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 S = 3 ⁢ V ⁢ R G ⁡ ( a - 2 , b - 2 , c - 2 ) 𝑆 3 𝑉 Carlson-integral-RG superscript 𝑎 2 superscript 𝑏 2 superscript 𝑐 2 {\displaystyle{\displaystyle S=3VR_{G}\left(a^{-2},b^{-2},c^{-2}\right)}}
S = 3V\CarlsonsymellintRG@{a^{-2}}{b^{-2}}{c^{-2}}		 

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 S 2 ⁢ π = c 2 + a ⁢ b sin ⁡ ϕ ⁢ ( E ⁡ ( ϕ , k ) ⁢ sin 2 ⁡ ϕ + F ⁡ ( ϕ , k ) ⁢ cos 2 ⁡ ϕ ) 𝑆 2 𝜋 superscript 𝑐 2 𝑎 𝑏 italic-ϕ elliptic-integral-second-kind-E italic-ϕ 𝑘 2 italic-ϕ elliptic-integral-first-kind-F italic-ϕ 𝑘 2 italic-ϕ {\displaystyle{\displaystyle\frac{S}{2\pi}=c^{2}+\frac{ab}{\sin\phi}\left(E% \left(\phi,k\right){\sin^{2}}\phi+F\left(\phi,k\right){\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)		 a ≥ b , b ≥ c formulae-sequence 𝑎 𝑏 𝑏 𝑐 {\displaystyle{\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 cos ⁡ ϕ = c a italic-ϕ 𝑐 𝑎 {\displaystyle{\displaystyle\cos\phi=\frac{c}{a}}}
\cos@@{\phi} = \frac{c}{a}		 

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 k 2 = a 2 ⁢ ( b 2 - c 2 ) b 2 ⁢ ( a 2 - c 2 ) superscript 𝑘 2 superscript 𝑎 2 superscript 𝑏 2 superscript 𝑐 2 superscript 𝑏 2 superscript 𝑎 2 superscript 𝑐 2 {\displaystyle{\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})}		 

(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 x 2 a 2 + λ + y 2 b 2 + λ + z 2 c 2 + λ = 1 superscript 𝑥 2 superscript 𝑎 2 𝜆 superscript 𝑦 2 superscript 𝑏 2 𝜆 superscript 𝑧 2 superscript 𝑐 2 𝜆 1 {\displaystyle{\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		 

((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 V ⁢ ( λ ) = Q ⁢ R F ⁡ ( a 2 + λ , b 2 + λ , c 2 + λ ) 𝑉 𝜆 𝑄 Carlson-integral-RF superscript 𝑎 2 𝜆 superscript 𝑏 2 𝜆 superscript 𝑐 2 𝜆 {\displaystyle{\displaystyle V(\lambda)=QR_{F}\left(a^{2}+\lambda,b^{2}+% \lambda,c^{2}+\lambda\right)}}
V(\lambda) = Q\CarlsonsymellintRF@{a^{2}+\lambda}{b^{2}+\lambda}{c^{2}+\lambda}		 

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 1 / C = R F ⁡ ( a 2 , b 2 , c 2 ) 1 𝐶 Carlson-integral-RF superscript 𝑎 2 superscript 𝑏 2 superscript 𝑐 2 {\displaystyle{\displaystyle 1/C=R_{F}\left(a^{2},b^{2},c^{2}\right)}}
1/C = \CarlsonsymellintRF@{a^{2}}{b^{2}}{c^{2}}		 

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 L c = 2 ⁢ π ⁢ a ⁢ b ⁢ c ⁢ ∫ 0 ∞ d λ ( a 2 + λ ) ⁢ ( b 2 + λ ) ⁢ ( c 2 + λ ) 3 subscript 𝐿 𝑐 2 𝜋 𝑎 𝑏 𝑐 superscript subscript 0 𝜆 superscript 𝑎 2 𝜆 superscript 𝑏 2 𝜆 superscript superscript 𝑐 2 𝜆 3 {\displaystyle{\displaystyle L_{c}=2\pi abc\int_{0}^{\infty}\frac{\mathrm{d}% \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}}}		 

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 2 ⁢ π ⁢ a ⁢ b ⁢ c ⁢ ∫ 0 ∞ d λ ( a 2 + λ ) ⁢ ( b 2 + λ ) ⁢ ( c 2 + λ ) 3 = V ⁢ R D ⁡ ( a 2 , b 2 , c 2 ) 2 𝜋 𝑎 𝑏 𝑐 superscript subscript 0 𝜆 superscript 𝑎 2 𝜆 superscript 𝑏 2 𝜆 superscript superscript 𝑐 2 𝜆 3 𝑉 Carlson-integral-RD superscript 𝑎 2 superscript 𝑏 2 superscript 𝑐 2 {\displaystyle{\displaystyle 2\pi abc\int_{0}^{\infty}\frac{\mathrm{d}\lambda}% {\sqrt{(a^{2}+\lambda)(b^{2}+\lambda)(c^{2}+\lambda)^{3}}}=VR_{D}\left(a^{2},b% ^{2},c^{2}\right)}}
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}}		 

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 L a + L b + L c = 4 ⁢ π subscript 𝐿 𝑎 subscript 𝐿 𝑏 subscript 𝐿 𝑐 4 𝜋 {\displaystyle{\displaystyle L_{a}+L_{b}+L_{c}=4\pi}}
L_{a}+L_{b}+L_{c} = 4\pi		 

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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