - | - | - | - | - | No translation possible. | Full data:
{
"id": 58,
"pid": 110,
"eid": "math.110.1",
"title": "Gauss\u2013Hermite quadrature",
"formulae": [
{
"id": "FORMULA_cdf8d887d4b5ad1a7724773d8eef8fd2",
"formula": "\\int_{-\\infty}^{+\\infty} e^{-x^2} f(x)\\,dx \\approx \\sum_{i=1}^n w_i f(x_i)",
"semanticFormula": "\\int_{-\\infty}^{+\\infty} \\expe^{-x^2} f(x) \\diff{x} \\approx \\sum_{i=1}^n w_i f(x_i)",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 0,
"sentence": 1,
"word": 3
}
],
"includes": [
"\\int_{-\\infty}^{+\\infty} e^{-x^2} f(x)\\,dx",
"n",
"x_{i}",
"w_{i}"
],
"isPartOf": [],
"definiens": [
{
"definition": "value of integral",
"score": 2
},
{
"definition": "form of Gaussian quadrature",
"score": 2
},
{
"definition": "Gauss -- Hermite quadrature",
"score": 2
},
{
"definition": "Hermite polynomial",
"score": 1
},
{
"definition": "associated weight",
"score": 2
}
]
}
]
} |
| Gold 59 | p_n(x;a,b,c,d|q) =(ab,ac,ad;q)_na^{-n}\;_{4}\phi_3 \left[\begin{matrix} q^{-n}&abcdq^{n-1}&ae^{i\theta}&ae^{-i\theta} \\ ab&ac&ad \end{matrix} ; q,q \right]
|
Askey–Wilson polynomials |
|
|
|
- | - | - | - | Could not extract the name Askey-Wilson polynomials. | Full data:
{
"id": 59,
"pid": 111,
"eid": "math.111.0",
"title": "Askey\u2013Wilson polynomials",
"formulae": [
{
"id": "FORMULA_cfe946a0547913234ac79d398f269607",
"formula": "p_n(x;a,b,c,d|q) =(ab,ac,ad;q)_na^{-n}\\;_{4}\\phi_3 \\left[\\begin{matrix} q^{-n}&abcdq^{n-1}&ae^{i\\theta}&ae^{-i\\theta} \\\\ ab&ac&ad \\end{matrix} ; q,q \\right]",
"semanticFormula": "\\AskeyWilsonpolyp{n}@{x}{a}{b}{c}{d}{q} = \\qmultiPochhammersym{ab , ac , ad}{q}{n} a^{-n} \\qgenhyperphi{4}{3}@{q^{-n} , abcdq^{n-1} , a\\expe^{\\iunit\\theta} , a\\expe^{-\\iunit\\theta}}{ab , ac , ad}{q}{q}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "p[n_, x_, a_, b_, c_, d_, q_] := Product[QPochhammer[Part[{a*b , a*c , a*d},i],q,n],{i,1,Length[{a*b , a*c , a*d}]}]*(a)^(- n)* QHypergeometricPFQ[{(q)^(- n), a*b*c*d*(q)^(n - 1), a*Exp[I*\\[Theta]], a*Exp[- I*\\[Theta]]},{a*b , a*c , a*d},q,q]"
}
},
"positions": [
{
"section": 0,
"sentence": 3,
"word": 4
}
],
"includes": [
"\\phi",
"_{n}",
"n"
],
"isPartOf": [],
"definiens": [
{
"definition": "basic hypergeometric function",
"score": 2
},
{
"definition": "q-Pochhammer symbol",
"score": 2
},
{
"definition": "Askey\u2013Wilson polynomials",
"score": 2
}
]
}
]
} |
| Gold 60 | Q_n(x;\alpha,\beta,N)= {}_3F_2(-n,-x,n+\alpha+\beta+1;\alpha+1,-N+1;1).
|
Hahn polynomials |
|
|
|
- | - | - |
|
- | Full data:
{
"id": 60,
"pid": 112,
"eid": "math.112.0",
"title": "Hahn polynomials",
"formulae": [
{
"id": "FORMULA_777007203448847310455e0b0eaaeb2c",
"formula": "Q_n(x;\\alpha,\\beta,N)= {}_3F_2(-n,-x,n+\\alpha+\\beta+1;\\alpha+1,-N+1;1).",
"semanticFormula": "\\HahnpolyQ{n}@{x}{\\alpha}{\\beta}{N} = \\genhyperF{3}{2}@{- n , - x , n + \\alpha + \\beta + 1}{\\alpha + 1 , - N + 1}{1}",
"confidence": 0.8953028732079359,
"translations": {
"Mathematica": {
"translation": "Q[n_, x_, \\[Alpha]_, \\[Beta]_, N_] := HypergeometricPFQ[{- n , - x , n + \\[Alpha]+ \\[Beta]+ 1}, {\\[Alpha]+ 1 , - N + 1}, 1]"
}
},
"positions": [
{
"section": 0,
"sentence": 3,
"word": 11
}
],
"includes": [
"R_{n}(x;\\gamma,\\delta,N)",
"S_{n}(x;a,b,c)"
],
"isPartOf": [],
"definiens": [
{
"definition": "Hahn polynomial",
"score": 2
},
{
"definition": "basic hypergeometric function",
"score": 2
},
{
"definition": "hypergeometric function",
"score": 2
}
]
}
]
} |
| Gold 61 | \sum_{x=0}^\infty \frac{\mu^x}{x!} C_n(x; \mu)C_m(x; \mu)=\mu^{-n} e^\mu n! \delta_{nm}, \quad \mu>0
|
Charlier polynomials |
|
|
- | - | - |
|
- | Did not found Charlier polynomial. | Full data:
{
"id": 61,
"pid": 113,
"eid": "math.113.2",
"title": "Charlier polynomials",
"formulae": [
{
"id": "FORMULA_b76bcf7237b989f6b5d90082fafa53f1",
"formula": "\\sum_{x=0}^\\infty \\frac{\\mu^x}{x!} C_n(x; \\mu)C_m(x; \\mu)=\\mu^{-n} e^\\mu n! \\delta_{nm}, \\quad \\mu>0",
"semanticFormula": "\\sum_{x=0}^\\infty \\frac{\\mu^x}{x!} \\CharlierpolyC{n}@{x}{\\mu} \\CharlierpolyC{m}@{x}{\\mu} = \\mu^{-n} \\expe^\\mu n! \\delta_{nm} , \\quad \\mu > 0",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "Sum[Divide[\\[Mu]^x, x!] * HypergeometricPFQ[{-n, -x}, {}, -Divide[1,\\[Mu]]] * HypergeometricPFQ[{-m, -x}, {}, -Divide[1,\\[Mu]]], {x, 0, Infinity}] == \\[Mu]^(-n)*Exp[\\[Mu]]*n!*Subscript[\\[Delta], n, m]"
}
},
"positions": [
{
"section": 0,
"sentence": 2,
"word": 5
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "orthogonality relation",
"score": 2
},
{
"definition": "Charlier polynomial",
"score": 2
}
]
}
]
} |
| Gold 62 | p_n(q^{-x}+q^{x+1}cd;a,b,c,d;q) = {}_4\phi_3\left[\begin{matrix} q^{-n} &abq^{n+1}&q^{-x}&q^{x+1}cd\\aq&bdq&cq\\ \end{matrix};q;q\right]
|
Q-Racah polynomials |
|
- | - | - | - | - |
|
Did not find q-Recah polynomial. Since it is not a definition, and q-Recah are not supported by Mathematica, there is no translation possible. | Full data:
{
"id": 62,
"pid": 114,
"eid": "math.114.0",
"title": "Q-Racah polynomials",
"formulae": [
{
"id": "FORMULA_51c23bddc19530680328afbf28235b90",
"formula": "p_n(q^{-x}+q^{x+1}cd;a,b,c,d;q) = {}_4\\phi_3\\left[\\begin{matrix} q^{-n} &abq^{n+1}&q^{-x}&q^{x+1}cd\\\\aq&bdq&cq\\\\ \\end{matrix};q;q\\right]",
"semanticFormula": "\\qRacahpolyR{n}@{q^{-x} + q^{x+1} cd}{a}{b}{c}{d}{q} = \\qgenhyperphi{4}{3}@{q^{-n}, abq^{n+1}, q^{-x}, q^{x+1}cd}{aq , bdq , cq}{q}{q}",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 15
