Elliptic Integrals - 19.26 Addition Theorems

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
19.26.E1 R F ⁡ ( x + λ , y + λ , z + λ ) + R F ⁡ ( x + μ , y + μ , z + μ ) = R F ⁡ ( x , y , z ) Carlson-integral-RF 𝑥 𝜆 𝑦 𝜆 𝑧 𝜆 Carlson-integral-RF 𝑥 𝜇 𝑦 𝜇 𝑧 𝜇 Carlson-integral-RF 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle R_{F}\left(x+\lambda,y+\lambda,z+\lambda\right)+R% _{F}\left(x+\mu,y+\mu,z+\mu\right)=R_{F}\left(x,y,z\right)}}
\CarlsonsymellintRF@{x+\lambda}{y+\lambda}{z+\lambda}+\CarlsonsymellintRF@{x+\mu}{y+\mu}{z+\mu} = \CarlsonsymellintRF@{x}{y}{z}		 

0.5*int(1/(sqrt(t+x + lambda)*sqrt(t+y + lambda)*sqrt(t+(x + y*I)+ lambda)), t = 0..infinity)+ 0.5*int(1/(sqrt(t+x + mu)*sqrt(t+y + mu)*sqrt(t+(x + y*I)+ mu)), t = 0..infinity) = 0.5*int(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+x + y*I)), t = 0..infinity)

EllipticF[ArcCos[Sqrt[x + \[Lambda]/(x + y*I)+ \[Lambda]]],((x + y*I)+ \[Lambda]-y + \[Lambda])/((x + y*I)+ \[Lambda]-x + \[Lambda])]/Sqrt[(x + y*I)+ \[Lambda]-x + \[Lambda]]+ EllipticF[ArcCos[Sqrt[x + \[Mu]/(x + y*I)+ \[Mu]]],((x + y*I)+ \[Mu]-y + \[Mu])/((x + y*I)+ \[Mu]-x + \[Mu])]/Sqrt[(x + y*I)+ \[Mu]-x + \[Mu]] == EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]
Aborted Failure Skipped - Because timed out
Failed [300 / 300]
Result: Complex[0.6992255245511445, -1.8246422705609677]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[μ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[1.2162365888422955, -0.7585970772170993]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[μ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.26.E2 x + μ = λ - 2 ⁢ ( ( x + λ ) ⁢ y ⁢ z + x ⁢ ( y + λ ) ⁢ ( z + λ ) ) 2 𝑥 𝜇 superscript 𝜆 2 superscript 𝑥 𝜆 𝑦 𝑧 𝑥 𝑦 𝜆 𝑧 𝜆 2 {\displaystyle{\displaystyle x+\mu=\lambda^{-2}\left(\sqrt{(x+\lambda)yz}+% \sqrt{x(y+\lambda)(z+\lambda)}\right)^{2}}}
x+\mu = \lambda^{-2}\left(\sqrt{(x+\lambda)yz}+\sqrt{x(y+\lambda)(z+\lambda)}\right)^{2}		 

x + mu = (lambda)^(- 2)*(sqrt((x + lambda)*y*(x + y*I))+sqrt(x*(y + lambda)*((x + y*I)+ lambda)))^(2)
 
x + \[Mu] == \[Lambda]^(- 2)*(Sqrt[(x + \[Lambda])*y*(x + y*I)]+Sqrt[x*(y + \[Lambda])*((x + y*I)+ \[Lambda])])^(2)
 Skipped - no semantic math Skipped - no semantic math - -
19.26#Ex1 ( ξ , η , ζ ) = ( x + λ , y + λ , z + λ ) 𝜉 𝜂 𝜁 𝑥 𝜆 𝑦 𝜆 𝑧 𝜆 {\displaystyle{\displaystyle(\xi,\eta,\zeta)=(x+\lambda,y+\lambda,z+\lambda)}}
(\xi,\eta,\zeta) = (x+\lambda,y+\lambda,z+\lambda)		 

(xi , eta , zeta) = (x + lambda , y + lambda ,(x + y*I)+ lambda)
 
(\[Xi], \[Eta], \[Zeta]) == (x + \[Lambda], y + \[Lambda],(x + y*I)+ \[Lambda])
 Skipped - no semantic math Skipped - no semantic math - -
19.26.E5 μ = λ - 2 ⁢ ( x ⁢ y ⁢ z + ( x + λ ) ⁢ ( y + λ ) ⁢ ( z + λ ) ) 2 - λ - x - y - z 𝜇 superscript 𝜆 2 superscript 𝑥 𝑦 𝑧 𝑥 𝜆 𝑦 𝜆 𝑧 𝜆 2 𝜆 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle\mu=\lambda^{-2}\left(\sqrt{xyz}+\sqrt{(x+\lambda)% (y+\lambda)(z+\lambda)}\right)^{2}-\lambda-x-y-z}}
\mu = \lambda^{-2}\left(\sqrt{xyz}+\sqrt{(x+\lambda)(y+\lambda)(z+\lambda)}\right)^{2}-\lambda-x-y-z		 

