Orthogonal Polynomials - 19.2 Definitions
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| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
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Numeric Mathematica |
|---|---|---|---|---|---|---|---|---|
| 19.2.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle r(s,t) = \frac{(p_{1}+p_{2}s)(p_{3}-p_{4}s)s}{(p_{3}+p_{4}s)(p_{3}-p_{4}s)s}}
r(s,t) = \frac{(p_{1}+p_{2}s)(p_{3}-p_{4}s)s}{(p_{3}+p_{4}s)(p_{3}-p_{4}s)s} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | r(s , t) = ((p[1]+ p[2]*s)*(p[3]- p[4]*s)*s)/((p[3]+ p[4]*s)*(p[3]- p[4]*s)*s) |
r[s , t] == Divide[(Subscript[p, 1]+ Subscript[p, 2]*s)*(Subscript[p, 3]- Subscript[p, 4]*s)*s,(Subscript[p, 3]+ Subscript[p, 4]*s)*(Subscript[p, 3]- Subscript[p, 4]*s)*s] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 19.2.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \incellintFk@{\phi}{k} = \int_{0}^{\phi}\frac{\diff{\theta}}{\sqrt{1-k^{2}\sin^{2}@@{\theta}}}}
\incellintFk@{\phi}{k} = \int_{0}^{\phi}\frac{\diff{\theta}}{\sqrt{1-k^{2}\sin^{2}@@{\theta}}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | EllipticF(sin(phi), k) = int((1)/(sqrt(1 - (k)^(2)* (sin(theta))^(2))), theta = 0..phi)
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EllipticF[\[Phi], (k)^2] == Integrate[Divide[1,Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)]], {\[Theta], 0, \[Phi]}, GenerateConditions->None]
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Failure | Aborted | Failed [6 / 30] Result: Float(infinity)
Test Values: {phi = -2, k = 1}
Result: .2e-9-.5175477340*I
Test Values: {phi = -2, k = 2}
... skip entries to safe data |
Skipped - Because timed out |
| 19.2.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\phi}\frac{\diff{\theta}}{\sqrt{1-k^{2}\sin^{2}@@{\theta}}} = \int_{0}^{\sin@@{\phi}}\frac{\diff{t}}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}}}
\int_{0}^{\phi}\frac{\diff{\theta}}{\sqrt{1-k^{2}\sin^{2}@@{\theta}}} = \int_{0}^{\sin@@{\phi}}\frac{\diff{t}}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | int((1)/(sqrt(1 - (k)^(2)* (sin(theta))^(2))), theta = 0..phi) = int((1)/(sqrt(1 - (t)^(2))*sqrt(1 - (k)^(2)* (t)^(2))), t = 0..sin(phi))
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Integrate[Divide[1,Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)]], {\[Theta], 0, \[Phi]}, GenerateConditions->None] == Integrate[Divide[1,Sqrt[1 - (t)^(2)]*Sqrt[1 - (k)^(2)* (t)^(2)]], {t, 0, Sin[\[Phi]]}, GenerateConditions->None]
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Failure | Aborted | Failed [6 / 30] Result: Float(-infinity)
Test Values: {phi = -2, k = 1}
Result: -.2e-9+.5175477340*I
Test Values: {phi = -2, k = 2}
... skip entries to safe data |
Skipped - Because timed out |
| 19.2.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \incellintEk@{\phi}{k} = \int_{0}^{\phi}\sqrt{1-k^{2}\sin^{2}@@{\theta}}\diff{\theta}\\}
\incellintEk@{\phi}{k} = \int_{0}^{\phi}\sqrt{1-k^{2}\sin^{2}@@{\theta}}\diff{\theta}\\ |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | EllipticE(sin(phi), k) = int(sqrt(1 - (k)^(2)* (sin(theta))^(2)), theta = 0..phi)
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EllipticE[\[Phi], (k)^2] == Integrate[Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)], {\[Theta], 0, \[Phi]}, GenerateConditions->None]
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Failure | Aborted | Skipped - Because timed out | Skipped - Because timed out |
| 19.2.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\phi}\sqrt{1-k^{2}\sin^{2}@@{\theta}}\diff{\theta}\\ = \int_{0}^{\sin@@{\phi}}\frac{\sqrt{1-k^{2}t^{2}}}{\sqrt{1-t^{2}}}\diff{t}}