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "term of basic hypergeometric function",
"score": 2
}
]
}
]
} |
| Gold 63 | \displaystyle c_n(q^{-x};a;q) = {}_2\phi_1(q^{-n},q^{-x};0;q,-q^{n+1}/a)
|
Q-Charlier polynomials |
|
- | - | - | - | - | - |
|
Full data:
{
"id": 63,
"pid": 115,
"eid": "math.115.0",
"title": "Q-Charlier polynomials",
"formulae": [
{
"id": "FORMULA_925d68ff3ddf733a69ec9936dfede5d6",
"formula": "\\displaystyle c_n(q^{-x};a;q) = {}_2\\phi_1(q^{-n},q^{-x};0;q,-q^{n+1}\/a)",
"semanticFormula": "c_n(q^{-x} ; a ; q) = \\qgenhyperphi{2}{1}@{q^{-n} , q^{-x}}{0}{q}{- q^{n+1} \/ a}",
"confidence": 0.5776294951318733,
"translations": {},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 15
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "q-Charlier polynomial",
"score": 2
},
{
"definition": "term of the basic hypergeometric function",
"score": 2
}
]
}
]
} |
| Gold 64 | M_n(x,\beta,\gamma) = \sum_{k=0}^n (-1)^k{n \choose k}{x\choose k}k!(x+\beta)_{n-k}\gamma^{-k}
|
Meixner polynomials |
|
|
- | - | - |
|
- | Did not find Meixner. | Full data:
{
"id": 64,
"pid": 116,
"eid": "math.116.0",
"title": "Meixner polynomials",
"formulae": [
{
"id": "FORMULA_29a1f82de004c5721c8dfc5dd1dc5b98",
"formula": "M_n(x,\\beta,\\gamma) = \\sum_{k=0}^n (-1)^k{n \\choose k}{x\\choose k}k!(x+\\beta)_{n-k}\\gamma^{-k}",
"semanticFormula": "\\MeixnerpolyM{n}@{x}{\\beta}{\\gamma} = \\sum_{k=0}^n(- 1)^k{n \\choose k}{x\\choose k} k! \\Pochhammersym{x + \\beta}{n-k} \\gamma^{-k}",
"confidence": 0.8953028732079359,
"translations": {
"Mathematica": {
"translation": "M[n_, x_, \\[Beta]_, \\[Gamma]_] := Sum[(- 1)^(k)*Binomial[n,k]*Binomial[x,k]*(k)!*Pochhammer[x + \\[Beta], n - k]*\\[Gamma]^(- k), {k, 0, n}, GenerateConditions->None]"
}
},
"positions": [
{
"section": 0,
"sentence": 1,
"word": 16
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "Meixner polynomial",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 1
},
{
"definition": "term of binomial coefficient",
"score": 1
}
]
}
]
} |
| Gold 65 | x(1-x) \frac {\partial^2F_1(x,y)} {\partial x^2} + y(1-x) \frac {\partial^2F_1(x,y)} {\partial x \partial y} + [c - (a+b_1+1) x] \frac {\partial F_1(x,y)} {\partial x} - b_1 y \frac {\partial F_1(x,y)} {\partial y} - a b_1 F_1(x,y) = 0
|
Appell series |
|
|
- |
|
- |
|
- | Cannot match hidden arguments of Appell F1 function. | Full data:
{
"id": 65,
"pid": 117,
"eid": "math.117.19",
"title": "Appell series",
"formulae": [
{
"id": "FORMULA_85014aaf0c7c1f4fe433115e796a03db",
"formula": "x(1-x) \\frac {\\partial^2F_1(x,y)} {\\partial x^2} + y(1-x) \\frac {\\partial^2F_1(x,y)} {\\partial x \\partial y} + [c - (a+b_1+1) x] \\frac {\\partial F_1(x,y)} {\\partial x} - b_1 y \\frac {\\partial F_1(x,y)} {\\partial y} - a b_1 F_1(x,y) = 0",
"semanticFormula": "x(1-x) \\deriv[2]{\\AppellF{1}@{a}{b_1}{b_2}{\\gamma}{x}{y}}{x} + y(1-x) \\frac{\\pdiff[2]{\\AppellF{1}@{a}{b_1}{b_2}{\\gamma}{x}{y}}}{\\pdiff{x}\\pdiff{y}} + [c - (a+b_1+1) x] \\deriv[1]{\\AppellF{1}@{a}{b_1}{b_2}{\\gamma}{x}{y}}{x} - b_1 y \\deriv[1]{\\AppellF{1}@{a}{b_1}{b_2}{\\gamma}{x}{y}}{y} - a b_1 \\AppellF{1}@{a}{b_1}{b_2}{\\gamma}{x}{y} = 0",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "x*(1-x) * D[AppellF[a, Subscript[b, 1], Subscript[b, 2], \\[Gamma], x, y], {x,2}] + y*(1-x) * D[AppellF[a, Subscript[b, 1], Subscript[b, 2], \\[Gamma], x, y], x, y] + (c - (a+Subscript[b, 1]+1)*x) * D[AppellF[a, Subscript[b, 1], Subscript[b, 2], \\[Gamma], x, y], x] - Subscript[b,1] * y * D[AppellF[a, Subscript[b, 1], Subscript[b, 2], \\[Gamma], x, y], y] - a*Subscript[b,1]*AppellF[a, Subscript[b, 1], Subscript[b, 2], \\[Gamma], x, y] == 0"
}
},
"positions": [
{
"section": 3,
"sentence": 0,
"word": 39
}
],
"includes": [
"y",
"x",
"F_{1}",
"F",
"_{1}F_{1}"
],
"isPartOf": [],
"definiens": [
{
"definition": "Appell",
"score": 2
},
{
"definition": "partial differential equation",
"score": 2
},
{
"definition": "system of differential equation",
"score": 1
},
{
"definition": "system of second-order differential equation",
"score": 2
},
{
"definition": "Appell series",
"score": 2
}
]
}
]
} |
| Gold 66 | \Theta_\Lambda(\tau) = \sum_{x\in\Lambda}e^{i\pi\tau\|x\|^2}\qquad\mathrm{Im}\,\tau > 0
|
Theta function of a lattice |
|
|
|
- | - | - | - | - | Full data:
{
"id": 66,
"pid": 118,
"eid": "math.118.0",
"title": "Theta function of a lattice",
"formulae": [
{
"id": "FORMULA_39f4baaa3543f22706b6f7701518f3eb",
"formula": "\\Theta_\\Lambda(\\tau) = \\sum_{x\\in\\Lambda}e^{i\\pi\\tau\\|x\\|^2}\\qquad\\mathrm{Im}\\,\\tau > 0",
"semanticFormula": "\\Theta_\\Lambda(\\tau) = \\sum_{x\\in\\Lambda} \\expe^{\\iunit \\cpi \\tau \\|x \\|^2} \\qquad \\imagpart \\tau > 0",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "\\[CapitalTheta][\\[CapitalLambda]_, \\[Tau]_] := Sum[Exp[I*Pi*\\[Tau]*(Norm[x])^(2)], {x, \\[CapitalLambda]}]"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 15
}
],
"includes": [
"\\Lambda"
],
"isPartOf": [],
"definiens": [
{
"definition": "theta function",
"score": 2
},
{
"definition": "lattice",
"score": 1
},
{
"definition": "Theta function of a lattice",
"score": 1
}
]
}
]
} |
| Gold 67 | \frac{d^2 S}{dz^2}+\left(\sum _{j=1}^N \frac{\gamma _j}{z - a_j} \right) \frac{dS}{dz} + \frac{V(z)}{\prod _{j=1}^N (z - a_j)}S = 0
|
Heine–Stieltjes polynomials |
|
- | - |
|
- | - | - | Mistakenly detected Stieltjes constant. No translation possible for S. | Full data:
{
"id": 67,
"pid": 119,
"eid": "math.119.0",
"title": "Heine\u2013Stieltjes polynomials",
"formulae": [
{
