Elliptic Integrals - 19.11 Addition Theorems

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
19.11.E1 F ⁡ ( θ , k ) + F ⁡ ( ϕ , k ) = F ⁡ ( ψ , k ) elliptic-integral-first-kind-F 𝜃 𝑘 elliptic-integral-first-kind-F italic-ϕ 𝑘 elliptic-integral-first-kind-F 𝜓 𝑘 {\displaystyle{\displaystyle F\left(\theta,k\right)+F\left(\phi,k\right)=F% \left(\psi,k\right)}}
\incellintFk@{\theta}{k}+\incellintFk@{\phi}{k} = \incellintFk@{\psi}{k}		 

EllipticF(sin(theta), k)+ EllipticF(sin(phi), k) = EllipticF(sin(psi), k)

EllipticF[\[Theta], (k)^2]+ EllipticF[\[Phi], (k)^2] == EllipticF[\[Psi], (k)^2]
Failure Failure
Failed [300 / 300]
Result: .8208700290+.6773780507*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 1}


Result: .4831883421+.7182528229*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.43180375739814203, 0.27142936483528934]
Test Values: {Rule[k, 1], Rule[θ, 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]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[0.3965687056216178, 0.33175091278780894]
Test Values: {Rule[k, 2], Rule[θ, 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]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.11.E2 E ⁡ ( θ , k ) + E ⁡ ( ϕ , k ) = E ⁡ ( ψ , k ) + k 2 ⁢ sin ⁡ θ ⁢ sin ⁡ ϕ ⁢ sin ⁡ ψ elliptic-integral-second-kind-E 𝜃 𝑘 elliptic-integral-second-kind-E italic-ϕ 𝑘 elliptic-integral-second-kind-E 𝜓 𝑘 superscript 𝑘 2 𝜃 italic-ϕ 𝜓 {\displaystyle{\displaystyle E\left(\theta,k\right)+E\left(\phi,k\right)=E% \left(\psi,k\right)+k^{2}\sin\theta\sin\phi\sin\psi}}
\incellintEk@{\theta}{k}+\incellintEk@{\phi}{k} = \incellintEk@{\psi}{k}+k^{2}\sin@@{\theta}\sin@@{\phi}\sin@@{\psi}		 

EllipticE(sin(theta), k)+ EllipticE(sin(phi), k) = EllipticE(sin(psi), k)+ (k)^(2)* sin(theta)*sin(phi)*sin(psi)

EllipticE[\[Theta], (k)^2]+ EllipticE[\[Phi], (k)^2] == EllipticE[\[Psi], (k)^2]+ (k)^(2)* Sin[\[Theta]]*Sin[\[Phi]]*Sin[\[Psi]]
Failure Failure
Failed [300 / 300]
Result: .5188815884-.3712110352*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -.324003006-2.889566484*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.41998937174924766, 0.11250711558240023]
Test Values: {Rule[k, 1], Rule[θ, 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]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[0.3908843789278109, -0.3018102404271388]
Test Values: {Rule[k, 2], Rule[θ, 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]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.11#Ex3 cos ⁡ ψ = cos ⁡ θ ⁢ cos ⁡ ϕ - ( sin ⁡ θ ⁢ sin ⁡ ϕ ) ⁢ Δ ⁢ ( θ ) ⁢ Δ ⁢ ( ϕ ) 1 - k 2 ⁢ sin 2 ⁡ θ ⁢ sin 2 ⁡ ϕ 𝜓 𝜃 italic-ϕ 𝜃 italic-ϕ Δ 𝜃 Δ italic-ϕ 1 superscript 𝑘 2 2 𝜃 2 italic-ϕ {\displaystyle{\displaystyle\cos\psi=\frac{\cos\theta\cos\phi-(\sin\theta\sin% \phi)\Delta(\theta)\Delta(\phi)}{1-k^{2}{\sin^{2}}\theta{\sin^{2}}\phi}}}
\cos@@{\psi} = \frac{\cos@@{\theta}\cos@@{\phi}-(\sin@@{\theta}\sin@@{\phi})\Delta(\theta)\Delta(\phi)}{1-k^{2}\sin^{2}@@{\theta}\sin^{2}@@{\phi}}		 

cos(psi) = (cos(theta)*cos(phi)-(sin(theta)*sin(phi))*(sqrt(1 - (k)^(2)* (sin(theta))^(2)))*Delta(phi))/(1 - (k)^(2)* (sin(theta))^(2)* (sin(phi))^(2))