\int_{0}^{\phi}\sqrt{1-k^{2}\sin^{2}@@{\theta}}\diff{\theta}\\ = \int_{0}^{\sin@@{\phi}}\frac{\sqrt{1-k^{2}t^{2}}}{\sqrt{1-t^{2}}}\diff{t} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | int(sqrt(1 - (k)^(2)* (sin(theta))^(2)), theta = 0..phi) = int((sqrt(1 - (k)^(2)* (t)^(2)))/(sqrt(1 - (t)^(2))), t = 0..sin(phi))
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Integrate[Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)], {\[Theta], 0, \[Phi]}, GenerateConditions->None] == Integrate[Divide[Sqrt[1 - (k)^(2)* (t)^(2)],Sqrt[1 - (t)^(2)]], {t, 0, Sin[\[Phi]]}, GenerateConditions->None]
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Failure | Aborted | Skipped - Because timed out | Skipped - Because timed out |
| 19.2.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \incellintDk@{\phi}{k} = \int_{0}^{\phi}\frac{\sin^{2}@@{\theta}\diff{\theta}}{\sqrt{1-k^{2}\sin^{2}@@{\theta}}}}
\incellintDk@{\phi}{k} = \int_{0}^{\phi}\frac{\sin^{2}@@{\theta}\diff{\theta}}{\sqrt{1-k^{2}\sin^{2}@@{\theta}}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | (EllipticF(sin(phi), k) - EllipticE(sin(phi), k))/(k)^2 = int(((sin(theta))^(2))/(sqrt(1 - (k)^(2)* (sin(theta))^(2))), theta = 0..phi)
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Divide[EllipticF[\[Phi], (k)^2] - EllipticE[\[Phi], (k)^2], (k)^4] == Integrate[Divide[(Sin[\[Theta]])^(2),Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)]], {\[Theta], 0, \[Phi]}, GenerateConditions->None]
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Failure | Aborted | Skipped - Because timed out | Skipped - Because timed out |
| 19.2.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\phi}\frac{\sin^{2}@@{\theta}\diff{\theta}}{\sqrt{1-k^{2}\sin^{2}@@{\theta}}} = \int_{0}^{\sin@@{\phi}}\frac{t^{2}\diff{t}}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}}}
\int_{0}^{\phi}\frac{\sin^{2}@@{\theta}\diff{\theta}}{\sqrt{1-k^{2}\sin^{2}@@{\theta}}} = \int_{0}^{\sin@@{\phi}}\frac{t^{2}\diff{t}}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | int(((sin(theta))^(2))/(sqrt(1 - (k)^(2)* (sin(theta))^(2))), theta = 0..phi) = int(((t)^(2))/(sqrt(1 - (t)^(2))*sqrt(1 - (k)^(2)* (t)^(2))), t = 0..sin(phi))
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Integrate[Divide[(Sin[\[Theta]])^(2),Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)]], {\[Theta], 0, \[Phi]}, GenerateConditions->None] == Integrate[Divide[(t)^(2),Sqrt[1 - (t)^(2)]*Sqrt[1 - (k)^(2)* (t)^(2)]], {t, 0, Sin[\[Phi]]}, GenerateConditions->None]
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Failure | Aborted | Skipped - Because timed out | Skipped - Because timed out |
| 19.2.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\sin@@{\phi}}\frac{t^{2}\diff{t}}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}} = (\incellintFk@{\phi}{k}-\incellintEk@{\phi}{k})/k^{2}}
\int_{0}^{\sin@@{\phi}}\frac{t^{2}\diff{t}}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}} = (\incellintFk@{\phi}{k}-\incellintEk@{\phi}{k})/k^{2} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | int(((t)^(2))/(sqrt(1 - (t)^(2))*sqrt(1 - (k)^(2)* (t)^(2))), t = 0..sin(phi)) = (EllipticF(sin(phi), k)- EllipticE(sin(phi), k))/(k)^(2)
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Integrate[Divide[(t)^(2),Sqrt[1 - (t)^(2)]*Sqrt[1 - (k)^(2)* (t)^(2)]], {t, 0, Sin[\[Phi]]}, GenerateConditions->None] == (EllipticF[\[Phi], (k)^2]- EllipticE[\[Phi], (k)^2])/(k)^(2)
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Failure | Aborted | Successful [Tested: 0] | Skipped - Because timed out |
| 19.2.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \incellintPik@{\phi}{\alpha^{2}}{k} = \int_{0}^{\phi}\frac{\diff{\theta}}{\sqrt{1-k^{2}\sin^{2}@@{\theta}}(1-\alpha^{2}\sin^{2}@@{\theta})}}
