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)

z[\[Phi], \[Rho]] == Integrate[Cos[\[Rho]*Divide[Cos[\[Theta]+ \[Phi]],(Cos[\[Theta]])^(2)]], {\[Theta], - Pi/2, Pi/2}, GenerateConditions->None]
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)

Subscript[\[Theta], +][\[Phi]] == Divide[1,2]*(ArcSin[3*Sin[\[Phi]]]- \[Phi])
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)))

Subscript[\[Theta], -][\[Phi]] == Divide[1,2]*(Pi - \[Phi]- ArcSin[3*Sin[\[Phi]]])
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]

Abs[\[Phi]] == Subscript[\[Phi], c]
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))

Subscript[\[Phi], c] == ArcSin[Divide[1,3]]
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 - -
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