"id": "FORMULA_d673cd2334542e8f83f099798c4027b3",
"formula": "\\frac{d^2 S}{dz^2}+\\left(\\sum _{j=1}^N \\frac{\\gamma _j}{z - a_j} \\right) \\frac{dS}{dz} + \\frac{V(z)}{\\prod _{j=1}^N (z - a_j)}S = 0",
"semanticFormula": "\\deriv [2]{S}{z} +(\\sum_{j=1}^N \\frac{\\gamma _j}{z - a_j}) \\deriv[]{S}{z} + \\frac{V(z)}{\\prod _{j=1}^N (z - a_j)} S = 0",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 0,
"sentence": 1,
"word": 6
}
],
"includes": [
"V(z)",
"S",
"V"
],
"isPartOf": [],
"definiens": [
{
"definition": "form",
"score": 0
},
{
"definition": "Fuchsian equation",
"score": 2
},
{
"definition": "polynomial",
"score": 1
},
{
"definition": "degree",
"score": 0
},
{
"definition": "Edward Burr Van Vleck",
"score": 0
},
{
"definition": "Heine",
"score": 1
},
{
"definition": "polynomial solution",
"score": 1
},
{
"definition": "Stieltjes polynomial",
"score": 1
},
{
"definition": "Van Vleck polynomial",
"score": 1
}
]
}
]
} |
| Gold 68 | w(x) = \frac{k}{\sqrt{\pi}} x^{-1/2} \exp(-k^2\log^2 x)
|
Stieltjes–Wigert polynomials |
|
|
|
- | - | - | - | - | Full data:
{
"id": 68,
"pid": 120,
"eid": "math.120.0",
"title": "Stieltjes\u2013Wigert polynomials",
"formulae": [
{
"id": "FORMULA_583d3b9e00bbd73091b01f368d1a82c7",
"formula": "w(x) = \\frac{k}{\\sqrt{\\pi}} x^{-1\/2} \\exp(-k^2\\log^2 x)",
"semanticFormula": "w(x) = \\frac{k}{\\sqrt{\\cpi}} x^{-1\/2} \\exp(- k^2 \\log^2 x)",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "w[x_] := Divide[k,Sqrt[Pi]]*(x)^(- 1\/2)* Exp[- (k)^(2)* (Log[x])^(2)]"
},
"Maple": {
"translation": "w := (x) -> (k)\/(sqrt(Pi))*(x)^(- 1\/2)* exp(- (k)^(2)* (log(x))^(2))"
}
},
"positions": [
{
"section": 0,
"sentence": 0,
"word": 38
}
],
"includes": [
"\\frac{k}{\\sqrt{\\pi}} x^{-1\/2} \\exp \\left(-k^2 \\log^2 x \\right)"
],
"isPartOf": [],
"definiens": [
{
"definition": "weight function",
"score": 2
},
{
"definition": "positive real line",
"score": 0
},
{
"definition": "basic Askey scheme",
"score": 1
},
{
"definition": "family of basic hypergeometric orthogonal polynomial",
"score": 1
},
{
"definition": "mathematics",
"score": 0
},
{
"definition": "Stieltjes -- Wigert polynomial",
"score": 2
},
{
"definition": "Thomas Jan Stieltjes",
"score": 0
},
{
"definition": "Carl Severin Wigert",
"score": 0
},
{
"definition": "example of such weight function",
"score": 0
}
]
}
]
} |
| Gold 69 | y^2=x(x-1)(x-\lambda)
|
Modular lambda function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 69,
"pid": 121,
"eid": "math.121.23",
"title": "Modular lambda function",
"formulae": [
{
"id": "FORMULA_4e5334aa6f5fa551b0718a2372816061",
"formula": "y^2=x(x-1)(x-\\lambda)",
"semanticFormula": "y^2=x(x-1)(x-\\lambda)",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "(y)^(2) == x*(x - 1)*(x - \\[Lambda])"
},
"Maple": {
"translation": "(y)^(2) = x*(x - 1)*(x - lambda)"
}
},
"positions": [
{
"section": 2,
"sentence": 5,
"word": 13
}
],
"includes": [
"\\lambda"
],
"isPartOf": [],
"definiens": [
{
"definition": "elliptic curve of Legendre form",
"score": 2
},
{
"definition": "relation to the j-invariant",
"score": 1
},
{
"definition": "relation to the j-invariant",
"score": 1
}
]
}
]
} |
| Gold 70 | P_1^{(\lambda)}(x;\phi)=2(\lambda\cos\phi + x\sin\phi)
|
Meixner–Pollaczek polynomials |
|
- | - | - | - | - |
|
- | Full data:
{
"id": 70,
"pid": 122,
"eid": "math.122.3",
"title": "Meixner\u2013Pollaczek polynomials",
"formulae": [
{
"id": "FORMULA_96d19b4b504f801548c69064d662043b",
"formula": "P_1^{(\\lambda)}(x;\\phi)=2(\\lambda\\cos\\phi + x\\sin\\phi)",
"semanticFormula": "\\MeixnerPollaczekpolyP{\\lambda}{1}@{x}{\\phi} = 2(\\lambda \\cos \\phi + x \\sin \\phi)",
"confidence": 0.8953028732079359,
"translations": {},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 9
}
],
"includes": [
"P_{m}^{(\\lambda)}(x;\\varphi)"
],
"isPartOf": [],
"definiens": [
{
"definition": "first few Meixner -- Pollaczek polynomial",
"score": 2
}
]
}
]
} |
| Gold 71 | P_n^{(\alpha,\beta)}(z)=\frac{(\alpha+1)_n}{n!}\,{}_2F_1\left(-n,1+\alpha+\beta+n;\alpha+1;\tfrac{1}{2}(1-z)\right)
|
Jacobi polynomials |
|
|
- | - | - | - | - | - | Full data:
{
"id": 71,
"pid": 123,
"eid": "math.123.0",
"title": "Jacobi polynomials",
"formulae": [
{
"id": "FORMULA_c8b5b9184e45bca39744427c45693115",
"formula": "P_n^{(\\alpha,\\beta)}(z)=\\frac{(\\alpha+1)_n}{n!}\\,{}_2F_1\\left(-n,1+\\alpha+\\beta+n;\\alpha+1;\\tfrac{1}{2}(1-z)\\right)",
"semanticFormula": "\\JacobipolyP{\\alpha}{\\beta}{n}@{z} = \\frac{\\Pochhammersym{\\alpha + 1}{n}}{n!} \\genhyperF{2}{1}@{- n , 1 + \\alpha + \\beta + n}{\\alpha + 1}{\\tfrac{1}{2}(1 - z)}",
"confidence": 0.7595006538205181,
"translations": {
"Mathematica": {
"translation": "JacobiP[n, \\[Alpha], \\[Beta], z] == Divide[Pochhammer[\\[Alpha]+ 1, n],(n)!]*HypergeometricPFQ[{- n , 1 + \\[Alpha]+ \\[Beta]+ n}, {\\[Alpha]+ 1}, Divide[1,2]*(1 - z)]"
},
"Maple": {
"translation": "JacobiP(n, alpha, beta, z) = (pochhammer(alpha + 1, n))\/(factorial(n))*hypergeom([- n , 1 + alpha + beta + n], [alpha + 1], (1)\/(2)*(1 - z))"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 12
}
],
"includes": [
"P_{n}^{(\\alpha, \\beta)}(x)",
"(\\alpha+1)_n",
"n",
"n + \\alpha + \\beta",
"P_{n}^{(\\alpha, \\beta)}",
"\\alpha,\\beta",
"z"
],
"isPartOf": [],
"definiens": [
{
"definition": "Pochhammer 's symbol",
"score": 2
},
{
"definition": "hypergeometric function",
"score": 2
},
{
"definition": "Jacobi polynomial",
"score": 2
}
]
}
]
} |
| Gold 72 | S_n(x^2;a,b,c)= {}_3F_2(-n,a+ix,a-ix;a+b,a+c;1).
