19.11

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
19.11.E1	${\displaystyle{\displaystyle F\left(\theta,k\right)+F\left(\phi,k\right)=F\left(\psi,k\right)}}$ \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	${\displaystyle{\displaystyle E\left(\theta,k\right)+E\left(\phi,k\right)=E\left(\psi,k\right)+k^{2}\sin\theta\sin\phi\sin\psi}}$ \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	${\displaystyle{\displaystyle\cos\psi=\frac{\cos\theta\cos\phi-(\sin\theta\sin\phi)\Delta(\theta)\Delta(\phi)}{1-k^{2}{\sin^{2}}\theta{\sin^{2}}\phi}}}$ \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	${\displaystyle{\displaystyle\tan\left(\tfrac{1}{2}\psi\right)=\frac{(\sin\theta)\Delta(\phi)+(\sin\phi)\Delta(\theta)}{\cos\theta+\cos\phi}}}$ \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	${\displaystyle{\displaystyle\Pi\left(\theta,\alpha^{2},k\right)+\Pi\left(\phi,\alpha^{2},k\right)=\Pi\left(\psi,\alpha^{2},k\right)-\alpha^{2}R_{C}\left(\gamma-\delta,\gamma\right)}}$ \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	${\displaystyle{\displaystyle R_{C}\left(\gamma-\delta,\gamma\right)=\frac{-1}{\sqrt{\delta}}\operatorname{arctan}\left(\frac{\sqrt{\delta}\sin\theta\sin\phi\sin\psi}{\alpha^{2}-1-\alpha^{2}\cos\theta\cos\phi\cos\psi}\right)}}$ \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	${\displaystyle{\displaystyle F\left(\phi,k\right)=K\left(k\right)-F\left(\theta,k\right)}}$ \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	${\displaystyle{\displaystyle E\left(\phi,k\right)=E\left(k\right)-E\left(\theta,k\right)+k^{2}\sin\theta\sin\phi}}$ \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	${\displaystyle{\displaystyle\tan\theta=1/(k^{\prime}\tan\phi)}}$ \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	${\displaystyle{\displaystyle\Pi\left(\phi,\alpha^{2},k\right)=\Pi\left(\alpha^{2},k\right)-\Pi\left(\theta,\alpha^{2},k\right)-\alpha^{2}R_{C}\left(\gamma-\delta,\gamma\right)}}$ \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	${\displaystyle{\displaystyle F\left(\psi,k\right)=2F\left(\theta,k\right)}}$ \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	${\displaystyle{\displaystyle E\left(\psi,k\right)=2E\left(\theta,k\right)-k^{2}{\sin^{2}}\theta\sin\psi}}$ \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	${\displaystyle{\displaystyle\cos\psi=(\cos\left(2\theta\right)+k^{2}{\sin^{4}}\theta)/(1-k^{2}{\sin^{4}}\theta)}}$ \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	${\displaystyle{\displaystyle\tan\left(\tfrac{1}{2}\psi\right)=(\tan\theta)\Delta(\theta)}}$ \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	${\displaystyle{\displaystyle\sin\theta=(\sin\psi)/\sqrt{(1+\cos\psi)(1+\Delta(\psi))}}}$ \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	${\displaystyle{\displaystyle\cos\theta=\sqrt{\frac{(\cos\psi)+\Delta(\psi)}{1+\Delta(\psi)}}}}$ \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	${\displaystyle{\displaystyle\tan\theta=\tan\left(\tfrac{1}{2}\psi\right)\sqrt{\frac{1+\cos\psi}{(\cos\psi)+\Delta(\psi)}}}}$ \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	${\displaystyle{\displaystyle\Pi\left(\psi,\alpha^{2},k\right)=2\Pi\left(\theta,\alpha^{2},k\right)+\alpha^{2}R_{C}\left(\gamma-\delta,\gamma\right)}}$ \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