Cos[\[Psi]] == Divide[Cos[\[Theta]]*Cos[\[Phi]]-(Sin[\[Theta]]*Sin[\[Phi]])*(Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)])*\[CapitalDelta][\[Phi]],1 - (k)^(2)* (Sin[\[Theta]])^(2)* (Sin[\[Phi]])^(2)]
Failure Failure
Failed [300 / 300]
Result: -.360132946e-1+.3498736067*I
Test Values: {Delta = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 1}


Result: .3023079579-.441042741e-1*I
Test Values: {Delta = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.06008432780660544, 0.09466439987688165]
Test Values: {Rule[k, 1], Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, 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]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[0.1274461431695849, -0.029704144406044533]
Test Values: {Rule[k, 2], Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, 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]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.11#Ex4 tan ⁡ ( 1 2 ⁢ ψ ) = ( sin ⁡ θ ) ⁢ Δ ⁢ ( ϕ ) + ( sin ⁡ ϕ ) ⁢ Δ ⁢ ( θ ) cos ⁡ θ + cos ⁡ ϕ 1 2 𝜓 𝜃 Δ italic-ϕ italic-ϕ Δ 𝜃 𝜃 italic-ϕ {\displaystyle{\displaystyle\tan\left(\tfrac{1}{2}\psi\right)=\frac{(\sin% \theta)\Delta(\phi)+(\sin\phi)\Delta(\theta)}{\cos\theta+\cos\phi}}}
\tan@{\tfrac{1}{2}\psi} = \frac{(\sin@@{\theta})\Delta(\phi)+(\sin@@{\phi})\Delta(\theta)}{\cos@@{\theta}+\cos@@{\phi}}		 

tan((1)/(2)*psi) = ((sin(theta))*Delta(phi)+(sin(phi))*(sqrt(1 - (k)^(2)* (sin(theta))^(2))))/(cos(theta)+ cos(phi))

Tan[Divide[1,2]*\[Psi]] == Divide[(Sin[\[Theta]])*\[CapitalDelta][\[Phi]]+(Sin[\[Phi]])*(Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)]),Cos[\[Theta]]+ Cos[\[Phi]]]
Translation Error Translation Error - -
19.11.E5 Π ⁡ ( θ , α 2 , k ) + Π ⁡ ( ϕ , α 2 , k ) = Π ⁡ ( ψ , α 2 , k ) - α 2 ⁢ R C ⁡ ( γ - δ , γ ) elliptic-integral-third-kind-Pi 𝜃 superscript 𝛼 2 𝑘 elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 𝑘 elliptic-integral-third-kind-Pi 𝜓 superscript 𝛼 2 𝑘 superscript 𝛼 2 Carlson-integral-RC 𝛾 𝛿 𝛾 {\displaystyle{\displaystyle\Pi\left(\theta,\alpha^{2},k\right)+\Pi\left(\phi,% \alpha^{2},k\right)=\Pi\left(\psi,\alpha^{2},k\right)-\alpha^{2}R_{C}\left(% \gamma-\delta,\gamma\right)}}
\incellintPik@{\theta}{\alpha^{2}}{k}+\incellintPik@{\phi}{\alpha^{2}}{k} = \incellintPik@{\psi}{\alpha^{2}}{k}-\alpha^{2}\CarlsonellintRC@{\gamma-\delta}{\gamma}		 