|
Continuous dual Hahn polynomials |
|
- | - | - | - | - |
|
- | Full data:
{
"id": 72,
"pid": 124,
"eid": "math.124.0",
"title": "Continuous dual Hahn polynomials",
"formulae": [
{
"id": "FORMULA_b0d448ba925dc6b2bf2ce32a1253dee4",
"formula": "S_n(x^2;a,b,c)= {}_3F_2(-n,a+ix,a-ix;a+b,a+c;1).",
"semanticFormula": "\\contdualHahnpolyS{n}@{x^2}{a}{b}{c} = \\genhyperF{3}{2}@{- n , a + \\iunit x , a - \\iunit x}{a + b , a + c}{1}",
"confidence": 0.7132263353695951,
"translations": {},
"positions": [
{
"section": 0,
"sentence": 1,
"word": 10
}
],
"includes": [
"R_{n}(x;\\gamma,\\delta,N)"
],
"isPartOf": [],
"definiens": [
{
"definition": "hypergeometric function",
"score": 1
},
{
"definition": "dual Hahn polynomial",
"score": 1
},
{
"definition": "continuous Hahn polynomial",
"score": 1
},
{
"definition": "continuous dual Hahn polynomial",
"score": 2
}
]
}
]
} |
| Gold 73 | P_n^{(\alpha,\beta)}=\lim_{t\to\infty}t^{-n}p_n\left(\tfrac12xt; \tfrac12(\alpha+1-it), \tfrac12(\beta+1+it), \tfrac12(\alpha+1+it), \tfrac12(\beta+1-it)\right)
|
Continuous Hahn polynomials |
|
|
- |
|
- |
|
- | Hidden argument cause mismatch. | Full data:
{
"id": 73,
"pid": 125,
"eid": "math.125.15",
"title": "Continuous Hahn polynomials",
"formulae": [
{
"id": "FORMULA_ff971744100fef3b34b2c93b6adc3efb",
"formula": "P_n^{(\\alpha,\\beta)}=\\lim_{t\\to\\infty}t^{-n}p_n\\left(\\tfrac12xt; \\tfrac12(\\alpha+1-it), \\tfrac12(\\beta+1+it), \\tfrac12(\\alpha+1+it), \\tfrac12(\\beta+1-it)\\right)",
"semanticFormula": "\\JacobipolyP{\\alpha}{\\beta}{n}@{x} = \\lim_{t\\to\\infty} t^{-n} \\contHahnpolyp{n}@{\\tfrac12 xt}{\\tfrac12(\\alpha + 1 - \\iunit t)}{\\tfrac12(\\beta + 1 + \\iunit t)}{\\tfrac12(\\alpha + 1 + \\iunit t)}{\\tfrac12(\\beta + 1 - \\iunit t)}",
"confidence": 0.9041995034970904,
"translations": {
"Mathematica": {
"translation": "JacobiP[n, \\[Alpha], \\[Beta], x] == Limit[(t)^(- n)* I^(n)*Divide[Pochhammer[Divide[1,2]*(\\[Alpha]+ 1 - I*t) + Divide[1,2]*(\\[Alpha]+ 1 + I*t), n]*Pochhammer[Divide[1,2]*(\\[Alpha]+ 1 - I*t) + Divide[1,2]*(\\[Beta]+ 1 - I*t), n], (n)!] * HypergeometricPFQ[{-(n), n + 2*Re[Divide[1,2]*(\\[Alpha]+ 1 - I*t) + Divide[1,2]*(\\[Beta]+ 1 + I*t)] - 1, Divide[1,2]*(\\[Alpha]+ 1 - I*t) + I*(Divide[1,2]*x*t)}, {Divide[1,2]*(\\[Alpha]+ 1 - I*t) + Divide[1,2]*(\\[Alpha]+ 1 + I*t), Divide[1,2]*(\\[Alpha]+ 1 - I*t) + Divide[1,2]*(\\[Beta]+ 1 - I*t)}, 1], t -> Infinity, GenerateConditions->None]"
}
},
"positions": [
{
"section": 5,
"sentence": 2,
"word": 20
}
],
"includes": [
"p_{n}(x;a,b,c,d)",
"F_{n}",
"P_{n}^{(\\alpha,\\beta)}"
],
"isPartOf": [],
"definiens": [
{
"definition": "case of the continuous Hahn polynomial",
"score": 1
},
{
"definition": "Jacobi polynomial",
"score": 2
},
{
"definition": "continuous Hahn polynomial",
"score": 2
}
]
}
]
} |
| Gold 74 | \sum^{b-1}_{s=a}w_n^{(c)}(s,a,b)w_m^{(c)}(s,a,b)\rho(s)[\Delta x(s-\frac{1}{2}) ]=\delta_{nm}d_n^2
|
Dual Hahn polynomials |
|
- | - |
|
- | - | - | Not standard notation for dual Hahn polynomial. DLMF uses R. Further, dual Hahn does not exist in Mathematica. | Full data:
{
"id": 74,
"pid": 126,
"eid": "math.126.7",
"title": "Dual Hahn polynomials",
"formulae": [
{
"id": "FORMULA_657ec9a2e460e61adc6857260291be56",
"formula": "\\sum^{b-1}_{s=a}w_n^{(c)}(s,a,b)w_m^{(c)}(s,a,b)\\rho(s)[\\Delta x(s-\\frac{1}{2}) ]=\\delta_{nm}d_n^2",
"semanticFormula": "\\sum_{s=a}^{b-1} \\dualHahnpolyR{n}@{c}{s}{a}{b} \\dualHahnpolyR{m}@{c}{s}{a}{b} \\rho(s) [\\Delta x(s - \\frac{1}{2})] = \\delta_{nm} d_n^2",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 8
}
],
"includes": [
"n"
],
"isPartOf": [],
"definiens": [
{
"definition": "Dual Hahn polynomial",
"score": 2
}
]
}
]
} |
| Gold 75 | p_n(x;a,b,c|q)=a^{-n}e^{-inu}(abe^{2iu},ac,ad;q)_n*_4\Phi_3(q^{-n},abcdq^{n-1},ae^{i{(t+2u)}},ae^{-it};abe^{2iu},ac,ad;q;q)
|
Continuous q-Hahn polynomials |
|
|
|
|
- | - | - | Asterisk has index. Wrong LaTeX from Wikipedia Editor. | Full data:
{
"id": 75,
"pid": 127,
"eid": "math.127.0",
"title": "Continuous q-Hahn polynomials",
"formulae": [
{
"id": "FORMULA_67e28846328978f4e08bb6b69fe6c549",
"formula": "p_n(x;a,b,c|q)=a^{-n}e^{-inu}(abe^{2iu},ac,ad;q)_n*_4\\Phi_3(q^{-n},abcdq^{n-1},ae^{i{(t+2u)}},ae^{-it};abe^{2iu},ac,ad;q;q)",
"semanticFormula": "p_n(x ; a , b , c|q) = a^{-n} \\expe^{-\\iunit nu} \\qmultiPochhammersym{ab\\expe^{2\\iunit u} , ac , ad}{q}{n} * \\qgenhyperphi{4}{3}@{q^{-n} , abcdq^{n-1} , a\\expe^{\\iunit{(t+2u)}} , a\\expe^{-\\iunit t}}{ab\\expe^{2\\iunit u} , ac , ad}{q}{q}",
"confidence": 0.8662724998444776,
"translations": {
"Mathematica": {
"translation": "p[n_, x_, a_, b_, c_, q_] := (a)^(- n)* Exp[- I*\\[Nu]]*Product[QPochhammer[Part[{a*b*Exp[2*I*u], a*c , a*d},i],q,n],{i,1,Length[{a*b*Exp[2*I*u], a*c , a*d}]}]* QHypergeometricPFQ[{(q)^(- n), a*b*c*d*(q)^(n - 1), a*Exp[I*(t + 2*u)], a*Exp[- I*t]},{a*b*Exp[2*I*u], a*c , a*d},q,q]"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 15
}
],
"includes": [
"q"
],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 1
},
{
"definition": "Pochhammer symbol",
"score": 1
},
{
"definition": "continuous FORMULA_7694f4a66316e53c8cdd9d9954bd611d - Hahn polynomial",
"score": 2
},
{
"definition": "q - Pochhammer symbol",
"score": 2
}
]
}
]
} |
| Gold 76 | p_n(x;a,b,c\mid q)=\frac{(ab,ac;q)_n}{a^n}\cdot {_3\Phi_2}(q^-n,ae^{i\theta},ae^{-i\theta}; ab, ac \mid q;q)
|
Continuous dual q-Hahn polynomials |
|
|
|
|
- | - | Wrong LaTeX. q^-n only puts into the subscript but not .
