|
|
| Line 3: |
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| |- | | |- |
| ! # !! Formula !! Title !! Evaluation data | | ! # !! 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#math.51.18| Bessel function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | | |-| 1 || <syntaxhighlight lang="tex" inline >\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}</syntaxhighlight> |
| | || [[Bessel function#math.51.18| Bessel function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ |
| "id": 1, | | "id": 1, |
| "pid": 51, | | "pid": 51, |
| Line 241: |
Line 242: |
| ] | | ] |
| }</syntaxhighlight> | | }</syntaxhighlight> |
| </div></div>|-| 2 || E(e) \,=\, \int_0^{\pi/2}\sqrt {1 - e^2 \sin^2\theta}\ d\theta || [[Ellipse#math.52.404| Ellipse]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | | </div></div>|- |
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| |
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| |
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| |
| "\\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"
| |
| }
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| |
| "\\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"
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| </div></div>|-| 3 || F(x;k) = u || [[Elliptic integral#math.53.6| Elliptic integral]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
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| "semanticFormula": "\\incellintFk@{\\asin@{\\Jacobiellsnk@@{u}{k}}}{k} = u",
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| |
| "F(x ; k) = \\int_{0}^{x} \\frac{\\mathrm{d}t}{\\sqrt{\\left(1 - t^2\\right)\\left(1 - k^2 t^2\\right)}}",
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| "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",
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| |
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| |
| "definition": "Jacobian elliptic function",
| |
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| |
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| |
| </div></div>|-| 4 || \frac{1}{\Gamma(z)} = \frac{i}{2\pi}\int_C (-t)^{-z}e^{-t}\,dt || [[Gamma function#math.54.195| Gamma function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
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| |
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| |
| "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,
| |
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| </div></div>|-| 5 || 2^{4} = 2 \times2 \times 2 \times 2 = 16 || [[Logarithm#| Logarithm]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
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| </div></div>|-| 6 || \psi(x) := \sum_{n=1}^\infty e^{-n^2 \pi x} || [[Riemann zeta function#math.56.40| Riemann zeta function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
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| "id": 6,
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| |
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| |
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| |
| },
| |
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| |
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| </div></div>|-| 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#math.57.2| Logarithmic integral function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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": {
| |
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| |
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| |
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| |
| "\\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"
| |
| }
| |
| }
| |
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| |
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| |
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| "\\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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 8 || w_{i} = \frac{1}{p'_{n}(x_{i})}\int_{a}^{b}\omega(x)\frac{p_{n}(x)}{x-x_{i}}dx || [[Gaussian quadrature#math.58.61| Gaussian quadrature]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 9 || \begin{align}x & =ue^u, \\[5pt]\frac{dx}{du} & =(u+1)e^u.\end{align} || [[Lambert W function#math.59.52| Lambert W function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.60.57| Legendre polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.61.27| Error function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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": [
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| {
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 12 || x_k = \cos\left(\frac{\pi(k+1/2)}{n}\right),\quad k=0,\ldots,n-1 || [[Chebyshev polynomials#math.62.44| Chebyshev polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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}"
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| |
| "definiens": [
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| |
| "definition": "root",
| |
| "score": 2
| |
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| |
| {
| |
| "definition": "one",
| |
| "score": 0
| |
| },
| |
| {
| |
| "definition": "trigonometric definition",
| |
| "score": 0
| |
| },
| |
| {
| |
| "definition": "fact",
| |
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| |
| },
| |
| {
| |
| "definition": "different simple root",
| |
| "score": 1
| |
| },
| |
| {
| |
| "definition": "Chebyshev polynomial of the first kind",
| |
| "score": 1
| |
| },
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| "definition": "Chebyshev polynomial",
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| "score": 1
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| |
| }
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| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.63.109| Hermite polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "id": 13,
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| "pid": 63,
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| "eid": "math.63.109",
| |
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| "formulae": [
| |
| {
| |
| "id": "FORMULA_249043719eb4dd70350b460363255e11",
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| "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))"
| |
| }
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| },
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| "H_{n}(x)",
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| "\\psi_{n}(x)",
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| "n",
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| "He_{n}(x)",
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| "H_{n}(y)"
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| "definition": "distributional identity",
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| "definition": "separable kernel",