\incellintPik@{\phi}{\alpha^{2}}{k} = \int_{0}^{\phi}\frac{\diff{\theta}}{\sqrt{1-k^{2}\sin^{2}@@{\theta}}(1-\alpha^{2}\sin^{2}@@{\theta})} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | EllipticPi(sin(phi), (alpha)^(2), k) = int((1)/(sqrt(1 - (k)^(2)* (sin(theta))^(2))*(1 - (alpha)^(2)* (sin(theta))^(2))), theta = 0..phi)
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EllipticPi[\[Alpha]^(2), \[Phi],(k)^2] == Integrate[Divide[1,Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)]*(1 - \[Alpha]^(2)* (Sin[\[Theta]])^(2))], {\[Theta], 0, \[Phi]}, GenerateConditions->None]
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Aborted | Aborted | Skipped - Because timed out | Skipped - Because timed out |
| 19.2.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\phi}\frac{\diff{\theta}}{\sqrt{1-k^{2}\sin^{2}@@{\theta}}(1-\alpha^{2}\sin^{2}@@{\theta})} = \int_{0}^{\sin@@{\phi}}\frac{\diff{t}}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}(1-\alpha^{2}t^{2})}}
\int_{0}^{\phi}\frac{\diff{\theta}}{\sqrt{1-k^{2}\sin^{2}@@{\theta}}(1-\alpha^{2}\sin^{2}@@{\theta})} = \int_{0}^{\sin@@{\phi}}\frac{\diff{t}}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}(1-\alpha^{2}t^{2})} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | int((1)/(sqrt(1 - (k)^(2)* (sin(theta))^(2))*(1 - (alpha)^(2)* (sin(theta))^(2))), theta = 0..phi) = int((1)/(sqrt(1 - (t)^(2))*sqrt(1 - (k)^(2)* (t)^(2))*(1 - (alpha)^(2)* (t)^(2))), t = 0..sin(phi))
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Integrate[Divide[1,Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)]*(1 - \[Alpha]^(2)* (Sin[\[Theta]])^(2))], {\[Theta], 0, \[Phi]}, GenerateConditions->None] == Integrate[Divide[1,Sqrt[1 - (t)^(2)]*Sqrt[1 - (k)^(2)* (t)^(2)]*(1 - \[Alpha]^(2)* (t)^(2))], {t, 0, Sin[\[Phi]]}, GenerateConditions->None]
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Aborted | Aborted | Skipped - Because timed out | Skipped - Because timed out |
| 19.2#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \compellintKk@{k} = \incellintFk@{\pi/2}{k}}
\compellintKk@{k} = \incellintFk@{\pi/2}{k} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | EllipticK(k) = EllipticF(sin(Pi/2), k)
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EllipticK[(k)^2] == EllipticF[Pi/2, (k)^2]
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Successful | Successful | - | Successful [Tested: 3] |
| 19.2#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \compellintEk@{k} = \incellintEk@{\pi/2}{k}}
\compellintEk@{k} = \incellintEk@{\pi/2}{k} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | EllipticE(k) = EllipticE(sin(Pi/2), k)
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EllipticE[(k)^2] == EllipticE[Pi/2, (k)^2]
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Successful | Successful | - | Successful [Tested: 3] |
| 19.2#Ex3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \compellintDk@{k} = \incellintDk@{\pi/2}{k}}
\compellintDk@{k} = \incellintDk@{\pi/2}{k} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | (EllipticK(k) - EllipticE(k))/(k)^2 = (EllipticF(sin(Pi/2), k) - EllipticE(sin(Pi/2), k))/(k)^2
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Divide[EllipticK[(k)^2] - EllipticE[(k)^2], (k)^4] == Divide[EllipticF[Pi/2, (k)^2] - EllipticE[Pi/2, (k)^2], (k)^4]
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Successful | Successful | - | Successful [Tested: 3] |
| 19.2#Ex3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \incellintDk@{\pi/2}{k} = (\compellintKk@{k}-\compellintEk@{k})/k^{2}}
\incellintDk@{\pi/2}{k} = (\compellintKk@{k}-\compellintEk@{k})/k^{2} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | (EllipticF(sin(Pi/2), k) - EllipticE(sin(Pi/2), k))/(k)^2 = (EllipticK(k)- EllipticE(k))/(k)^(2)
