Orthogonal Polynomials - 18.10 Integral Representations
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| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
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| 18.10.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{\JacobipolyP{\alpha}{\alpha}{n}@{\cos@@{\theta}}}{\JacobipolyP{\alpha}{\alpha}{n}@{1}} = \frac{\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{\cos@@{\theta}}}{\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{1}}}
\frac{\JacobipolyP{\alpha}{\alpha}{n}@{\cos@@{\theta}}}{\JacobipolyP{\alpha}{\alpha}{n}@{1}} = \frac{\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{\cos@@{\theta}}}{\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{1}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 < \theta, \theta < \pi, \alpha > -\tfrac{1}{2}} | (JacobiP(n, alpha, alpha, cos(theta)))/(JacobiP(n, alpha, alpha, 1)) = (GegenbauerC(n, alpha +(1)/(2), cos(theta)))/(GegenbauerC(n, alpha +(1)/(2), 1))
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Divide[JacobiP[n, \[Alpha], \[Alpha], Cos[\[Theta]]],JacobiP[n, \[Alpha], \[Alpha], 1]] == Divide[GegenbauerC[n, \[Alpha]+Divide[1,2], Cos[\[Theta]]],GegenbauerC[n, \[Alpha]+Divide[1,2], 1]]
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Successful | Successful | Skip - symbolical successful subtest | Successful [Tested: 27] |
| 18.10.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{\cos@@{\theta}}}{\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{1}} = \frac{2^{\alpha+\frac{1}{2}}\EulerGamma@{\alpha+1}}{\pi^{\frac{1}{2}}\EulerGamma@{\alpha+\frac{1}{2}}}(\sin@@{\theta})^{-2\alpha}\int_{0}^{\theta}\frac{\cos@{(n+\alpha+\tfrac{1}{2})\phi}}{(\cos@@{\phi}-\cos@@{\theta})^{-\alpha+\frac{1}{2}}}\diff{\phi}}
\frac{\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{\cos@@{\theta}}}{\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{1}} = \frac{2^{\alpha+\frac{1}{2}}\EulerGamma@{\alpha+1}}{\pi^{\frac{1}{2}}\EulerGamma@{\alpha+\frac{1}{2}}}(\sin@@{\theta})^{-2\alpha}\int_{0}^{\theta}\frac{\cos@{(n+\alpha+\tfrac{1}{2})\phi}}{(\cos@@{\phi}-\cos@@{\theta})^{-\alpha+\frac{1}{2}}}\diff{\phi} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 < \theta, \theta < \pi, \alpha > -\tfrac{1}{2}, \realpart@@{(\alpha+1)} > 0, \realpart@@{(\alpha+\frac{1}{2})} > 0} | (GegenbauerC(n, alpha +(1)/(2), cos(theta)))/(GegenbauerC(n, alpha +(1)/(2), 1)) = ((2)^(alpha +(1)/(2))* GAMMA(alpha + 1))/((Pi)^((1)/(2))* GAMMA(alpha +(1)/(2)))*(sin(theta))^(- 2*alpha)* int((cos((n + alpha +(1)/(2))*phi))/((cos(phi)- cos(theta))^(- alpha +(1)/(2))), phi = 0..theta)
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Divide[GegenbauerC[n, \[Alpha]+Divide[1,2], Cos[\[Theta]]],GegenbauerC[n, \[Alpha]+Divide[1,2], 1]] == Divide[(2)^(\[Alpha]+Divide[1,2])* Gamma[\[Alpha]+ 1],(Pi)^(Divide[1,2])* Gamma[\[Alpha]+Divide[1,2]]]*(Sin[\[Theta]])^(- 2*\[Alpha])* Integrate[Divide[Cos[(n + \[Alpha]+Divide[1,2])*\[Phi]],(Cos[\[Phi]]- Cos[\[Theta]])^(- \[Alpha]+Divide[1,2])], {\[Phi], 0, \[Theta]}, GenerateConditions->None]
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Failure | Aborted | Successful [Tested: 27] | Skipped - Because timed out |
| 18.10.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LegendrepolyP{n}@{\cos@@{\theta}} = \frac{2^{\frac{1}{2}}}{\pi}\int_{0}^{\theta}\frac{\cos@{(n+\tfrac{1}{2})\phi}}{(\cos@@{\phi}-\cos@@{\theta})^{\frac{1}{2}}}\diff{\phi}}
