Poject:GoldData

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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
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 } ] } ] }
2	E(e) \,=\, \int_0^{\pi/2}\sqrt {1 - e^2 \sin^2\theta}\ d\theta	Ellipse			-	N	-	-	-	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 } ] } ] }
3	F(x;k) = u	Elliptic integral			N	-	-	-	-	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 } ] } ] }
4	\frac{1}{\Gamma(z)} = \frac{i}{2\pi}\int_C (-t)^{-z}e^{-t}\,dt	Gamma function		-	-	N	-	-	-	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 } ] } ] }
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 } ] } ] }
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 } ] } ] }
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 } ] } ] }
8	w_{i} = \frac{1}{p'_{n}(x_{i})}\int_{a}^{b}\omega(x)\frac{p_{n}(x)}{x-x_{i}}dx	Gaussian quadrature			-	-	N	-	-	-	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 } ] } ] }
9	\begin{align}x & =ue^u, \\[5pt]\frac{dx}{du} & =(u+1)e^u.\end{align}	Lambert W function			N	-	-	-	-	-	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 } ] } ] }
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			-	-	N	-	-	-	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 } ] } ] }
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			-	-	N	-	-	${\mathrm{erf}}^{(k)}$ 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 } ] } ] }
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 } ] } ] }
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 } ] } ] }
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 } ] } ] }
15	E_n=2^nE_n(\tfrac{1}{2})	Bernoulli polynomials			-	N	-	-	-	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 } ] } ] }
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 } ] } ] }
17	f(z)=\frac{1}{\Beta(x,y)}	Beta function			N	-	-	N	-	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 } ] } ] }
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			-	N	-	-	-	Matrix argument of ${}_1{F}_1$ 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 } ] } ] }
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			-	-	-	N	-	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 } ] } ] }
20	{}_1F_0(1;;z) = \sum_{n \geqslant 0} z^n = (1-z)^{-1}	Generalized hypergeometric function			-	N	-	-	-	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 } ] } ] }
21	\chi(-1) = 1	Dirichlet L-function			-	-	-	N	-	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 } ] } ] }
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		-	-	-	N	-	-	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 } ] } ] }
23	F'(y)=1-2yF(y)	Dawson function			-	-	N	N	-	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 } ] } ] }
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 } ] } ] }
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 } ] } ] }
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 } ] } ] }
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 } ] } ] }
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 } ] } ] }
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 } ] } ] }
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 } ] } ] }
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			-	-	-	N	-	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 } ] } ] }
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 } ] } ] }
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 } ] } ] }
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			-	-	-	-	N	-	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 } ] } ] }
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			-	-	-	-	N	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 } ] } ] }
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 } ] } ] }
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 } ] } ] }
38	\frac{d^2f}{dz^2} + \left(\tilde{a}z^2+\tilde{b}z+\tilde{c}\right)f=0	Parabolic cylinder function			-	-	-	N	-	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 } ] } ] }
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 } ] } ] }
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 } ] } ] }
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 } ] } ] }
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": [] } ] }
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 } ] } ] }
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		-	-	N	-	-	-	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 } ] } ] }
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 } ] } ] }
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 } ] } ] }
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 } ] } ] }
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 } ] } ] }
49	\begin{Bmatrix} i & j & \ell\\ k & m & n \end{Bmatrix}= (\Phi_{i,j}^{k,m})_{\ell,n}	6-j symbol		-	-	-	-	N	-	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 } ] } ] }
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 } ] } ] }
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			-	-	-	N	-	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 } ] } ] }
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			N	-	-	-	-	-	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 } ] } ] }
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			-	-	-	N	-	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 } ] } ] }
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			-	N	-	-	-	Arguments of ${}_1{F}_2$ 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 } ] } ] }