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Divide[EllipticF[Pi/2, (k)^2] - EllipticE[Pi/2, (k)^2], (k)^4] == (EllipticK[(k)^2]- EllipticE[(k)^2])/(k)^(2)
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Successful | Failure | - | Failed [3 / 3]
Result: Indeterminate
Test Values: {Rule[k, 1]}
Result: Complex[-0.08185805455243832, 0.4541460103381725]
Test Values: {Rule[k, 2]}
... skip entries to safe data |
| 19.2#Ex4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \compellintPik@{\alpha^{2}}{k} = \incellintPik@{\pi/2}{\alpha^{2}}{k}}
\compellintPik@{\alpha^{2}}{k} = \incellintPik@{\pi/2}{\alpha^{2}}{k} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | EllipticPi((alpha)^(2), k) = EllipticPi(sin(Pi/2), (alpha)^(2), k)
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EllipticPi[\[Alpha]^(2), (k)^2] == EllipticPi[\[Alpha]^(2), Pi/2,(k)^2]
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Successful | Successful | - | Successful [Tested: 9] |
| 19.2#Ex5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ccompellintKk@{k} = \compellintKk@{k^{\prime}}}
\ccompellintKk@{k} = \compellintKk@{k^{\prime}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | EllipticCK(k) = EllipticK(sqrt(1 - (k)^(2)))
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EllipticK[1-(k)^2] == EllipticK[(Sqrt[1 - (k)^(2)])^2]
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Successful | Successful | - | Successful [Tested: 3] |
| 19.2#Ex6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ccompellintEk@{k} = \compellintEk@{k^{\prime}}}
\ccompellintEk@{k} = \compellintEk@{k^{\prime}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | EllipticCE(k) = EllipticE(sqrt(1 - (k)^(2)))
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EllipticE[1-(k)^2] == EllipticE[(Sqrt[1 - (k)^(2)])^2]
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Successful | Successful | - | Successful [Tested: 3] |
| 19.2#Ex7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle k^{\prime} = \sqrt{1-k^{2}}}
k^{\prime} = \sqrt{1-k^{2}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | sqrt(1 - (k)^(2)) = sqrt(1 - (k)^(2))
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Sqrt[1 - (k)^(2)] == Sqrt[1 - (k)^(2)]
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Successful | Successful | - | Successful [Tested: 3] |
| 19.2#Ex8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \incellintFk@{m\pi+\phi}{k} = 2m\compellintKk@{k}+\incellintFk@{\phi}{k}}
\incellintFk@{m\pi+\phi}{k} = 2m\compellintKk@{k}+\incellintFk@{\phi}{k} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | EllipticF(sin(m*Pi + phi), k) = 2*m*EllipticK(k)+ EllipticF(sin(phi), k)
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EllipticF[m*Pi + \[Phi], (k)^2] == 2*m*EllipticK[(k)^2]+ EllipticF[\[Phi], (k)^2]
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Failure | Failure | Error | Failed [30 / 90]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[m, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[m, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}
... skip entries to safe data |
| 19.2#Ex8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \incellintFk@{m\pi-\phi}{k} = 2m\compellintKk@{k}-\incellintFk@{\phi}{k}}
\incellintFk@{m\pi-\phi}{k} = 2m\compellintKk@{k}-\incellintFk@{\phi}{k} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | EllipticF(sin(m*Pi - phi), k) = 2*m*EllipticK(k)- EllipticF(sin(phi), k)
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EllipticF[m*Pi - \[Phi], (k)^2] == 2*m*EllipticK[(k)^2]- EllipticF[\[Phi], (k)^2]
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Failure | Failure | Error | Failed [30 / 90]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[m, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[m, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}
