Integrals with Coalescing Saddles - 36.13 Kelvin’s Ship-Wave Pattern
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| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|---|
| 36.13.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle z(\phi,\rho) = \int_{-\pi/2}^{\pi/2}\cos@{\rho\frac{\cos@{\theta+\phi}}{\cos^{2}@@{\theta}}}\diff{\theta}}
z(\phi,\rho) = \int_{-\pi/2}^{\pi/2}\cos@{\rho\frac{\cos@{\theta+\phi}}{\cos^{2}@@{\theta}}}\diff{\theta} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | z(phi , rho) = int(cos(rho*(cos(theta + phi))/((cos(theta))^(2))), theta = - Pi/2..Pi/2)
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z[\[Phi], \[Rho]] == Integrate[Cos[\[Rho]*Divide[Cos[\[Theta]+ \[Phi]],(Cos[\[Theta]])^(2)]], {\[Theta], - Pi/2, Pi/2}, GenerateConditions->None]
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Failure | Failure | Skipped - Because timed out | Error |
| 36.13.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \rho = \ifrac{gr}{V^{2}}}
\rho = \ifrac{gr}{V^{2}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | rho = (g*r)/((V)^(2)) |
\[Rho] == Divide[g*r,(V)^(2)] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 36.13#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \theta_{+}(\phi) = \tfrac{1}{2}(\asin@{3\sin@@{\phi}}-\phi)}
\theta_{+}(\phi) = \tfrac{1}{2}(\asin@{3\sin@@{\phi}}-\phi) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | theta[+](phi) = (1)/(2)*(arcsin(3*sin(phi))- phi)
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Subscript[\[Theta], +][\[Phi]] == Divide[1,2]*(ArcSin[3*Sin[\[Phi]]]- \[Phi])
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Error | Failure | - | Error |
| 36.13#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \theta_{-}(\phi) = \tfrac{1}{2}(\pi-\phi-\asin@{3\sin@@{\phi}})}
\theta_{-}(\phi) = \tfrac{1}{2}(\pi-\phi-\asin@{3\sin@@{\phi}}) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | theta[-](phi) = (1)/(2)*(Pi - phi - arcsin(3*sin(phi)))
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Subscript[\[Theta], -][\[Phi]] == Divide[1,2]*(Pi - \[Phi]- ArcSin[3*Sin[\[Phi]]])
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Error | Failure | - | Error |
| 36.13.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\phi| = \phi_{c}}
|\phi| = \phi_{c} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | abs(phi) = phi[c]
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Abs[\[Phi]] == Subscript[\[Phi], c]
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Failure | Failure | Failed [282 / 300] Result: .1339745960-.5000000000*I
Test Values: {c = -3/2, phi = 1/2*3^(1/2)+1/2*I, phi[c] = 1/2*3^(1/2)+1/2*I}
Result: 1.500000000-.8660254040*I
Test Values: {c = -3/2, phi = 1/2*3^(1/2)+1/2*I, phi[c] = -1/2+1/2*I*3^(1/2)}
... skip entries to safe data |
Failed [282 / 300]
Result: Complex[0.1339745962155613, -0.49999999999999994]
Test Values: {Rule[c, -1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, c], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Complex[1.4999999999999998, -0.8660254037844387]
Test Values: {Rule[c, -1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, c], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
| 36.13.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \phi_{c} = \asin@{\tfrac{1}{3}}}
\phi_{c} = \asin@{\tfrac{1}{3}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | phi[c] = arcsin((1)/(3))
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Subscript[\[Phi], c] == ArcSin[Divide[1,3]]
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Failure | Failure | Failed [300 / 300] Result: .5261884946+.5000000000*I
Test Values: {c = -3/2, phi = 1/2*3^(1/2)+1/2*I, phi[c] = 1/2*3^(1/2)+1/2*I}
Result: -.8398369094+.8660254040*I
Test Values: {c = -3/2, phi = 1/2*3^(1/2)+1/2*I, phi[c] = -1/2+1/2*I*3^(1/2)}
... skip entries to safe data |
Failed [300 / 300]
Result: Complex[0.5261884943303168, 0.49999999999999994]
Test Values: {Rule[c, -1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, c], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Complex[-0.8398369094541217, 0.8660254037844387]
Test Values: {Rule[c, -1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, c], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
| 36.13.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \omega(\mathbf{k}) = \sqrt{gk}+\mathbf{V}\cdot\mathbf{k}}
\omega(\mathbf{k}) = \sqrt{gk}+\mathbf{V}\cdot\mathbf{k} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | omega(k) = sqrt(g*k)+ V * k |
\[Omega][k] == Sqrt[g*k]+ V * k |
Skipped - no semantic math | Skipped - no semantic math | - | - |