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 } ] } ] }
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			-	-	-	N	-	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 } ] } ] }
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 } ] } ] }
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 } ] } ] }
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			N	-	-	-	-	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 } ] } ] }
60	Q_n(x;\alpha,\beta,N)= {}_3F_2(-n,-x,n+\alpha+\beta+1;\alpha+1,-N+1;1).	Hahn polynomials			N	-	-	-	N	-	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 } ] } ] }
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			-	-	-	N	-	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 } ] } ] }
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		-	-	-	-	-	N	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 } ] } ] }
63	\displaystyle c_n(q^{-x};a;q) = {}_2\phi_1(q^{-n},q^{-x};0;q,-q^{n+1}/a)	Q-Charlier polynomials		-	-	-	-	-	-	N	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 } ] } ] }
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			-	-	-	N	-	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 } ] } ] }
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			-	N	-	N	-	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 } ] } ] }
66	\Theta_\Lambda(\tau) = \sum_{x\in\Lambda}e^{i\pi\tau\|x\|^2}\qquad\mathrm{Im}\,\tau > 0	Theta function of a lattice			N	-	-	-	-	-	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 } ] } ] }
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		-	-	N	-	-	-	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 } ] } ] }
68	w(x) = \frac{k}{\sqrt{\pi}} x^{-1/2} \exp(-k^2\log^2 x)	Stieltjes–Wigert polynomials			N	-	-	-	-	-	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 } ] } ] }
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 } ] } ] }
70	P_1^{(\lambda)}(x;\phi)=2(\lambda\cos\phi + x\sin\phi)	Meixner–Pollaczek polynomials		-	-	-	-	-	N	-	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 } ] } ] }
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 } ] } ] }
72	S_n(x^2;a,b,c)= {}_3F_2(-n,a+ix,a-ix;a+b,a+c;1).	Continuous dual Hahn polynomials		-	-	-	-	-	N	-	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 } ] } ] }
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			-	N	-	N	-	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 } ] } ] }
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		-	-	N	-	-	-	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 } ] } ] }
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			N	N	-	-	-	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 } ] } ] }
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			N	N	-	-	Wrong LaTeX. q^-n only puts $-$ into the subscript but not $-n$.	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 , ae^{\\iunit \\theta} , ae^{- \\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*(e)^(I*\\[Theta]), a*(e)^(- 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 } ] } ] }
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			N	-	-	-	-	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 } ] } ] }
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 } ] } ] }
79	\Phi_n^*(z)=z^n\overline{\Phi_n(1/\overline{z})}	Orthogonal polynomials on the unit circle			-	N	-	-	-	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 } ] } ] }
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 } ] } ] }
81	\displaystyle p_n(x;a,b;q) = {}_2\phi_1(q^{-n},abq^{n+1};aq;q,xq)	Little q-Jacobi polynomials			N	-	-	-	N	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 } ] } ] }
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			N	-	-	-	N	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 } ] } ] }
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			-	N	-	-	-	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 } ] } ] }
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		-	-	N	-	-	-	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 } ] } ] }
85	P_{n}^{(\alpha)}(x|q)=\frac{(q^\alpha+1;q)_{n}}{(q;q)_{n}}	Continuous q-Laguerre polynomials			N	N	-	-	-	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 } ] } ] }
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			N	N	-	-	-	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 } ] } ] }
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			N	N	-	-	-	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 } ] } ] }
88	h_n(ix;q^{-1}) = i^n\hat h_n(x;q)	Discrete q-Hermite polynomials		-	-	N	-	-	-	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 } ] } ] }
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			N	N	-	-	-	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 } ] } ] }
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			N	-	-	-	N	-	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 } ] } ] }
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		-	-	N	-	-	-	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 } ] } ] }
92	w^{\prime\prime}+\xi\sin(2z)w^{\prime}+(\eta-p\xi\cos(2z))w=0.	Ince equation			-	-	N	N	-	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 } ] } ] }
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			-	N	-	N	-	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 } ] } ] }
94	H_{-v}^{(1)}(z,w)=e^{v\pi i}H_v^{(1)}(z,w)	Incomplete Bessel functions		-	-	-	-	-	-	N	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 } ] } ] }
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			N	-	-	-	-	-	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 } ] } ] }