... skip entries to safe data |
| 19.2#Ex9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \incellintEk@{m\pi+\phi}{k} = 2m\compellintEk@{k}+\incellintEk@{\phi}{k}}
\incellintEk@{m\pi+\phi}{k} = 2m\compellintEk@{k}+\incellintEk@{\phi}{k} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | EllipticE(sin(m*Pi + phi), k) = 2*m*EllipticE(k)+ EllipticE(sin(phi), k)
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EllipticE[m*Pi + \[Phi], (k)^2] == 2*m*EllipticE[(k)^2]+ EllipticE[\[Phi], (k)^2]
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Failure | Failure | Failed [90 / 90] Result: -3.717960670-.6751929261*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, k = 1, m = 1}
Result: -4.000000000-.3e-9*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, k = 1, m = 2}
... skip entries to safe data |
Successful [Tested: 90] |
| 19.2#Ex9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \incellintEk@{m\pi-\phi}{k} = 2m\compellintEk@{k}-\incellintEk@{\phi}{k}}
\incellintEk@{m\pi-\phi}{k} = 2m\compellintEk@{k}-\incellintEk@{\phi}{k} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | EllipticE(sin(m*Pi - phi), k) = 2*m*EllipticE(k)- EllipticE(sin(phi), k)
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EllipticE[m*Pi - \[Phi], (k)^2] == 2*m*EllipticE[(k)^2]- EllipticE[\[Phi], (k)^2]
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Failure | Failure | Failed [90 / 90] Result: -.2820393315+.6751929264*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, k = 1, m = 1}
Result: -4.000000000-.4e-9*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, k = 1, m = 2}
... skip entries to safe data |
Successful [Tested: 90] |
| 19.2#Ex10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \incellintDk@{m\pi+\phi}{k} = 2m\compellintDk@{k}+\incellintDk@{\phi}{k}}
\incellintDk@{m\pi+\phi}{k} = 2m\compellintDk@{k}+\incellintDk@{\phi}{k} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | (EllipticF(sin(m*Pi + phi), k) - EllipticE(sin(m*Pi + phi), k))/(k)^2 = 2*m*(EllipticK(k) - EllipticE(k))/(k)^2 + (EllipticF(sin(phi), k) - EllipticE(sin(phi), k))/(k)^2
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Divide[EllipticF[m*Pi + \[Phi], (k)^2] - EllipticE[m*Pi + \[Phi], (k)^2], (k)^4] == 2*m*Divide[EllipticK[(k)^2] - EllipticE[(k)^2], (k)^4]+ Divide[EllipticF[\[Phi], (k)^2] - EllipticE[\[Phi], (k)^2], (k)^4]
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Failure | Failure | Error | Failed [30 / 90]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[m, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[m, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}
... skip entries to safe data |
| 19.2#Ex10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \incellintDk@{m\pi-\phi}{k} = 2m\compellintDk@{k}-\incellintDk@{\phi}{k}}
\incellintDk@{m\pi-\phi}{k} = 2m\compellintDk@{k}-\incellintDk@{\phi}{k} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | (EllipticF(sin(m*Pi - phi), k) - EllipticE(sin(m*Pi - phi), k))/(k)^2 = 2*m*(EllipticK(k) - EllipticE(k))/(k)^2 - (EllipticF(sin(phi), k) - EllipticE(sin(phi), k))/(k)^2
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Divide[EllipticF[m*Pi - \[Phi], (k)^2] - EllipticE[m*Pi - \[Phi], (k)^2], (k)^4] == 2*m*Divide[EllipticK[(k)^2] - EllipticE[(k)^2], (k)^4]- Divide[EllipticF[\[Phi], (k)^2] - EllipticE[\[Phi], (k)^2], (k)^4]
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Failure | Failure | Error | Failed [30 / 90]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[m, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[m, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}
... skip entries to safe data |