Error

EllipticPi[\[Alpha]^(2), \[Theta],(k)^2]+ EllipticPi[\[Alpha]^(2), \[Phi],(k)^2] == EllipticPi[\[Alpha]^(2), \[Psi],(k)^2]- \[Alpha]^(2)* 1/Sqrt[\[Gamma]]*Hypergeometric2F1[1/2,1/2,3/2,1-(\[Gamma]-(\[Alpha]^(2)*(1 - \[Alpha]^(2))*(\[Alpha]^(2)- (k)^(2))))/(\[Gamma])]
Missing Macro Error Failure -
Failed [300 / 300]
Result: Complex[2.431737700775111, 0.07689658395417326]
Test Values: {Rule[k, 1], Rule[α, 1.5], Rule[γ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, 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]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[1.648685299290325, -1.4197583822626343]
Test Values: {Rule[k, 2], Rule[α, 1.5], Rule[γ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, 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]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.11.E6_5 R C ⁡ ( γ - δ , γ ) = - 1 δ ⁢ arctan ⁡ ( δ ⁢ sin ⁡ θ ⁢ sin ⁡ ϕ ⁢ sin ⁡ ψ α 2 - 1 - α 2 ⁢ cos ⁡ θ ⁢ cos ⁡ ϕ ⁢ cos ⁡ ψ ) Carlson-integral-RC 𝛾 𝛿 𝛾 1 𝛿 𝛿 𝜃 italic-ϕ 𝜓 superscript 𝛼 2 1 superscript 𝛼 2 𝜃 italic-ϕ 𝜓 {\displaystyle{\displaystyle R_{C}\left(\gamma-\delta,\gamma\right)=\frac{-1}{% \sqrt{\delta}}\operatorname{arctan}\left(\frac{\sqrt{\delta}\sin\theta\sin\phi% \sin\psi}{\alpha^{2}-1-\alpha^{2}\cos\theta\cos\phi\cos\psi}\right)}}
\CarlsonellintRC@{\gamma-\delta}{\gamma} = \frac{-1}{\sqrt{\delta}}\atan@{\frac{\sqrt{\delta}\sin@@{\theta}\sin@@{\phi}\sin@@{\psi}}{\alpha^{2}-1-\alpha^{2}\cos@@{\theta}\cos@@{\phi}\cos@@{\psi}}}		 

Error

1/Sqrt[\[Gamma]]*Hypergeometric2F1[1/2,1/2,3/2,1-(\[Gamma]-(\[Alpha]^(2)*(1 - \[Alpha]^(2))*(\[Alpha]^(2)- (k)^(2))))/(\[Gamma])] == Divide[- 1,Sqrt[\[Alpha]^(2)*(1 - \[Alpha]^(2))*(\[Alpha]^(2)- (k)^(2))]]*ArcTan[Divide[Sqrt[\[Alpha]^(2)*(1 - \[Alpha]^(2))*(\[Alpha]^(2)- (k)^(2))]*Sin[\[Theta]]*Sin[\[Phi]]*Sin[\[Psi]],\[Alpha]^(2)- 1 - \[Alpha]^(2)* Cos[\[Theta]]*Cos[\[Phi]]*Cos[\[Psi]]]]
Missing Macro Error Translation Error - -
19.11.E7 F ⁡ ( ϕ , k ) = K ⁡ ( k ) - F ⁡ ( θ , k ) elliptic-integral-first-kind-F italic-ϕ 𝑘 complete-elliptic-integral-first-kind-K 𝑘 elliptic-integral-first-kind-F 𝜃 𝑘 {\displaystyle{\displaystyle F\left(\phi,k\right)=K\left(k\right)-F\left(% \theta,k\right)}}
\incellintFk@{\phi}{k} = \compellintKk@{k}-\incellintFk@{\theta}{k}		 

EllipticF(sin(phi), k) = EllipticK(k)- EllipticF(sin(theta), k)