|
Underscore mismatch. | Full data:
{
"id": 76,
"pid": 128,
"eid": "math.128.0",
"title": "Continuous dual q-Hahn polynomials",
"formulae": [
{
"id": "FORMULA_95daf919f18506606090e49a38d1c1a6",
"formula": "p_n(x;a,b,c\\mid q)=\\frac{(ab,ac;q)_n}{a^n}\\cdot {_3\\Phi_2}(q^-n,ae^{i\\theta},ae^{-i\\theta}; ab, ac \\mid q;q)",
"semanticFormula": "p_n(x ; a , b , c \mid q) = \frac{\qmultiPochhammersym{ab , ac}{q}{n}}{a^n} \cdot \qgenhyperphi{3}{2}@{q^{- n} , a\expe^{\iunit \theta} , a\expe^{- \iunit \theta}}{ab , ac}{q}{q}",
"confidence": 0.8662724998444776,
"translations": {
"Mathematica": {
"translation": "p[n_, x_, a_, b_, c_, q_] := Divide[Product[QPochhammer[Part[{a*b , a*c},i],q,n],{i,1,Length[{a*b , a*c}]}],(a)^(n)] * QHypergeometricPFQ[{(q)^(- n), a*Exp[I*\[Theta]], a*Exp[- I*\[Theta]]},{a*b , a*c},q,q]"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 15
}
],
"includes": [
"q"
],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 1
},
{
"definition": "continuous dual FORMULA_7694f4a66316e53c8cdd9d9954bd611d - Hahn polynomial",
"score": 2
}
]
}
]
} |
| Gold 77 | Q_n(x;a,b,N;q)=\;_{3}\phi_2\left[\begin{matrix} q^-n & abq^n+1 & x \\ aq & q^-N \end{matrix} ; q,q \right]
|
Q-Hahn polynomials |
|
|
|
- | - | - | - | Cannot detect name of function. | Full data:
{
"id": 77,
"pid": 129,
"eid": "math.129.0",
"title": "Q-Hahn polynomials",
"formulae": [
{
"id": "FORMULA_b3a9ac90714e1e705d2a88b30e79cca0",
"formula": "Q_n(x;a,b,N;q)=\\;_{3}\\phi_2\\left[\\begin{matrix} q^-n & abq^n+1 & x \\\\ aq & q^-N \\end{matrix} ; q,q \\right]",
"semanticFormula": "\\qHahnpolyQ{n}@{x}{a}{b}{N}{q} = \\qgenhyperphi{3}{2}@{q^-n , abq^n+1 , x}{aq , q^-N}{q}{q}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "Q[n_, x_, a_, b_, N_, q_] := QHypergeometricPFQ[{(q)^(-)* n , a*b*(q)^(n)+ 1 , x},{a*q , (q)^(-)* N},q,q]"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 15
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "q - Hahn polynomial",
"score": 2
},
{
"definition": "polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 0
}
]
}
]
} |
| Gold 78 | x=
|
Al-Salam–Chihara polynomials |
|
|
- | - | - | - | - | Wrong math detection. | Full data:
{
"id": 78,
"pid": 131,
"eid": "math.131.0",
"title": "Al-Salam\u2013Chihara polynomials",
"formulae": [
{
"id": "FORMULA_52a07ce46212cbc2298415c5fca6e075",
"formula": "x=",
"semanticFormula": "x=\\cos@{\\theta}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "x = Cos[\\[Theta]]"
},
"Maple": {
"translation": "x = cos(theta)"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 20
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "cosine function",
"score": 2
},
{
"definition": "substitution",
"score": 2
}
]
}
]
} |
| Gold 79 | \Phi_n^*(z)=z^n\overline{\Phi_n(1/\overline{z})}
|
Orthogonal polynomials on the unit circle |
|
|
- |
|
- | - | - | Nested overline didnt match (bug). | Full data:
{
"id": 79,
"pid": 132,
"eid": "math.132.7",
"title": "Orthogonal polynomials on the unit circle",
"formulae": [
{
"id": "FORMULA_f2d41903301a99a3fade5f2f49450694",
"formula": "\\Phi_n^*(z)=z^n\\overline{\\Phi_n(1\/\\overline{z})}",
"semanticFormula": "\\Phi_n^*(z) = z^n{\\conj{\\Phi_n(1 \/ \\conj{z})}}",
"confidence": 0.7579553437219001,
"translations": {
"Mathematica": {
"translation": "\\[CapitalPhi]\\[Prima][n_, z_] := z^n*Conjugate[\\[CapitalPhi][n, Divide[1, Conjugate[z]]]]"
}
},
"positions": [
{
"section": 2,
"sentence": 0,
"word": 8
}
],
"includes": [
"\\Phi_n(z)",
"z^n",
"\\alpha_n"
],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 2
}
]
}
]
} |
| Gold 80 | P_n(x) = c_n \, \det \begin{bmatrix}m_0 & m_1 & m_2 &\cdots & m_n \\m_1 & m_2 & m_3 &\cdots & m_{n+1} \\&&\vdots&& \vdots \\m_{n-1} &m_n& m_{n+1} &\cdots &m_{2n-1}\\1 & x & x^2 & \cdots & x^n\end{bmatrix}
|
Orthogonal polynomials |
|
- | - | - | - | - | - | No direct translation possible (indef number of arguments). | Full data:
{
"id": 80,
"pid": 133,
"eid": "math.133.8",
"title": "Orthogonal polynomials",
"formulae": [
{
"id": "FORMULA_c0641714ec593f58211623652c4a34f0",
"formula": "P_n(x) = c_n \\, \\det \\begin{bmatrix}m_0 & m_1 & m_2 &\\cdots & m_n \\\\m_1 & m_2 & m_3 &\\cdots & m_{n+1} \\\\&&\\vdots&& \\vdots \\\\m_{n-1} &m_n& m_{n+1} &\\cdots &m_{2n-1}\\\\1 & x & x^2 & \\cdots & x^n\\end{bmatrix}",
"semanticFormula": "P_n(x) = c_n \\det \\begin{bmatrix}m_0 & m_1 & m_2 &\\cdots & m_n \\\\m_1 & m_2 & m_3 &\\cdots & m_{n+1} \\\\&&\\vdots&& \\vdots \\\\m_{n-1} &m_n& m_{n+1} &\\cdots &m_{2n-1}\\\\1 & x & x^2 & \\cdots & x^n\\end{bmatrix}",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 5,
"sentence": 0,
"word": 16
}
],
"includes": [
"P_{n}",
"c_{n}",
"P_{m}"
],
"isPartOf": [],
"definiens": [
{
"definition": "constant",
"score": 0
},
{
"definition": "normalisation",
"score": 0
},
{
"definition": "orthogonal polynomial",
"score": 2
},
{
"definition": "term of the moment",
"score": 0
}
]
}
]
} |
| Gold 81 | \displaystyle p_n(x;a,b;q) = {}_2\phi_1(q^{-n},abq^{n+1};aq;q,xq)
|
Little q-Jacobi polynomials |
|
|
|
- | - | - |
|
No translation for \littleJacobipolyp
|
Full data:
{
"id": 81,
"pid": 134,
"eid": "math.134.0",
"title": "Little q-Jacobi polynomials",
"formulae": [
{
"id": "FORMULA_c492265e4cd4beeeb776dad843dc1f73",
"formula": "\\displaystyle p_n(x;a,b;q) = {}_2\\phi_1(q^{-n},abq^{n+1};aq;q,xq)",
"semanticFormula": "\\littleqJacobipolyp{n}@{x}{a}{b}{q} = \\qgenhyperphi{2}{1}@{q^{-n} , abq^{n+1}}{aq}{q}{xq}",
"confidence": 0.7229065246531701,
"translations": {
"Mathematica": {
"translation": "p[n_, x_, a_, b_, q_] := QHypergeometricPFQ[{(q)^(- n), a*b*(q)^(n + 1)},{a*q},q,x*q]"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 19
}
],
"includes": [
"q",
"p_{n}(x;a,b;q)"
],
"isPartOf": [],
"definiens": [
{
"definition": "Jacobi polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 0
},
{
"definition": "little q - Jacobi polynomial",
