Integrals with Coalescing Saddles - 36.13 Kelvin’s Ship-Wave Pattern

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
36.13.E1	${\displaystyle{\displaystyle z(\phi,\rho)=\int_{-\pi/2}^{\pi/2}\cos\left(\rho\frac{\cos\left(\theta+\phi\right)}{{\cos^{2}}\theta}\right)\mathrm{d}\theta}}$ z(\phi,\rho) = \int_{-\pi/2}^{\pi/2}\cos@{\rho\frac{\cos@{\theta+\phi}}{\cos^{2}@@{\theta}}}\diff{\theta}		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	${\displaystyle{\displaystyle\rho=\ifrac{gr}{V^{2}}}}$ \rho = \ifrac{gr}{V^{2}}		rho = (g*r)/((V)^(2))	\[Rho] == Divide[g*r,(V)^(2)]	Skipped - no semantic math	Skipped - no semantic math	-	-
36.13#Ex1	${\displaystyle{\displaystyle\theta_{+}(\phi)=\tfrac{1}{2}(\operatorname{arcsin}\left(3\sin\phi\right)-\phi)}}$ \theta_{+}(\phi) = \tfrac{1}{2}(\asin@{3\sin@@{\phi}}-\phi)		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	${\displaystyle{\displaystyle\theta_{-}(\phi)=\tfrac{1}{2}(\pi-\phi-\operatorname{arcsin}\left(3\sin\phi\right))}}$ \theta_{-}(\phi) = \tfrac{1}{2}(\pi-\phi-\asin@{3\sin@@{\phi}})		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	${\displaystyle{\displaystyle|\phi|=\phi_{c}}}$ |\phi| = \phi_{c}		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	${\displaystyle{\displaystyle\phi_{c}=\operatorname{arcsin}\left(\tfrac{1}{3}\right)}}$ \phi_{c} = \asin@{\tfrac{1}{3}}		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	${\displaystyle{\displaystyle\omega(\mathbf{k})=\sqrt{gk}+\mathbf{V}\cdot\mathbf{k}}}$ \omega(\mathbf{k}) = \sqrt{gk}+\mathbf{V}\cdot\mathbf{k}		omega(k) = sqrt(g*k)+ V * k	\[Omega][k] == Sqrt[g*k]+ V * k	Skipped - no semantic math	Skipped - no semantic math	-	-