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| "score": 1
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| },
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| {
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 14 || x = \pm 1 || [[Legendre function#math.64.8| Legendre function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
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| "id": 14,
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| "formulae": [
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| {
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| "id": "FORMULA_06f9b7b1d3f141742ad1c582b55056ba",
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| "formula": "x = \\pm 1",
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| "semanticFormula": "x = \\pm 1",
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| "confidence": 0,
| |
| "translations": {
| |
| "Mathematica": {
| |
| "translation": "x == \\[PlusMinus]1",
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| "translationInformation": {
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| "x = + 1",
| |
| "x = - 1"
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| "tokenTranslations": {
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| "\\pm": "was translated to: \\[PlusMinus]"
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| }
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| }
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| },
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| "translation": "x = &+- 1",
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| "translationInformation": {
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| "subEquations": [
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| "x = + 1",
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| "x = - 1"
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| "definition": "value",
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| "score": 2
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| }
| |
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| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 15 || E_n=2^nE_n(\tfrac{1}{2}) || [[Bernoulli polynomials#math.65.27| Bernoulli polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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": [
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| {
| |
| "section": 8,
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| "sentence": 4,
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| "word": 6
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| "B_{n}",
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| "n",
| |
| "E_{k}"
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| ],
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| "isPartOf": [],
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| "definiens": [
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| {
| |
| "definition": "Euler number",
| |
| "score": 2
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 16 || \operatorname{Si}(ix) = i\operatorname{Shi}(x) || [[Trigonometric integral#math.66.8| Trigonometric integral]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
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| "id": 16,
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| "pid": 66,
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| "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]"
| |
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| |
| "\\iunit": "Imaginary unit was translated to: I",
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| "\\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)"
| |
| ],
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| }
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| {
| |
| "definition": "ordinary sine",
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| "score": 1
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| |
| "definition": "Trigonometric integral",
| |
| "score": 2
| |
| },
| |
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| |
| "definition": "hyperbolic sine integral",
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| |
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| |
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| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 17 || f(z)=\frac{1}{\Beta(x,y)} || [[Beta function#math.67.29| Beta function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
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| "id": 17,
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| "formulae": [
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| {
| |
| "id": "FORMULA_5f59825d73d63a9990498edca7222261",
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| "formula": "f(z)=\\frac{1}{\\Beta(x,y)}",
| |
| "semanticFormula": "f(x, y) = \\frac{1}{\\EulerBeta@{x}{y}}",
| |
| "confidence": 0.8953028732079359,
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| "translations": {
| |
| "Mathematica": {
| |
| "translation": "f[x_, y_] := Divide[1,Beta[x, y]]"
| |
| },
| |
| "Maple": {
| |
| "translation": "f := (x,y) -> (1)\/(Beta(x, y))"
| |
| }
| |
| },
| |
| "positions": [
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| |
| "definition": "definite integral of trigonometric function",
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| "definition": "integral representation",
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| |
| </div></div>|-| 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#math.68.51| Fresnel integral]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "id": 18,
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| "eid": "math.68.51",
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| "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))"
| |
| }
| |
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| |
| "definition": "incomplete gamma function",
| |
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| |
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| |
| {
| |
| "definition": "confluent hypergeometric function",
| |
| "score": 2
| |
| },
| |
| {
| |
| "definition": "Fresnel integral",
| |
| "score": 1
| |
| },
| |
| {
| |
| "definition": "imaginary part",
| |
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| |
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| |
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| |
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| |
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| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.69.117| Classical orthogonal polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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)])"
| |
| }
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| |
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| |