EllipticF[\[Phi], (k)^2] == EllipticK[(k)^2]- EllipticF[\[Theta], (k)^2]
Failure Failure Error
Failed [300 / 300]
Result: DirectedInfinity[]
Test Values: {Rule[k, 1], Rule[θ, 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[-0.04973776616306258, 1.7417596493254397]
Test Values: {Rule[k, 2], Rule[θ, 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
19.11.E8 E ⁡ ( ϕ , k ) = E ⁡ ( k ) - E ⁡ ( θ , k ) + k 2 ⁢ sin ⁡ θ ⁢ sin ⁡ ϕ elliptic-integral-second-kind-E italic-ϕ 𝑘 complete-elliptic-integral-second-kind-E 𝑘 elliptic-integral-second-kind-E 𝜃 𝑘 superscript 𝑘 2 𝜃 italic-ϕ {\displaystyle{\displaystyle E\left(\phi,k\right)=E\left(k\right)-E\left(% \theta,k\right)+k^{2}\sin\theta\sin\phi}}
\incellintEk@{\phi}{k} = \compellintEk@{k}-\incellintEk@{\theta}{k}+k^{2}\sin@@{\theta}\sin@@{\phi}		 

EllipticE(sin(phi), k) = EllipticE(k)- EllipticE(sin(theta), k)+ (k)^(2)* sin(theta)*sin(phi)

EllipticE[\[Phi], (k)^2] == EllipticE[(k)^2]- EllipticE[\[Theta], (k)^2]+ (k)^(2)* Sin[\[Theta]]*Sin[\[Phi]]
Failure Failure
Failed [295 / 300]
Result: .940848258e-1+.952154806e-1*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -.829018303-3.772436995*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [297 / 300]
Result: Complex[-0.2691514567553243, 0.26012051423236426]
Test Values: {Rule[k, 1], Rule[θ, 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[-0.06105092961961717, -1.8070495799711206]
Test Values: {Rule[k, 2], Rule[θ, 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
19.11.E9 tan ⁡ θ = 1 / ( k ′ ⁢ tan ⁡ ϕ ) 𝜃 1 superscript 𝑘 ′ italic-ϕ {\displaystyle{\displaystyle\tan\theta=1/(k^{\prime}\tan\phi)}}
\tan@@{\theta} = 1/(k^{\prime}\tan@@{\phi})		 

tan(theta) = 1/(sqrt(1 - (k)^(2))*tan(phi))

Tan[\[Theta]] == 1/(Sqrt[1 - (k)^(2)]*Tan[\[Phi]])
Failure Failure
Failed [300 / 300]
Result: Float(infinity)+Float(infinity)*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 1}


Result: 1.112198033+1.184536461*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: DirectedInfinity[]
Test Values: {Rule[k, 1], Rule[θ, 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[1.0561283793604441, 1.210195136063891]
Test Values: {Rule[k, 2], Rule[θ, 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
19.11.E10 Π ⁡ ( ϕ , α 2 , k ) = Π ⁡ ( α 2 , k ) - Π ⁡ ( θ , α 2 , k ) - α 2 ⁢ R C ⁡ ( γ - δ , γ ) elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 𝑘 complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 𝑘 elliptic-integral-third-kind-Pi 𝜃 superscript 𝛼 2 𝑘 superscript 𝛼 2 Carlson-integral-RC 𝛾 𝛿 𝛾 {\displaystyle{\displaystyle\Pi\left(\phi,\alpha^{2},k\right)=\Pi\left(\alpha^% {2},k\right)-\Pi\left(\theta,\alpha^{2},k\right)-\alpha^{2}R_{C}\left(\gamma-% \delta,\gamma\right)}}
\incellintPik@{\phi}{\alpha^{2}}{k} = \compellintPik@{\alpha^{2}}{k}-\incellintPik@{\theta}{\alpha^{2}}{k}-\alpha^{2}\CarlsonellintRC@{\gamma-\delta}{\gamma}		 