"score": 2
}
]
}
]
} |
| Gold 82 | \displaystyle P_n(x;a,b,c;q)={}_3\phi_2(q^{-n},abq^{n+1},x;aq,cq;q,q)
|
Big q-Jacobi polynomials |
|
|
|
- | - | - |
|
No translation for \bigqJacobipolyP
|
Full data:
{
"id": 82,
"pid": 135,
"eid": "math.135.0",
"title": "Big q-Jacobi polynomials",
"formulae": [
{
"id": "FORMULA_0680f701a101288f89487a7a3fabefb1",
"formula": "\\displaystyle P_n(x;a,b,c;q)={}_3\\phi_2(q^{-n},abq^{n+1},x;aq,cq;q,q)",
"semanticFormula": "\\bigqJacobipolyP{n}@{x}{a}{b}{c}{q} = \\qgenhyperphi{3}{2}@{q^{-n} , abq^{n+1} , x}{aq , cq}{q}{q}",
"confidence": 0.7424814142326033,
"translations": {
"Mathematica": {
"translation": "p[n_, x_, a_, b_, c_, q_] := QHypergeometricPFQ[{(q)^(- n), a*b*(q)^(n + 1), x},{a*q , c*q},q,q]"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 11
}
],
"includes": [
"P_{n}(x;a,b,c;q)",
"q"
],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 2
},
{
"definition": "big q - Jacobi polynomial",
"score": 2
}
]
}
]
} |
| Gold 83 | P_n(x;a,b;q)=\frac{1}{(b^{-1}*q^{-n};q,n)}*_2\Phi_1(q^{-n},aqx^{-1};aq|q;\frac{x}{b})
|
Big q-Laguerre polynomials |
|
|
- |
|
- | - | - | Again, invalid latex. The asterisk has the underscore. | Full data:
{
"id": 83,
"pid": 137,
"eid": "math.137.0",
"title": "Big q-Laguerre polynomials",
"formulae": [
{
"id": "FORMULA_aa5a6972c7e8327e316eddc8fd8e9b08",
"formula": "P_n(x;a,b;q)=\\frac{1}{(b^{-1}*q^{-n};q,n)}*_2\\Phi_1(q^{-n},aqx^{-1};aq|q;\\frac{x}{b})",
"semanticFormula": "P_n(x;a,b;q) =\\frac{1}{\\qmultiPochhammersym{b^{-1}*q^{-n}}{q}{n}} * \\qgenhyperphi{2}{1}@{q^{-n},aqx^{-1}}{aq}{q}{\\frac{x}{b}}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "P[n_, x_, a_, b_, q_] := Divide[1,Product[QPochhammer[Part[{(b)^(- 1)* (q)^(- n)},i],q,n],{i,1,Length[{(b)^(- 1)* (q)^(- n)}]}]]* QHypergeometricPFQ[{(q)^(- n), a*q*(x)^(- 1)},{a*q},q,Divide[x,b]]"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 15
}
],
"includes": [
"q"
],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 1
},
{
"definition": "Pochhammer symbol",
"score": 1
},
{
"definition": "q - Pochhammer symbol",
"score": 1
},
{
"definition": "big q - Laguerre polynomial",
"score": 2
}
]
}
]
} |
| Gold 84 | K_n(\lambda(x);c,N|q)=_3\Phi_2(q^{-n},q^{-x},cq^{x-N};q^{-N},0|q;q)
|
Dual q-Krawtchouk polynomials |
|
- | - |
|
- | - | - | Illegal LaTeX. Equal sign has underscore 3 (which is wrong). Further, dual q-Krawtchouk do not exist in Mathematica. | Full data:
{
"id": 84,
"pid": 138,
"eid": "math.138.0",
"title": "Dual q-Krawtchouk polynomials",
"formulae": [
{
"id": "FORMULA_9221dfda453868628eb8bbcd2d414fdf",
"formula": "K_n(\\lambda(x);c,N|q)=_3\\Phi_2(q^{-n},q^{-x},cq^{x-N};q^{-N},0|q;q)",
"semanticFormula": "K_n(\\lambda(x);c,N|q) = \\qgenhyperphi{3}{2}@{q^{-n},q^{-x},cq^{x-N}}{q^{-N},0}{q}{q}",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 15
}
],
"includes": [
"q"
],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 0
},
{
"definition": "dual q - Krawtchouk polynomial",
"score": 2
}
]
}
]
} |
| Gold 85 | P_{n}^{(\alpha)}(x|q)=\frac{(q^\alpha+1;q)_{n}}{(q;q)_{n}}
|
Continuous q-Laguerre polynomials |
|
|
|
|
- | - | - | Did not detect q-multi Pochhammer symbol. | Full data:
{
"id": 85,
"pid": 139,
"eid": "math.139.0",
"title": "Continuous q-Laguerre polynomials",
"formulae": [
{
"id": "FORMULA_8c9e3af3c57272f3a6ddabba68ab4d3e",
"formula": "P_{n}^{(\\alpha)}(x|q)=\\frac{(q^\\alpha+1;q)_{n}}{(q;q)_{n}}",
"semanticFormula": "P_{n}^{(\\alpha)}(x|q) = \\frac{\\qmultiPochhammersym{q^\\alpha+1}{q}{n}}{\\qPochhammer{q}{q}{n}} \\qgenhyperphi{3}{2}@{q^{-n},q^{\\alpha\/2+1\/4}\\expe^{\\iunit\\theta},q^{\\alpha\/2+1\/4}*\\expe^{-\\iunit\\theta}}{q^{\\alpha+1},0}{q}{q}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "P[n_, \\[Alpha]_, x_, q_] := Divide[Product[QPochhammer[Part[{(q)^\\[Alpha]+ 1},i],q,n],{i,1,Length[{(q)^\\[Alpha]+ 1}]}],QPochhammer[q, q, n]]*QHypergeometricPFQ[{(q)^(- n), (q)^(\\[Alpha]\/2 + 1\/4)* Exp[I*\\[Theta]], (q)^(\\[Alpha]\/2 + 1\/4)* Exp[- I*\\[Theta]]},{(q)^(\\[Alpha]+ 1), 0},q,q]"
}
},
"positions": [
{
"section": 1,
"sentence": 1,
"word": 0
}
],
"includes": [
"q"
],
"isPartOf": [],
"definiens": [
{
"definition": "continuous q - Laguerre polynomial",
"score": 2
},
{
"definition": "family of basic hypergeometric orthogonal polynomial",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 2
}
]
}
]
} |
| Gold 86 | \displaystyle p_n(x;a|q) = {}_2\phi_1(q^{-n},0;aq;q,qx) = \frac{1}{(a^{-1}q^{-n};q)_n}{}_2\phi_0(q^{-n},x^{-1};;q,x/a)
|
Little q-Laguerre polynomials |
|
|
|
|
- | - | - | Could not match empty arguments (bug). | Full data:
{
"id": 86,
"pid": 142,
"eid": "math.142.0",
"title": "Little q-Laguerre polynomials",
"formulae": [
{
"id": "FORMULA_4e548bca196e13d5af0eaadf2ea725d1",
"formula": "\\displaystyle p_n(x;a|q) = {}_2\\phi_1(q^{-n},0;aq;q,qx) = \\frac{1}{(a^{-1}q^{-n};q)_n}{}_2\\phi_0(q^{-n},x^{-1};;q,x\/a)",
"semanticFormula": "p_n(x ; a|q) = \\qgenhyperphi{2}{1}@{q^{-n} , 0}{aq}{q}{qx} = \\frac{1}{\\qmultiPochhammersym{a^{-1} q^{-n}}{q}{n}} \\qgenhyperphi{2}{0}@{q^{-n} , x^{-1}}{}{q}{x\/a}",
"confidence": 0.7219509974881755,
"translations": {
"Mathematica": "p[n_, x_, a_, q_] := QHypergeometricPFQ[{(q)^(- n), 0},{a*q},q,q*x] == Divide[1,Product[QPochhammer[Part[{(a)^(- 1)* (q)^(- n)},i],q,n],{i,1,Length[{(a)^(- 1)* (q)^(- n)}]}]]*QHypergeometricPFQ[{(q)^(- n), (x)^(- 1)},{},q,x\/a]"
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 15
}
],
"includes": [
"q",
"p_{n}(x;a|q)"
],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 1
},
{
"definition": "little q - Laguerre polynomial",
"score": 2
},
{
"definition": "q - Pochhammer symbol",
"score": 2
}
]
}
]
} |
| Gold 87 | y_{n}(x;a;q)=\;_{2}\phi_1 \left(\begin{matrix} q^{-N} & -aq^{n} \\ 0 \end{matrix} ; q,qx \right)
|
Q-Bessel polynomials |
|
|
|
|
- | - | - | Wrong LaTeX. Equal sign has subsript 2. | Full data:
{