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| |
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| |
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| |
| "definition": "orthogonal polynomial",
| |
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| |
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| |
| {
| |
| "definition": "Chebyshev polynomials of the second kind",
| |
| "score": 1
| |
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| |
| {
| |
| "definition": "classical orthogonal polynomial",
| |
| "score": 1
| |
| },
| |
| {
| |
| "definition": "Chebyshev polynomial",
| |
| "score": 2
| |
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| |
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| |
| "definition": "Gamma function",
| |
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| |
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| |
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| |
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| |
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| |
| }</syntaxhighlight>
| |
| </div></div>|-| 20 || {}_1F_0(1;;z) = \sum_{n \geqslant 0} z^n = (1-z)^{-1} || [[Generalized hypergeometric function#math.70.58| Generalized hypergeometric function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "id": 20,
| |
| "pid": 70,
| |
| "eid": "math.70.58",
| |
| "title": "Generalized hypergeometric function",
| |
| "formulae": [
| |
| {
| |
| "id": "FORMULA_699b5f465d21dd6af7212cd8414f60c6",
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| "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}",
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| "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,
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| "sentence": 2,
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| "word": 0
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| }
| |
| ],
| |
| "includes": [
| |
| "z",
| |
| "n",
| |
| "z)",
| |
| "_{p}F_{q}",
| |
| "^{n}"
| |
| ],
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| "isPartOf": [],
| |
| "definiens": [
| |
| {
| |
| "definition": "geometric series with ratio",
| |
| "score": 2
| |
| },
| |
| {
| |
| "definition": "coefficient",
| |
| "score": 0
| |
| },
| |
| {
| |
| "definition": "hypergeometric function",
| |
| "score": 2
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 21 || \chi(-1) = 1 || [[Dirichlet L-function#math.71.1| Dirichlet L-function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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,
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| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.72.15| Airy function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 23 || F'(y)=1-2yF(y) || [[Dawson function#math.73.41| Dawson function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 24 || s\not =1 || [[Hurwitz zeta function#math.74.0| Hurwitz zeta function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
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| ],
| |
| "includes": [
| |
| "s",
| |
| "1",
| |
| "\\not = 1"
| |
| ],
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| "isPartOf": [],
| |
| "definiens": [
| |
| {
| |
| "definition": "value",
| |
| "score": 2
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 25 || q = e^{i\pi\tau} || [[Theta function#math.75.6| Theta function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 26 || \frac{\mathrm{d}}{\mathrm{d}z} \operatorname{dn}(z) = - k^2 \operatorname{sn}(z) \operatorname{cn}(z) || [[Jacobi elliptic functions#math.76.155| Jacobi elliptic functions]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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,
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| "sentence": 0,
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| ],
| |
| "includes": [
| |
| "k",
| |
| "^{2}"
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| ],
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.77.118| Incomplete gamma function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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": [
| |
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| |
| "section": 25,
| |
| "sentence": 1,
| |
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| |
| }
| |
| ],
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 28 || _{1}(z) = || [[Polylogarithm#math.78.0| Polylogarithm]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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,
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| }
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| ],
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 29 || \int_{-\infty}^\infty \operatorname{sinc}(t) \, e^{-i 2 \pi f t}\,dt = \operatorname{rect}(f) || [[Sinc function#math.79.11| Sinc function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 30 || N=1 || [[Exponential integral#math.80.26| Exponential integral]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.81.84| Laguerre polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 32 || c_{lm} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!} || [[Associated Legendre polynomials#math.82.8| Associated Legendre polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 33 || \mathrm{Gi}(x) = \frac{1}{\pi} \int_0^\infty \sin\left(\frac{t^3}{3} + xt\right)\, dt || [[Scorer's function#math.83.3| Scorer's function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.84.31| Voigt profile]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.85.57| Lerch zeta function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 36 || M(1,2,z)=(e^z-1)/z,\ \ M(1,3,z)=2!(e^z-1-z)/z^2 || [[Confluent hypergeometric function#math.86.44| Confluent hypergeometric function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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,
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| </div></div>|-| 37 || \sigma = \pm 1 || [[Mathieu function#math.87.54| Mathieu function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
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| </div></div>|-| 38 || \frac{d^2f}{dz^2} + \left(\tilde{a}z^2+\tilde{b}z+\tilde{c}\right)f=0 || [[Parabolic cylinder function#math.88.0| Parabolic cylinder function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
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| |
| </div></div>|-| 39 || c=\infty || [[Painlevé transcendents#math.89.23| Painlevé transcendents]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
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| </div></div>|-| 40 || c = a + 1 || [[Hypergeometric function#math.90.7| Hypergeometric function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
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| </div></div>|-| 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#math.91.47| Barnes G-function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