Error

EllipticPi[\[Alpha]^(2), \[Phi],(k)^2] == EllipticPi[\[Alpha]^(2), (k)^2]- EllipticPi[\[Alpha]^(2), \[Theta],(k)^2]- \[Alpha]^(2)* 1/Sqrt[\[Gamma]]*Hypergeometric2F1[1/2,1/2,3/2,1-(\[Gamma]-(\[Alpha]^(2)*(1 - \[Alpha]^(2))*(\[Alpha]^(2)- (k)^(2))))/(\[Gamma])]
Missing Macro Error Failure -
Failed [300 / 300]
Result: DirectedInfinity[]
Test Values: {Rule[k, 1], Rule[α, 1.5], Rule[γ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, 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[2.2835000786563655, -0.476202278380103]
Test Values: {Rule[k, 2], Rule[α, 1.5], Rule[γ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, 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
19.11.E12 F ⁡ ( ψ , k ) = 2 ⁢ F ⁡ ( θ , k ) elliptic-integral-first-kind-F 𝜓 𝑘 2 elliptic-integral-first-kind-F 𝜃 𝑘 {\displaystyle{\displaystyle F\left(\psi,k\right)=2F\left(\theta,k\right)}}
\incellintFk@{\psi}{k} = 2\incellintFk@{\theta}{k}		 

EllipticF(sin(psi), k) = 2*EllipticF(sin(theta), k)

EllipticF[\[Psi], (k)^2] == 2*EllipticF[\[Theta], (k)^2]
Failure Failure
Failed [300 / 300]
Result: -.8208700290-.6773780507*I
Test Values: {psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -.4831883421-.7182528229*I
Test Values: {psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.43180375739814203, -0.27142936483528934]
Test Values: {Rule[k, 1], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-0.3965687056216178, -0.33175091278780894]
Test Values: {Rule[k, 2], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.11.E13 E ⁡ ( ψ , k ) = 2 ⁢ E ⁡ ( θ , k ) - k 2 ⁢ sin 2 ⁡ θ ⁢ sin ⁡ ψ elliptic-integral-second-kind-E 𝜓 𝑘 2 elliptic-integral-second-kind-E 𝜃 𝑘 superscript 𝑘 2 2 𝜃 𝜓 {\displaystyle{\displaystyle E\left(\psi,k\right)=2E\left(\theta,k\right)-k^{2% }{\sin^{2}}\theta\sin\psi}}
\incellintEk@{\psi}{k} = 2\incellintEk@{\theta}{k}-k^{2}\sin^{2}@@{\theta}\sin@@{\psi}		 

EllipticE(sin(psi), k) = 2*EllipticE(sin(theta), k)- (k)^(2)* (sin(theta))^(2)* sin(psi)

EllipticE[\[Psi], (k)^2] == 2*EllipticE[\[Theta], (k)^2]- (k)^(2)* (Sin[\[Theta]])^(2)* Sin[\[Psi]]
Failure Failure
Failed [300 / 300]
Result: -.5188815884+.3712110352*I
Test Values: {psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 1}


Result: .324003006+2.889566484*I
Test Values: {psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [298 / 300]
Result: Complex[-0.41998937174924766, -0.11250711558240023]
Test Values: {Rule[k, 1], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-0.3908843789278109, 0.3018102404271388]
Test Values: {Rule[k, 2], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.11#Ex9 cos ⁡ ψ = ( cos ⁡ ( 2 ⁢ θ ) + k 2 ⁢ sin 4 ⁡ θ ) / ( 1 - k 2 ⁢ sin 4 ⁡ θ ) 𝜓 2 𝜃 superscript 𝑘 2 4 𝜃 1 superscript 𝑘 2 4 𝜃 {\displaystyle{\displaystyle\cos\psi=(\cos\left(2\theta\right)+k^{2}{\sin^{4}}% \theta)/(1-k^{2}{\sin^{4}}\theta)}}
\cos@@{\psi} = (\cos@{2\theta}+k^{2}\sin^{4}@@{\theta})/(1-k^{2}\sin^{4}@@{\theta})		 

cos(psi) = (cos(2*theta)+ (k)^(2)* (sin(theta))^(4))/(1 - (k)^(2)* (sin(theta))^(4))