"id": 87,
"pid": 143,
"eid": "math.143.0",
"title": "Q-Bessel polynomials",
"formulae": [
{
"id": "FORMULA_c89da2fda6f9f6411ed4292f6d845f52",
"formula": "y_{n}(x;a;q)=\\;_{2}\\phi_1 \\left(\\begin{matrix} q^{-N} & -aq^{n} \\\\ 0 \\end{matrix} ; q,qx \\right)",
"semanticFormula": "y_{n}(x;a;q) = \\qgenhyperphi{2}{1}@{q^{-N} , -aq^{n}}{0}{q}{qx}",
"confidence": 0.6264217257193126,
"translations": {
"Mathematica": "y[n_, x_, a_, q_] := QHypergeometricPFQ[{(q)^(- N), - a*(q)^(n)},{0},q,q*x]"
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 16
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 0
},
{
"definition": "q - Bessel polynomial",
"score": 1
}
]
}
]
} |
| Gold 88 | h_n(ix;q^{-1}) = i^n\hat h_n(x;q)
|
Discrete q-Hermite polynomials |
|
- | - |
|
- | - | - | We correctly identified \discqHermitepolyhI but were not able to distinguish it from discqHermitepolyhII from RHS. Neither of them is translatable though.
|
Full data:
{
"id": 88,
"pid": 144,
"eid": "math.144.2",
"title": "Discrete q-Hermite polynomials",
"formulae": [
{
"id": "FORMULA_b9974285610b7a82c94b6a504726df8c",
"formula": "h_n(ix;q^{-1}) = i^n\\hat h_n(x;q)",
"semanticFormula": "\\discqHermitepolyhI{n}@{\\iunit x}{q^{-1}} = \\iunit^n \\discqHermitepolyhII{n}@{x}{q}",
"confidence": 0.8429359579302446,
"translations": {},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 27
}
],
"includes": [
"\\hat{h}_{n}(x;q)",
"q",
"h_{n}(x;q)"
],
"isPartOf": [],
"definiens": [
{
"definition": "Hermite polynomial",
"score": 2
},
{
"definition": "term of basic hypergeometric function",
"score": 1
},
{
"definition": "Carlitz polynomial",
"score": 1
},
{
"definition": "Al-Salam",
"score": 1
},
{
"definition": "discrete q - Hermite polynomial",
"score": 2
}
]
}
]
} |
| Gold 89 | P_{n}(x;a\mid q) = a^{-n} e^{in\phi} \frac{a^2;q_n}{(q;q)_n} {_3}\Phi_2(q^-n, ae^{i(\theta+2\phi)}, ae^{-i\theta}; a^2, 0 \mid q; q)
|
Q-Meixner–Pollaczek polynomials |
|
|
|
|
- | - | - | Did not match underscore {_3}
|
Full data:
{
"id": 89,
"pid": 145,
"eid": "math.145.0",
"title": "Q-Meixner\u2013Pollaczek polynomials",
"formulae": [
{
"id": "FORMULA_fa6650cad7aed4d975716018ef03068f",
"formula": "P_{n}(x;a\\mid q) = a^{-n} e^{in\\phi} \\frac{a^2;q_n}{(q;q)_n} {_3}\\Phi_2(q^-n, ae^{i(\\theta+2\\phi)}, ae^{-i\\theta}; a^2, 0 \\mid q; q)",
"semanticFormula": "P_{n}(x ; a \\mid q) = a^{-n} \\expe^{\\iunit n\\phi} \\frac{\\qmultiPochhammersym{a^2}{q}{n}}{\\qmultiPochhammersym{q}{q}{n}} \\qgenhyperphi{3}{2}@{q^- n , a\\expe^{\\iunit(\\theta + 2 \\phi)} , a\\expe^{- \\iunit \\theta}}{a^2, 0}{q}{q}",
"confidence": 0.8662724998444776,
"translations": {
"Mathematica": {
"translation": "P[n_, x_, a_, q_] := (a)^(- n)* Exp[I*n*\\[Phi]]*Divide[Product[QPochhammer[Part[{(a)^(2)},i],q,n],{i,1,Length[{(a)^(2)}]}],Product[QPochhammer[Part[{q},i],q,n],{i,1,Length[{q}]}]]*QHypergeometricPFQ[{(q)^(-)* n , a*Exp[I*(\\[Theta]+ 2*\\[Phi])], a*Exp[- I*\\[Theta]]},{(a)^(2), 0},q,q]"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 16
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 1
},
{
"definition": "q - Pochhammer symbol",
"score": 2
},
{
"definition": "Q Meixner \u2013 Pollaczek polynomials",
"score": 2
}
]
}
]
} |
| Gold 90 | \displaystyle L_n^{(\alpha)}(x;q) = \frac{(q^{\alpha+1};q)_n}{(q;q)_n} {}_1\phi_1(q^{-n};q^{\alpha+1};q,-q^{n+\alpha+1}x)
|
Q-Laguerre polynomials |
|
|
|
- | - | - |
|
- | Full data:
{
"id": 90,
"pid": 149,
"eid": "math.149.0",
"title": "Q-Laguerre polynomials",
"formulae": [
{
"id": "FORMULA_dea0af895f73964b98741e71bc0635cb",
"formula": "\\displaystyle L_n^{(\\alpha)}(x;q) = \\frac{(q^{\\alpha+1};q)_n}{(q;q)_n} {}_1\\phi_1(q^{-n};q^{\\alpha+1};q,-q^{n+\\alpha+1}x)",
"semanticFormula": "\\qLaguerrepolyL{\\alpha}{n}@{x}{q} = \\frac{\\qmultiPochhammersym{q^{\\alpha+1}}{q}{n}}{\\qmultiPochhammersym{q}{q}{n}} \\qgenhyperphi{1}{1}@{q^{-n}}{q^{\\alpha+1}}{q}{- q^{n+\\alpha+1} x}",
"confidence": 0.779734956061429,
"translations": {
"Mathematica": {
"translation": "L[n_, \\[Alpha]_, x_, q_] := Divide[Product[QPochhammer[Part[{(q)^(\\[Alpha]+ 1)},i],q,n],{i,1,Length[{(q)^(\\[Alpha]+ 1)}]}],Product[QPochhammer[Part[{q},i],q,n],{i,1,Length[{q}]}]]*QHypergeometricPFQ[{(q)^(- n)},{(q)^(\\[Alpha]+ 1)},q,- (q)^(n + \\[Alpha]+ 1)* x]"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 18
}
],
"includes": [
"q",
"P_{n}^{(\\alpha)}(x;q)"
],
"isPartOf": [],
"definiens": [
{
"definition": "Laguerre polynomial",
"score": 2
},
{
"definition": "q - Laguerre polynomial",
"score": 2
},
{
"definition": "term of basic hypergeometric function",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 1
},
{
"definition": "q - Pochhammer symbol",
"score": 2
}
]
}
]
} |
| Gold 91 | \sum_{n=0}^\infty H_n(x \mid q) \frac{t^n}{(q;q)_n} = \frac{1}{\left( t e^{i \theta},t e^{-i \theta};q \right)_\infty}
|
Continuous q-Hermite polynomials |
|
- | - |
|
- | - | - | Mistakenly detect Hermite polynomial but was continuous q-Hermite polynomial. | Full data:
{
"id": 91,
"pid": 150,
"eid": "math.150.3",
"title": "Continuous q-Hermite polynomials",
"formulae": [
{
"id": "FORMULA_a10dc9de9b2b618ad2f2e96dc9eb0207",
"formula": "\\sum_{n=0}^\\infty H_n(x \\mid q) \\frac{t^n}{(q;q)_n} = \\frac{1}{\\left( t e^{i \\theta},t e^{-i \\theta};q \\right)_\\infty}",
"semanticFormula": "\\sum_{n=0}^\\infty \\contqHermitepolyH{n}@{x}{q} \\frac{t^n}{\\qmultiPochhammersym{q}{q}{n}} = \\frac{1}{\\qmultiPochhammersym{t \\expe^{\\iunit \\theta} , t \\expe^{- \\iunit \\theta}}{q}{\\infty}}",
"confidence": 0.7796357038819148,
"translations": {},
"positions": [
{
"section": 3,
"sentence": 0,
"word": 0
}
],
"includes": [
"q"
],
"isPartOf": [],
"definiens": [
{
"definition": "continuous q - Hermite polynomial",
"score": 2
},
{
"definition": "q - Pochhammer symbol",
"score": 2
}
]
}
]
} |
| Gold 92 | w^{\prime\prime}+\xi\sin(2z)w^{\prime}+(\eta-p\xi\cos(2z))w=0.
