Poject:GoldData
| # | Formula | Title | 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
}
]
}
]
}
|