Cos[\[Psi]] == (Cos[2*\[Theta]]+ (k)^(2)* (Sin[\[Theta]])^(4))/(1 - (k)^(2)* (Sin[\[Theta]])^(4))
Failure Failure
Failed [300 / 300]
Result: .6382547213-.68319321e-2*I
Test Values: {psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 1}


Result: 1.291602175-.5372399851*I
Test Values: {psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.22600457397095797, 0.19313483829287414]
Test Values: {Rule[k, 1], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[0.33144266284556045, -0.05654646036238595]
Test Values: {Rule[k, 2], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.11#Ex10 tan ⁡ ( 1 2 ⁢ ψ ) = ( tan ⁡ θ ) ⁢ Δ ⁢ ( θ ) 1 2 𝜓 𝜃 Δ 𝜃 {\displaystyle{\displaystyle\tan\left(\tfrac{1}{2}\psi\right)=(\tan\theta)% \Delta(\theta)}}
\tan@{\tfrac{1}{2}\psi} = (\tan@@{\theta})\Delta(\theta)		 

tan((1)/(2)*psi) = (tan(theta))*(sqrt(1 - (k)^(2)* (sin(theta))^(2)))

Tan[Divide[1,2]*\[Psi]] == (Tan[\[Theta]])*(Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)])
Failure Failure
Failed [300 / 300]
Result: -.4299370879-.441018886e-1*I
Test Values: {psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -1.378631246+.6589669897*I
Test Values: {psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.21639778041374116, -0.09902593860776912]
Test Values: {Rule[k, 1], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-0.2801868441200064, 0.09163936360272593]
Test Values: {Rule[k, 2], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.11#Ex11 sin ⁡ θ = ( sin ⁡ ψ ) / ( 1 + cos ⁡ ψ ) ⁢ ( 1 + Δ ⁢ ( ψ ) ) 𝜃 𝜓 1 𝜓 1 Δ 𝜓 {\displaystyle{\displaystyle\sin\theta=(\sin\psi)/\sqrt{(1+\cos\psi)(1+\Delta(% \psi))}}}
\sin@@{\theta} = (\sin@@{\psi})/\sqrt{(1+\cos@@{\psi})(1+\Delta(\psi))}		 

sin(theta) = (sin(psi))/(sqrt((1 + cos(psi))*(1 + Delta(psi))))

Sin[\[Theta]] == (Sin[\[Psi]])/(Sqrt[(1 + Cos[\[Psi]])*(1 + \[CapitalDelta][\[Psi]])])
Failure Failure
Failed [300 / 300]
Result: .3459933254+.2199626413*I
Test Values: {Delta = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I}


Result: -1.183718368+.7410028953*I
Test Values: {Delta = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, theta = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.13267626462165183, 0.09545710280323466]
Test Values: {Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[0.6075958421397494, -0.12937331954381406]
Test Values: {Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.11#Ex12 cos ⁡ θ = ( cos ⁡ ψ ) + Δ ⁢ ( ψ ) 1 + Δ ⁢ ( ψ ) 𝜃 𝜓 Δ 𝜓 1 Δ 𝜓 {\displaystyle{\displaystyle\cos\theta=\sqrt{\frac{(\cos\psi)+\Delta(\psi)}{1+% \Delta(\psi)}}}}
\cos@@{\theta} = \sqrt{\frac{(\cos@@{\psi})+\Delta(\psi)}{1+\Delta(\psi)}}		 

cos(theta) = sqrt(((cos(psi))+ Delta(psi))/(1 + Delta(psi)))

Cos[\[Theta]] == Sqrt[Divide[(Cos[\[Psi]])+ \[CapitalDelta][\[Psi]],1 + \[CapitalDelta][\[Psi]]]]
Failure Failure
Failed [300 / 300]
Result: -.1386531520-.3275237699*I
Test Values: {Delta = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I}