| 19.2.E16 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\atan@@{x}}\frac{\diff{\theta}}{(\cos^{2}@@{\theta}+p\sin^{2}@@{\theta})\sqrt{\cos^{2}@@{\theta}+k_{c}^{2}\sin^{2}@@{\theta}}} = \incellintPik@{\atan@@{x}}{1-p}{k}}
\int_{0}^{\atan@@{x}}\frac{\diff{\theta}}{(\cos^{2}@@{\theta}+p\sin^{2}@@{\theta})\sqrt{\cos^{2}@@{\theta}+k_{c}^{2}\sin^{2}@@{\theta}}} = \incellintPik@{\atan@@{x}}{1-p}{k} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x^{2} \neq -1/p} | int((1)/(((cos(theta))^(2)+ p*(sin(theta))^(2))*sqrt((cos(theta))^(2)+ (k[c])^(2)*(sin(theta))^(2))), theta = 0..arctan(x)) = EllipticPi(sin(arctan(x)), 1 - p, k)
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Integrate[Divide[1,((Cos[\[Theta]])^(2)+ p*(Sin[\[Theta]])^(2))*Sqrt[(Cos[\[Theta]])^(2)+ (Subscript[k, c])^(2)*(Sin[\[Theta]])^(2)]], {\[Theta], 0, ArcTan[x]}, GenerateConditions->None] == EllipticPi[1 - p, ArcTan[x],(k)^2]
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Error | Aborted | - | Skipped - Because timed out |
| 19.2.E17 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \CarlsonellintRC@{x}{y} = \frac{1}{2}\int_{0}^{\infty}\frac{\diff{t}}{\sqrt{t+x}(t+y)}}
\CarlsonellintRC@{x}{y} = \frac{1}{2}\int_{0}^{\infty}\frac{\diff{t}}{\sqrt{t+x}(t+y)} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Error
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1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)] == Divide[1,2]*Integrate[Divide[1,Sqrt[t + x]*(t + y)], {t, 0, Infinity}, GenerateConditions->None]
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Missing Macro Error | Failure | - | Failed [12 / 18]
Result: Complex[-1.0177225554447185, 0.0]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}
Result: Indeterminate
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}
... skip entries to safe data |
| 19.2.E18 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \CarlsonellintRC@{x}{y} = \frac{1}{\sqrt{y-x}}\atan@@{\sqrt{\frac{y-x}{x}}}}
\CarlsonellintRC@{x}{y} = \frac{1}{\sqrt{y-x}}\atan@@{\sqrt{\frac{y-x}{x}}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 \leq x, x < y} | Error
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1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)] == Divide[1,Sqrt[y - x]]*ArcTan[Sqrt[Divide[y - x,x]]]
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Missing Macro Error | Failure | Skip - symbolical successful subtest | Successful [Tested: 3] |
| 19.2.E18 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{\sqrt{y-x}}\atan@@{\sqrt{\frac{y-x}{x}}} = \frac{1}{\sqrt{y-x}}\acos@@{\sqrt{x/y}}}
\frac{1}{\sqrt{y-x}}\atan@@{\sqrt{\frac{y-x}{x}}} = \frac{1}{\sqrt{y-x}}\acos@@{\sqrt{x/y}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 \leq x, x < y} | (1)/(sqrt(y - x))*arctan(sqrt((y - x)/(x))) = (1)/(sqrt(y - x))*arccos(sqrt(x/y)) |
Divide[1,Sqrt[y - x]]*ArcTan[Sqrt[Divide[y - x,x]]] == Divide[1,Sqrt[y - x]]*ArcCos[Sqrt[x/y]] |
Failure | Failure | Successful [Tested: 3] | Successful [Tested: 3] |
| 19.2.E19 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \CarlsonellintRC@{x}{y} = \frac{1}{\sqrt{x-y}}\atanh@@{\sqrt{\frac{x-y}{x}}}}
\CarlsonellintRC@{x}{y} = \frac{1}{\sqrt{x-y}}\atanh@@{\sqrt{\frac{x-y}{x}}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 < y, y < x} | Error |
1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)] == Divide[1,Sqrt[x - y]]*ArcTanh[Sqrt[Divide[x - y,x]]] |
Missing Macro Error | Failure | Skip - symbolical successful subtest | Successful [Tested: 3] |
| 19.2.E19 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{\sqrt{x-y}}\atanh@@{\sqrt{\frac{x-y}{x}}} = \frac{1}{\sqrt{x-y}}\ln@@{\frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{y}}}}