\LegendrepolyP{n}@{\cos@@{\theta}} = \frac{2^{\frac{1}{2}}}{\pi}\int_{0}^{\theta}\frac{\cos@{(n+\tfrac{1}{2})\phi}}{(\cos@@{\phi}-\cos@@{\theta})^{\frac{1}{2}}}\diff{\phi} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 < \theta, \theta < \pi} | LegendreP(n, cos(theta)) = ((2)^((1)/(2)))/(Pi)*int((cos((n +(1)/(2))*phi))/((cos(phi)- cos(theta))^((1)/(2))), phi = 0..theta)
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LegendreP[n, Cos[\[Theta]]] == Divide[(2)^(Divide[1,2]),Pi]*Integrate[Divide[Cos[(n +Divide[1,2])*\[Phi]],(Cos[\[Phi]]- Cos[\[Theta]])^(Divide[1,2])], {\[Phi], 0, \[Theta]}, GenerateConditions->None]
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Failure | Aborted | Successful [Tested: 9] | Skipped - Because timed out |
| 18.10.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\frac{\JacobipolyP{\alpha}{\alpha}{n}@{\cos@@{\theta}}}{\JacobipolyP{\alpha}{\alpha}{n}@{1}}=\frac{\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{\cos@@{\theta}}}{\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{1}}} = \frac{\EulerGamma@{\alpha+1}}{\pi^{\frac{1}{2}}\EulerGamma{(\alpha+\tfrac{1}{2})}}\*{\int_{0}^{\pi}(\cos@@{\theta}+i\sin@@{\theta}\cos@@{\phi})^{n}\*(\sin@@{\phi})^{2\alpha}\diff{\phi}}}
{\frac{\JacobipolyP{\alpha}{\alpha}{n}@{\cos@@{\theta}}}{\JacobipolyP{\alpha}{\alpha}{n}@{1}}=\frac{\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{\cos@@{\theta}}}{\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{1}}} = \frac{\EulerGamma@{\alpha+1}}{\pi^{\frac{1}{2}}\EulerGamma{(\alpha+\tfrac{1}{2})}}\*{\int_{0}^{\pi}(\cos@@{\theta}+i\sin@@{\theta}\cos@@{\phi})^{n}\*(\sin@@{\phi})^{2\alpha}\diff{\phi}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \alpha > -\frac{1}{2}, \realpart@@{(\alpha+1)} > 0} | (JacobiP(n, alpha, alpha, cos(theta)))/(JacobiP(n, alpha, alpha, 1)) = (GegenbauerC(n, alpha +(1)/(2), cos(theta)))/(GegenbauerC(n, alpha +(1)/(2), 1)) = (GAMMA(alpha + 1))/((Pi)^((1)/(2))* GAMMA(alpha +(1)/(2)))*int((cos(theta)+ I*sin(theta)*cos(phi))^(n)*(sin(phi))^(2*alpha), phi = 0..Pi)
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Divide[JacobiP[n, \[Alpha], \[Alpha], Cos[\[Theta]]],JacobiP[n, \[Alpha], \[Alpha], 1]] == Divide[GegenbauerC[n, \[Alpha]+Divide[1,2], Cos[\[Theta]]],GegenbauerC[n, \[Alpha]+Divide[1,2], 1]] == Divide[Gamma[\[Alpha]+ 1],(Pi)^(Divide[1,2])* Gamma[\[Alpha]+Divide[1,2]]]*Integrate[(Cos[\[Theta]]+ I*Sin[\[Theta]]*Cos[\[Phi]])^(n)*(Sin[\[Phi]])^(2*\[Alpha]), {\[Phi], 0, Pi}, GenerateConditions->None]
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Failure | Aborted | Skipped - Because timed out | Skipped - Because timed out |
| 18.10.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LegendrepolyP{n}@{\cos@@{\theta}} = \frac{1}{\pi}\int_{0}^{\pi}(\cos@@{\theta}+i\sin@@{\theta}\cos@@{\phi})^{n}\diff{\phi}}
\LegendrepolyP{n}@{\cos@@{\theta}} = \frac{1}{\pi}\int_{0}^{\pi}(\cos@@{\theta}+i\sin@@{\theta}\cos@@{\phi})^{n}\diff{\phi} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | LegendreP(n, cos(theta)) = (1)/(Pi)*int((cos(theta)+ I*sin(theta)*cos(phi))^(n), phi = 0..Pi)
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LegendreP[n, Cos[\[Theta]]] == Divide[1,Pi]*Integrate[(Cos[\[Theta]]+ I*Sin[\[Theta]]*Cos[\[Phi]])^(n), {\[Phi], 0, Pi}, GenerateConditions->None]
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Failure | Aborted | Successful [Tested: 30] | Skipped - Because timed out |
| 18.10.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \HermitepolyH{n}@{x} = \frac{2^{n}}{\pi^{\frac{1}{2}}}\int_{-\infty}^{\infty}(x+it)^{n}e^{-t^{2}}\diff{t}}
\HermitepolyH{n}@{x} = \frac{2^{n}}{\pi^{\frac{1}{2}}}\int_{-\infty}^{\infty}(x+it)^{n}e^{-t^{2}}\diff{t} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | HermiteH(n, x) = ((2)^(n))/((Pi)^((1)/(2)))*int((x + I*t)^(n)* exp(- (t)^(2)), t = - infinity..infinity)