|
Ince equation |
|
|
- | - |
|
|
- | ODE. | Full data:
{
"id": 92,
"pid": 151,
"eid": "math.151.0",
"title": "Ince equation",
"formulae": [
{
"id": "FORMULA_ce9ed9f979f486263028e3d86b63ac60",
"formula": "w^{\\prime\\prime}+\\xi\\sin(2z)w^{\\prime}+(\\eta-p\\xi\\cos(2z))w=0. ",
"semanticFormula": "w^{\\prime\\prime}+\\xi\\sin(2z)w^{\\prime}+(\\eta-p\\xi\\cos(2z))w=0",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "D[w[z], {z, 2}] + \\[Xi]*Sin[2*z]*D[w[z], {z, 1}] + (\\[Eta]-p*\\[Xi]*Cos[2*z])*w[z] == 0"
}
},
"positions": [
{
"section": 0,
"sentence": 0,
"word": 19
}
],
"includes": [
"p"
],
"isPartOf": [],
"definiens": [
{
"definition": "differential equation",
"score": 2
},
{
"definition": "Ince equation",
"score": 2
},
{
"definition": "mathematics",
"score": 0
},
{
"definition": "non-negative integer",
"score": 0
},
{
"definition": "Edward Lindsay Ince",
"score": 0
},
{
"definition": "polynomial solution",
"score": 0
},
{
"definition": "Ince polynomial",
"score": 1
}
]
}
]
} |
| Gold 93 | Q_v^\mu(x)= \cos(\mu\pi)\left(\frac{1+x}{1-x}\right)^{\mu/2}\frac{F(v+1,-v;1-\mu;1/2-2/x)} {\Gamma(1-\mu ) }
|
Ferrers function |
|
|
- |
|
- |
|
- | No information about gamma fuction and hypergeometric function. | Full data:
{
"id": 93,
"pid": 152,
"eid": "math.152.1",
"title": "Ferrers function",
"formulae": [
{
"id": "FORMULA_b5ab87b9cd2da05be00884345889d9e3",
"formula": "Q_v^\\mu(x)= \\cos(\\mu\\pi)\\left(\\frac{1+x}{1-x}\\right)^{\\mu\/2}\\frac{F(v+1,-v;1-\\mu;1\/2-2\/x)} {\\Gamma(1-\\mu ) }",
"semanticFormula": "\\FerrersQ[\\mu]{v}@{x} = \\cos(\\mu \\cpi)(\\frac{1+x}{1-x})^{\\mu\/2} \\frac{\\hyperF@{v+1}{-v}{1-\\mu}{1\/2-2\/x}}{\\EulerGamma@{1-\\mu}}",
"confidence": 0.8133162393162393,
"translations": {
"Mathematica": {
"translation": "LegendreQ[v, \\[Mu], x] == Cos[(\\[Mu]*Pi)*]*(Divide[1 + x,1 - x])^(\\[Mu]\/2)*Divide[Hypergeometric2F1[v + 1, - v, 1 - \\[Mu], 1\/2 - 2\/x],Gamma[1 - \\[Mu]]]"
},
"Maple": {
"translation": "LegendreQ(v, mu, x) = cos((mu*Pi)*)*((1 + x)\/(1 - x))^(mu\/2)*(hypergeom([v + 1, - v], [1 - mu], 1\/2 - 2\/x))\/(GAMMA(1 - mu))"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 13
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "Ferrers function of the second kind",
"score": 2
},
{
"definition": "Ferrers function of the first kind",
"score": 1
},
{
"definition": "Gamma function",
"score": 2
},
{
"definition": "hypergeometric function",
"score": 2
}
]
}
]
} |
| Gold 94 | H_{-v}^{(1)}(z,w)=e^{v\pi i}H_v^{(1)}(z,w)
|
Incomplete Bessel functions |
|
- | - | - | - | - | - |
|
Full data:
{
"id": 94,
"pid": 153,
"eid": "math.153.27",
"title": "Incomplete Bessel functions",
"formulae": [
{
"id": "FORMULA_35ab66efafff0de40d98c0778ebb63c3",
"formula": "H_{-v}^{(1)}(z,w)=e^{v\\pi i}H_v^{(1)}(z,w)",
"semanticFormula": "H_{-v}^{(1)}(z,w) = \\expe^{v \\cpi \\iunit} H_v^{(1)}(z , w)",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 2,
"sentence": 0,
"word": 16
}
],
"includes": [
"v",
"w",
"H_v^{(1)}(z,w)"
],
"isPartOf": [],
"definiens": [
{
"definition": "incomplete Bessel function",
"score": 2
}
]
}
]
} |
| Gold 95 | K_v(x,y)=\int_1^\infty\frac{e^{-xt-\frac{y}{t}}}{t^{v+1}}dt
|
Incomplete Bessel K function/generalized incomplete gamma function |
|
|
|
- | - | - | - | - | Full data:
{
"id": 95,
"pid": 154,
"eid": "math.154.0",
"title": "Incomplete Bessel K function\/generalized incomplete gamma function",
"formulae": [
{
"id": "FORMULA_c333a7966510ed0b8f4de3147eabe47a",
"formula": "K_v(x,y)=\\int_1^\\infty\\frac{e^{-xt-\\frac{y}{t}}}{t^{v+1}}dt",
"semanticFormula": "K_v(x , y) = \\int_1^\\infty \\frac{\\expe^{-xt-\\frac{y}{t}}}{t^{v+1}} \\diff{t}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "K[v_, x_, y_] := Integrate[Divide[Exp[- x*t -Divide[y,t]],(t)^(v + 1)], {t, 1, Infinity}, GenerateConditions->None]"
}
},
"positions": [
{
"section": 0,
"sentence": 0,
"word": 18
}
],
"includes": [
"K_v(x,y)"
],
"isPartOf": [],
"definiens": [
{
"definition": "mathematician",
"score": 0
},
{
"definition": "type incomplete-version of Bessel function",
"score": 2
},
{
"definition": "type generalized-version of incomplete gamma function",
"score": 0
}
]
}
]
} |