\frac{1}{\sqrt{x-y}}\atanh@@{\sqrt{\frac{x-y}{x}}} = \frac{1}{\sqrt{x-y}}\ln@@{\frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{y}}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 < y, y < x} | (1)/(sqrt(x - y))*arctanh(sqrt((x - y)/(x))) = (1)/(sqrt(x - y))*ln((sqrt(x)+sqrt(x - y))/(sqrt(y))) |
Divide[1,Sqrt[x - y]]*ArcTanh[Sqrt[Divide[x - y,x]]] == Divide[1,Sqrt[x - y]]*Log[Divide[Sqrt[x]+Sqrt[x - y],Sqrt[y]]] |
Failure | Failure | Successful [Tested: 3] | Successful [Tested: 3] |
| 19.2.E20 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \CarlsonellintRC@{x}{y} = \sqrt{\frac{x}{x-y}}\CarlsonellintRC@{x-y}{-y}}
\CarlsonellintRC@{x}{y} = \sqrt{\frac{x}{x-y}}\CarlsonellintRC@{x-y}{-y} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle y < 0, 0 \leq x} | Error |
1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)] == Sqrt[Divide[x,x - y]]*1/Sqrt[- y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x - y)/(- y)] |
Missing Macro Error | Failure | Skip - symbolical successful subtest | Failed [9 / 9]
Result: Complex[-1.0177225554447187, -0.906899682117109]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}Result: Complex[-1.862459718905424, -1.1107207345395915]
Test Values: {Rule[x, 1.5], Rule[y, -0.5]}... skip entries to safe data |
| 19.2.E20 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sqrt{\frac{x}{x-y}}\CarlsonellintRC@{x-y}{-y} = \frac{1}{\sqrt{x-y}}\atanh@@{\sqrt{\frac{x}{x-y}}}}
\sqrt{\frac{x}{x-y}}\CarlsonellintRC@{x-y}{-y} = \frac{1}{\sqrt{x-y}}\atanh@@{\sqrt{\frac{x}{x-y}}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle y < 0, 0 \leq x} | Error |
Sqrt[Divide[x,x - y]]*1/Sqrt[- y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x - y)/(- y)] == Divide[1,Sqrt[x - y]]*ArcTanh[Sqrt[Divide[x,x - y]]] |
Missing Macro Error | Failure | Skip - symbolical successful subtest | Successful [Tested: 9] |
| 19.2.E20 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{\sqrt{x-y}}\atanh@@{\sqrt{\frac{x}{x-y}}} = \frac{1}{\sqrt{x-y}}\ln@@{\frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{-y}}}}
\frac{1}{\sqrt{x-y}}\atanh@@{\sqrt{\frac{x}{x-y}}} = \frac{1}{\sqrt{x-y}}\ln@@{\frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{-y}}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle y < 0, 0 \leq x} | (1)/(sqrt(x - y))*arctanh(sqrt((x)/(x - y))) = (1)/(sqrt(x - y))*ln((sqrt(x)+sqrt(x - y))/(sqrt(- y))) |
Divide[1,Sqrt[x - y]]*ArcTanh[Sqrt[Divide[x,x - y]]] == Divide[1,Sqrt[x - y]]*Log[Divide[Sqrt[x]+Sqrt[x - y],Sqrt[- y]]] |
Failure | Failure | Successful [Tested: 9] | Successful [Tested: 9] |
| 19.2.E21 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \CarlsonellintRC@{x}{y} = \int_{0}^{1}(v^{2}x+(1-v^{2})y)^{-1/2}\diff{v}}
\CarlsonellintRC@{x}{y} = \int_{0}^{1}(v^{2}x+(1-v^{2})y)^{-1/2}\diff{v} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Error |
1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)] == Integrate[((v)^(2)* x +(1 - (v)^(2))*y)^(- 1/2), {v, 0, 1}, GenerateConditions->None] |
Missing Macro Error | Aborted | - | Skipped - Because timed out |
| 19.2.E22 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \CarlsonellintRC@{x}{y} = \frac{2}{\pi}\int_{0}^{\pi/2}\CarlsonellintRC@{y}{x\cos^{2}@@{\theta}+y\sin^{2}@@{\theta}}\diff{\theta}}
\CarlsonellintRC@{x}{y} = \frac{2}{\pi}\int_{0}^{\pi/2}\CarlsonellintRC@{y}{x\cos^{2}@@{\theta}+y\sin^{2}@@{\theta}}\diff{\theta} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Error |
1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)] == Divide[2,Pi]*Integrate[1/Sqrt[x*(Cos[\[Theta]])^(2)+ y*(Sin[\[Theta]])^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-(y)/(x*(Cos[\[Theta]])^(2)+ y*(Sin[\[Theta]])^(2))], {\[Theta], 0, Pi/2}, GenerateConditions->None] |
Missing Macro Error | Aborted | - | Skipped - Because timed out |