Result: .3585693461+.5385011568*I
Test Values: {Delta = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, theta = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.027928525698177165, -0.06433717895055871]
Test Values: {Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-0.11337825659380207, -0.16573354274294425]
Test Values: {Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.11#Ex13 tan ⁡ θ = tan ⁡ ( 1 2 ⁢ ψ ) ⁢ 1 + cos ⁡ ψ ( cos ⁡ ψ ) + Δ ⁢ ( ψ ) 𝜃 1 2 𝜓 1 𝜓 𝜓 Δ 𝜓 {\displaystyle{\displaystyle\tan\theta=\tan\left(\tfrac{1}{2}\psi\right)\sqrt{% \frac{1+\cos\psi}{(\cos\psi)+\Delta(\psi)}}}}
\tan@@{\theta} = \tan@{\tfrac{1}{2}\psi}\sqrt{\frac{1+\cos@@{\psi}}{(\cos@@{\psi})+\Delta(\psi)}}		 

tan(theta) = tan((1)/(2)*psi)*sqrt((1 + cos(psi))/((cos(psi))+ Delta(psi)))

Tan[\[Theta]] == Tan[Divide[1,2]*\[Psi]]*Sqrt[Divide[1 + Cos[\[Psi]],(Cos[\[Psi]])+ \[CapitalDelta][\[Psi]]]]
 Failure Failure
Failed [300 / 300]
Result: .1382279959+.6687205345*I
Test Values: {Delta = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I}

Result: -.8192630216+.6110829935*I
Test Values: {Delta = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, theta = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.12433851209893465, 0.1415108829927562]
Test Values: {Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

Result: Complex[0.5756669065605976, -0.05657247148971478]
Test Values: {Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.11.E16 Π ⁡ ( ψ , α 2 , k ) = 2 ⁢ Π ⁡ ( θ , α 2 , k ) + α 2 ⁢ R C ⁡ ( γ - δ , γ ) elliptic-integral-third-kind-Pi 𝜓 superscript 𝛼 2 𝑘 2 elliptic-integral-third-kind-Pi 𝜃 superscript 𝛼 2 𝑘 superscript 𝛼 2 Carlson-integral-RC 𝛾 𝛿 𝛾 {\displaystyle{\displaystyle\Pi\left(\psi,\alpha^{2},k\right)=2\Pi\left(\theta% ,\alpha^{2},k\right)+\alpha^{2}R_{C}\left(\gamma-\delta,\gamma\right)}}
\incellintPik@{\psi}{\alpha^{2}}{k} = 2\incellintPik@{\theta}{\alpha^{2}}{k}+\alpha^{2}\CarlsonellintRC@{\gamma-\delta}{\gamma}		 

Error
 
EllipticPi[\[Alpha]^(2), \[Psi],(k)^2] == 2*EllipticPi[\[Alpha]^(2), \[Theta],(k)^2]+ \[Alpha]^(2)* 1/Sqrt[(((Csc[\[Theta]])^(2))- \[Alpha]^(2))^(2)*(((Csc[\[Psi]])^(2))- \[Alpha]^(2))]*Hypergeometric2F1[1/2,1/2,3/2,1-(((((Csc[\[Theta]])^(2))- \[Alpha]^(2))^(2)*(((Csc[\[Psi]])^(2))- \[Alpha]^(2)))- \[Delta])/((((Csc[\[Theta]])^(2))- \[Alpha]^(2))^(2)*(((Csc[\[Psi]])^(2))- \[Alpha]^(2)))]
 Missing Macro Error Aborted -
Failed [300 / 300]
Result: Complex[-0.6318505653554005, -0.11296244472006367]
Test Values: {Rule[k, 1], Rule[α, 1.5], Rule[δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

Result: Complex[-0.5728350059366992, -0.1614996009729338]
Test Values: {Rule[k, 2], Rule[α, 1.5], Rule[δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
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