Elliptic Integrals - 19.33 Triaxial Ellipsoids

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
19.33.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle S = 3V\CarlsonsymellintRG@{a^{-2}}{b^{-2}}{c^{-2}}}
S = 3V\CarlsonsymellintRG@{a^{-2}}{b^{-2}}{c^{-2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
Error

S == 3*V*Sqrt[(c)^(- 2)-(a)^(- 2)]*(EllipticE[ArcCos[Sqrt[(a)^(- 2)/(c)^(- 2)]],((c)^(- 2)-(b)^(- 2))/((c)^(- 2)-(a)^(- 2))]+(Cot[ArcCos[Sqrt[(a)^(- 2)/(c)^(- 2)]]])^2*EllipticF[ArcCos[Sqrt[(a)^(- 2)/(c)^(- 2)]],((c)^(- 2)-(b)^(- 2))/((c)^(- 2)-(a)^(- 2))]+Cot[ArcCos[Sqrt[(a)^(- 2)/(c)^(- 2)]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[(a)^(- 2)/(c)^(- 2)]]]^2])
Missing Macro Error Failure -
Failed [300 / 300]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[S, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[V, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[S, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[V, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.33.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{S}{2\pi} = c^{2}+\frac{ab}{\sin@@{\phi}}\left(\incellintEk@{\phi}{k}\sin^{2}@@{\phi}+\incellintFk@{\phi}{k}\cos^{2}@@{\phi}\right)}
\frac{S}{2\pi} = c^{2}+\frac{ab}{\sin@@{\phi}}\left(\incellintEk@{\phi}{k}\sin^{2}@@{\phi}+\incellintFk@{\phi}{k}\cos^{2}@@{\phi}\right)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle a \geq b, b \geq c} 
(S)/(2*Pi) = (c)^(2)+(a*b)/(sin(phi))*(EllipticE(sin(phi), k)*(sin(phi))^(2)+ EllipticF(sin(phi), k)*(cos(phi))^(2))

Divide[S,2*Pi] == (c)^(2)+Divide[a*b,Sin[\[Phi]]]*(EllipticE[\[Phi], (k)^2]*(Sin[\[Phi]])^(2)+ EllipticF[\[Phi], (k)^2]*(Cos[\[Phi]])^(2))
Failure Failure
Failed [300 / 300]
Result: -4.910443424-.9759333290e-1*I
Test Values: {S = 1/2*3^(1/2)+1/2*I, a = -3/2, b = -3/2, c = -3/2, phi = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -5.505002077-.4622644670e-1*I
Test Values: {S = 1/2*3^(1/2)+1/2*I, a = -3/2, b = -3/2, c = -3/2, phi = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-4.54039506540302, -0.09283854764917886]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[k, 1], Rule[S, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-4.634568996487559, -0.31545051747139075]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[k, 2], Rule[S, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.33#Ex1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \cos@@{\phi} = \frac{c}{a}}
\cos@@{\phi} = \frac{c}{a}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
cos(phi) = (c)/(a)

Cos[\[Phi]] == Divide[c,a]
Failure Failure
Failed [300 / 300]
Result: -.2694569811-.3969495503*I
Test Values: {a = -3/2, c = -3/2, phi = 1/2*3^(1/2)+1/2*I}


Result: .227765517+.4690753764*I
Test Values: {a = -3/2, c = -3/2, phi = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.06378043051909243, -0.10599798465255418]
Test Values: {Rule[a, -1.5], Rule[c, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[0.061176166972244816, 0.11050836582743673]
Test Values: {Rule[a, -1.5], Rule[c, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.33#Ex2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle k^{2} = \frac{a^{2}(b^{2}-c^{2})}{b^{2}(a^{2}-c^{2})}}
k^{2} = \frac{a^{2}(b^{2}-c^{2})}{b^{2}(a^{2}-c^{2})}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(k)^(2) = ((a)^(2)*((b)^(2)- (c)^(2)))/((b)^(2)*((a)^(2)- (c)^(2)))
 
(k)^(2) == Divide[(a)^(2)*((b)^(2)- (c)^(2)),(b)^(2)*((a)^(2)- (c)^(2))]
 Skipped - no semantic math Skipped - no semantic math - -
19.33.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{x^{2}}{a^{2}+\lambda}+\frac{y^{2}}{b^{2}+\lambda}+\frac{z^{2}}{c^{2}+\lambda} = 1}
\frac{x^{2}}{a^{2}+\lambda}+\frac{y^{2}}{b^{2}+\lambda}+\frac{z^{2}}{c^{2}+\lambda} = 1		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
((x)^(2))/((a)^(2)+ lambda)+((y)^(2))/((b)^(2)+ lambda)+((x + y*I)^(2))/((c)^(2)+ lambda) = 1
 