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| </div></div>|-| 42 || 192/24 = 8 = 2 \times 4 || [[Heun function#math.92.1| Heun function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
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| </div></div>|-| 43 || =2 || [[Gegenbauer polynomials#math.93.0| Gegenbauer polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
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| "\\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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.94.4| Basic hypergeometric series]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.95.0| Whittaker function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 46 || e_1=\tfrac12,\qquad e_2=0,\qquad e_3=-\tfrac12 || [[Lemniscatic elliptic function#math.96.1| Lemniscatic elliptic function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 47 || \gamma> 0,n-p=m-q> 0 || [[Meijer G-function#math.98.53| Meijer G-function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.99.30| 3-j symbol]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 49 || \begin{Bmatrix} i & j & \ell\\ k & m & n \end{Bmatrix}= (\Phi_{i,j}^{k,m})_{\ell,n} || [[6-j symbol#math.100.14| 6-j symbol]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.101.32| 9-j symbol]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.102.5| Kravchuk polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.103.8| Kelvin functions]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.104.2| Lommel function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.105.18| Struve function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 55 || f(t+p) = f(t) || [[Hill differential equation#math.106.7| Hill differential equation]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.108.3| Anger function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 57 || (\operatorname{Ec})^'_{2K} = (\operatorname{Ec})^'_0 = 0, \;\; (\operatorname{Es})^'_{2K} = (\operatorname{Es})^'_0 = 0 || [[Lamé function#math.109.27| Lamé function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 58 || \int_{-\infty}^{+\infty} e^{-x^2} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i) || [[Gauss–Hermite quadrature#math.110.1| Gauss–Hermite quadrature]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.111.0| Askey–Wilson polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 60 || Q_n(x;\alpha,\beta,N)= {}_3F_2(-n,-x,n+\alpha+\beta+1;\alpha+1,-N+1;1). || [[Hahn polynomials#math.112.0| Hahn polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.113.2| Charlier polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.114.0| Q-Racah polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 63 || \displaystyle c_n(q^{-x};a;q) = {}_2\phi_1(q^{-n},q^{-x};0;q,-q^{n+1}/a) || [[Q-Charlier polynomials#math.115.0| Q-Charlier polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.116.0| Meixner polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.117.19| Appell series]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 66 || \Theta_\Lambda(\tau) = \sum_{x\in\Lambda}e^{i\pi\tau\|x\|^2}\qquad\mathrm{Im}\,\tau > 0 || [[Theta function of a lattice#math.118.0| Theta function of a lattice]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.119.0| Heine–Stieltjes polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 68 || w(x) = \frac{k}{\sqrt{\pi}} x^{-1/2} \exp(-k^2\log^2 x) || [[Stieltjes–Wigert polynomials#math.120.0| Stieltjes–Wigert polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 69 || y^2=x(x-1)(x-\lambda) || [[Modular lambda function#math.121.23| Modular lambda function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 70 || P_1^{(\lambda)}(x;\phi)=2(\lambda\cos\phi + x\sin\phi) || [[Meixner–Pollaczek polynomials#math.122.3| Meixner–Pollaczek polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.123.0| Jacobi polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 72 || S_n(x^2;a,b,c)= {}_3F_2(-n,a+ix,a-ix;a+b,a+c;1). || [[Continuous dual Hahn polynomials#math.124.0| Continuous dual Hahn polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.125.15| Continuous Hahn polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.126.7| Dual Hahn polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.127.0| Continuous q-Hahn polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.128.0| Continuous dual q-Hahn polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.129.0| Q-Hahn polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 78 || x= || [[Al-Salam–Chihara polynomials#math.131.0| Al-Salam–Chihara polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 79 || \Phi_n^*(z)=z^n\overline{\Phi_n(1/\overline{z})} || [[Orthogonal polynomials on the unit circle#math.132.7| Orthogonal polynomials on the unit circle]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.133.8| Orthogonal polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 81 || \displaystyle p_n(x;a,b;q) = {}_2\phi_1(q^{-n},abq^{n+1};aq;q,xq) || [[Little q-Jacobi polynomials#math.134.0| Little q-Jacobi polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.135.0| Big q-Jacobi polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.137.0| Big q-Laguerre polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.138.0| Dual q-Krawtchouk polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 85 || P_{n}^{(\alpha)}(x|q)=\frac{(q^\alpha+1;q)_{n}}{(q;q)_{n}} || [[Continuous q-Laguerre polynomials#math.139.0| Continuous q-Laguerre polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.142.0| Little q-Laguerre polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.143.0| Q-Bessel polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 88 || h_n(ix;q^{-1}) = i^n\hat h_n(x;q) || [[Discrete q-Hermite polynomials#math.144.2| Discrete q-Hermite polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.145.0| Q-Meixner–Pollaczek polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.149.0| Q-Laguerre polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.150.3| Continuous q-Hermite polynomials]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 92 || w^{\prime\prime}+\xi\sin(2z)w^{\prime}+(\eta-p\xi\cos(2z))w=0. || [[Ince equation#math.151.0| Ince equation]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.152.1| Ferrers function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 94 || H_{-v}^{(1)}(z,w)=e^{v\pi i}H_v^{(1)}(z,w) || [[Incomplete Bessel functions#math.153.27| Incomplete Bessel functions]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-| 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#math.154.0| Incomplete Bessel K function/generalized incomplete gamma function]] || <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{
| |
| "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
| |
| }
| |
| ]
| |
| }
| |
| ]
| |
| }</syntaxhighlight>
| |
| </div></div>|-|}
| |