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HermiteH[n, x] == Divide[(2)^(n),(Pi)^(Divide[1,2])]*Integrate[(x + I*t)^(n)* Exp[- (t)^(2)], {t, - Infinity, Infinity}, GenerateConditions->None]
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Failure | Failure | Successful [Tested: 9] | Failed [9 / 9]
Result: Indeterminate
Test Values: {Rule[n, 1], Rule[x, 1.5]}
Result: Indeterminate
Test Values: {Rule[n, 2], Rule[x, 1.5]}
... skip entries to safe data |
| 18.10.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LaguerrepolyL[\alpha]{n}@{x} = \frac{e^{x}x^{-\frac{1}{2}\alpha}}{n!}\int_{0}^{\infty}e^{-t}t^{n+\frac{1}{2}\alpha}\BesselJ{\alpha}@{2\sqrt{xt}}\diff{t}}
\LaguerrepolyL[\alpha]{n}@{x} = \frac{e^{x}x^{-\frac{1}{2}\alpha}}{n!}\int_{0}^{\infty}e^{-t}t^{n+\frac{1}{2}\alpha}\BesselJ{\alpha}@{2\sqrt{xt}}\diff{t} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \alpha > -1, \realpart@@{((\alpha)+k+1)} > 0} | LaguerreL(n, alpha, x) = (exp(x)*(x)^(-(1)/(2)*alpha))/(factorial(n))*int(exp(- t)*(t)^(n +(1)/(2)*alpha)* BesselJ(alpha, 2*sqrt(x*t)), t = 0..infinity)
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LaguerreL[n, \[Alpha], x] == Divide[Exp[x]*(x)^(-Divide[1,2]*\[Alpha]),(n)!]*Integrate[Exp[- t]*(t)^(n +Divide[1,2]*\[Alpha])* BesselJ[\[Alpha], 2*Sqrt[x*t]], {t, 0, Infinity}, GenerateConditions->None]
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Missing Macro Error | Aborted | - | Skipped - Because timed out |
| 18.10.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \HermitepolyH{n}@{x} = \frac{(-2i)^{n}e^{x^{2}}}{\pi^{\frac{1}{2}}}\int_{-\infty}^{\infty}e^{-t^{2}}t^{n}e^{2ixt}\diff{t}}
\HermitepolyH{n}@{x} = \frac{(-2i)^{n}e^{x^{2}}}{\pi^{\frac{1}{2}}}\int_{-\infty}^{\infty}e^{-t^{2}}t^{n}e^{2ixt}\diff{t} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | HermiteH(n, x) = ((- 2*I)^(n)* exp((x)^(2)))/((Pi)^((1)/(2)))*int(exp(- (t)^(2))*(t)^(n)* exp(2*I*x*t), t = - infinity..infinity)
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HermiteH[n, x] == Divide[(- 2*I)^(n)* Exp[(x)^(2)],(Pi)^(Divide[1,2])]*Integrate[Exp[- (t)^(2)]*(t)^(n)* Exp[2*I*x*t], {t, - Infinity, Infinity}, GenerateConditions->None]
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Failure | Failure | Successful [Tested: 9] | Successful [Tested: 9] |
| 18.10.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{(-2i)^{n}e^{x^{2}}}{\pi^{\frac{1}{2}}}\int_{-\infty}^{\infty}e^{-t^{2}}t^{n}e^{2ixt}\diff{t} = \frac{2^{n+1}}{\pi^{\frac{1}{2}}}e^{x^{2}}\int_{0}^{\infty}e^{-t^{2}}t^{n}\cos@{2xt-\tfrac{1}{2}n\pi}\diff{t}}
\frac{(-2i)^{n}e^{x^{2}}}{\pi^{\frac{1}{2}}}\int_{-\infty}^{\infty}e^{-t^{2}}t^{n}e^{2ixt}\diff{t} = \frac{2^{n+1}}{\pi^{\frac{1}{2}}}e^{x^{2}}\int_{0}^{\infty}e^{-t^{2}}t^{n}\cos@{2xt-\tfrac{1}{2}n\pi}\diff{t} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | ((- 2*I)^(n)* exp((x)^(2)))/((Pi)^((1)/(2)))*int(exp(- (t)^(2))*(t)^(n)* exp(2*I*x*t), t = - infinity..infinity) = ((2)^(n + 1))/((Pi)^((1)/(2)))*exp((x)^(2))*int(exp(- (t)^(2))*(t)^(n)* cos(2*x*t -(1)/(2)*n*Pi), t = 0..infinity)
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Divide[(- 2*I)^(n)* Exp[(x)^(2)],(Pi)^(Divide[1,2])]*Integrate[Exp[- (t)^(2)]*(t)^(n)* Exp[2*I*x*t], {t, - Infinity, Infinity}, GenerateConditions->None] == Divide[(2)^(n + 1),(Pi)^(Divide[1,2])]*Exp[(x)^(2)]*Integrate[Exp[- (t)^(2)]*(t)^(n)* Cos[2*x*t -Divide[1,2]*n*Pi], {t, 0, Infinity}, GenerateConditions->None]
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Failure | Aborted | Successful [Tested: 9] | Successful [Tested: 9] |