Divide[(x)^(2),(a)^(2)+ \[Lambda]]+Divide[(y)^(2),(b)^(2)+ \[Lambda]]+Divide[(x + y*I)^(2),(c)^(2)+ \[Lambda]] == 1
 Skipped - no semantic math Skipped - no semantic math - -
19.33.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle V(\lambda) = Q\CarlsonsymellintRF@{a^{2}+\lambda}{b^{2}+\lambda}{c^{2}+\lambda}}
V(\lambda) = Q\CarlsonsymellintRF@{a^{2}+\lambda}{b^{2}+\lambda}{c^{2}+\lambda}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
V(lambda) = Q*0.5*int(1/(sqrt(t+(a)^(2)+ lambda)*sqrt(t+(b)^(2)+ lambda)*sqrt(t+(c)^(2)+ lambda)), t = 0..infinity)

V[\[Lambda]] == Q*EllipticF[ArcCos[Sqrt[(a)^(2)+ \[Lambda]/(c)^(2)+ \[Lambda]]],((c)^(2)+ \[Lambda]-(b)^(2)+ \[Lambda])/((c)^(2)+ \[Lambda]-(a)^(2)+ \[Lambda])]/Sqrt[(c)^(2)+ \[Lambda]-(a)^(2)+ \[Lambda]]
Aborted Failure Skipped - Because timed out
Failed [300 / 300]
Result: Complex[-0.01914487900157147, 0.6670953471925876]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[Q, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[V, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-0.08207662518407155, 0.5134467292285442]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[Q, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[V, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.33.E6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 1/C = \CarlsonsymellintRF@{a^{2}}{b^{2}}{c^{2}}}
1/C = \CarlsonsymellintRF@{a^{2}}{b^{2}}{c^{2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
1/C = 0.5*int(1/(sqrt(t+(a)^(2))*sqrt(t+(b)^(2))*sqrt(t+(c)^(2))), t = 0..infinity)

1/C == EllipticF[ArcCos[Sqrt[(a)^(2)/(c)^(2)]],((c)^(2)-(b)^(2))/((c)^(2)-(a)^(2))]/Sqrt[(c)^(2)-(a)^(2)]
Aborted Failure Skipped - Because timed out
Failed [300 / 300]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[C, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[C, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.33.E7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle L_{c} = 2\pi abc\int_{0}^{\infty}\frac{\diff{\lambda}}{\sqrt{(a^{2}+\lambda)(b^{2}+\lambda)(c^{2}+\lambda)^{3}}}}
L_{c} = 2\pi abc\int_{0}^{\infty}\frac{\diff{\lambda}}{\sqrt{(a^{2}+\lambda)(b^{2}+\lambda)(c^{2}+\lambda)^{3}}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
L[c] = 2*Pi*a*b*c*int((1)/(sqrt(((a)^(2)+ lambda)*((b)^(2)+ lambda)*((c)^(2)+ lambda)^(3))), lambda = 0..infinity)

Subscript[L, c] == 2*Pi*a*b*c*Integrate[Divide[1,Sqrt[((a)^(2)+ \[Lambda])*((b)^(2)+ \[Lambda])*((c)^(2)+ \[Lambda])^(3)]], {\[Lambda], 0, Infinity}, GenerateConditions->None]
Aborted Aborted Skipped - Because timed out Skipped - Because timed out
19.33.E7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 2\pi abc\int_{0}^{\infty}\frac{\diff{\lambda}}{\sqrt{(a^{2}+\lambda)(b^{2}+\lambda)(c^{2}+\lambda)^{3}}} = V\CarlsonsymellintRD@{a^{2}}{b^{2}}{c^{2}}}
2\pi abc\int_{0}^{\infty}\frac{\diff{\lambda}}{\sqrt{(a^{2}+\lambda)(b^{2}+\lambda)(c^{2}+\lambda)^{3}}} = V\CarlsonsymellintRD@{a^{2}}{b^{2}}{c^{2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
Error

2*Pi*a*b*c*Integrate[Divide[1,Sqrt[((a)^(2)+ \[Lambda])*((b)^(2)+ \[Lambda])*((c)^(2)+ \[Lambda])^(3)]], {\[Lambda], 0, Infinity}, GenerateConditions->None] == V*3*(EllipticF[ArcCos[Sqrt[(a)^(2)/(c)^(2)]],((c)^(2)-(b)^(2))/((c)^(2)-(a)^(2))]-EllipticE[ArcCos[Sqrt[(a)^(2)/(c)^(2)]],((c)^(2)-(b)^(2))/((c)^(2)-(a)^(2))])/(((c)^(2)-(b)^(2))*((c)^(2)-(a)^(2))^(1/2))
Missing Macro Error Aborted Skip - symbolical successful subtest Skipped - Because timed out
19.33.E8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle L_{a}+L_{b}+L_{c} = 4\pi}
L_{a}+L_{b}+L_{c} = 4\pi		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
L[a]+ L[b]+ L[c] = 4*Pi
 
Subscript[L, a]+ Subscript[L, b]+ Subscript[L, c] == 4*Pi
 Skipped - no semantic math Skipped - no semantic math - -
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