mu = (lambda)^(- 2)*(sqrt(x*y*(x + y*I))+sqrt((x + lambda)*(y + lambda)*((x + y*I)+ lambda)))^(2)- lambda - x - y -(x + y*I)
 
\[Mu] == \[Lambda]^(- 2)*(Sqrt[x*y*(x + y*I)]+Sqrt[(x + \[Lambda])*(y + \[Lambda])*((x + y*I)+ \[Lambda])])^(2)- \[Lambda]- x - y -(x + y*I)
 Skipped - no semantic math Skipped - no semantic math - -
19.26.E6 ( λ ⁢ μ - x ⁢ y - x ⁢ z - y ⁢ z ) 2 = 4 ⁢ x ⁢ y ⁢ z ⁢ ( λ + μ + x + y + z ) superscript 𝜆 𝜇 𝑥 𝑦 𝑥 𝑧 𝑦 𝑧 2 4 𝑥 𝑦 𝑧 𝜆 𝜇 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle(\lambda\mu-xy-xz-yz)^{2}=4xyz(\lambda+\mu+x+y+z)}}
(\lambda\mu-xy-xz-yz)^{2} = 4xyz(\lambda+\mu+x+y+z)		 

(lambda*mu - x*y - x*(x + y*I)- y*(x + y*I))^(2) = 4*x*y*(x + y*I)*(lambda + mu + x + y +(x + y*I))
 
(\[Lambda]*\[Mu]- x*y - x*(x + y*I)- y*(x + y*I))^(2) == 4*x*y*(x + y*I)*(\[Lambda]+ \[Mu]+ x + y +(x + y*I))
 Skipped - no semantic math Skipped - no semantic math - -
19.26.E7 R D ⁡ ( x + λ , y + λ , z + λ ) + R D ⁡ ( x + μ , y + μ , z + μ ) = R D ⁡ ( x , y , z ) - 3 z ⁢ ( z + λ ) ⁢ ( z + μ ) Carlson-integral-RD 𝑥 𝜆 𝑦 𝜆 𝑧 𝜆 Carlson-integral-RD 𝑥 𝜇 𝑦 𝜇 𝑧 𝜇 Carlson-integral-RD 𝑥 𝑦 𝑧 3 𝑧 𝑧 𝜆 𝑧 𝜇 {\displaystyle{\displaystyle R_{D}\left(x+\lambda,y+\lambda,z+\lambda\right)+R% _{D}\left(x+\mu,y+\mu,z+\mu\right)=R_{D}\left(x,y,z\right)-\frac{3}{\sqrt{z(z+% \lambda)(z+\mu)}}}}
\CarlsonsymellintRD@{x+\lambda}{y+\lambda}{z+\lambda}+\CarlsonsymellintRD@{x+\mu}{y+\mu}{z+\mu} = \CarlsonsymellintRD@{x}{y}{z}-\frac{3}{\sqrt{z(z+\lambda)(z+\mu)}}		 

Error

3*(EllipticF[ArcCos[Sqrt[x + \[Lambda]/(x + y*I)+ \[Lambda]]],((x + y*I)+ \[Lambda]-y + \[Lambda])/((x + y*I)+ \[Lambda]-x + \[Lambda])]-EllipticE[ArcCos[Sqrt[x + \[Lambda]/(x + y*I)+ \[Lambda]]],((x + y*I)+ \[Lambda]-y + \[Lambda])/((x + y*I)+ \[Lambda]-x + \[Lambda])])/(((x + y*I)+ \[Lambda]-y + \[Lambda])*((x + y*I)+ \[Lambda]-x + \[Lambda])^(1/2))+ 3*(EllipticF[ArcCos[Sqrt[x + \[Mu]/(x + y*I)+ \[Mu]]],((x + y*I)+ \[Mu]-y + \[Mu])/((x + y*I)+ \[Mu]-x + \[Mu])]-EllipticE[ArcCos[Sqrt[x + \[Mu]/(x + y*I)+ \[Mu]]],((x + y*I)+ \[Mu]-y + \[Mu])/((x + y*I)+ \[Mu]-x + \[Mu])])/(((x + y*I)+ \[Mu]-y + \[Mu])*((x + y*I)+ \[Mu]-x + \[Mu])^(1/2)) == 3*(EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticE[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/((x + y*I-y)*(x + y*I-x)^(1/2))-Divide[3,Sqrt[(x + y*I)*((x + y*I)+ \[Lambda])*((x + y*I)+ \[Mu])]]
Missing Macro Error Aborted -
Failed [300 / 300]
Result: Complex[-0.4984590390126629, 1.2092907867192135]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[μ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[0.01924185171185039, 1.9974068077017313]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[μ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.26.E8 2 ⁢ R G ⁡ ( x + λ , y + λ , z + λ ) + 2 ⁢ R G ⁡ ( x + μ , y + μ , z + μ ) = 2 ⁢ R G ⁡ ( x , y , z ) + λ ⁢ R F ⁡ ( x + λ , y + λ , z + λ ) + μ ⁢ R F ⁡ ( x + μ , y + μ , z + μ ) + λ + μ + x + y + z 2 Carlson-integral-RG 𝑥 𝜆 𝑦 𝜆 𝑧 𝜆 2 Carlson-integral-RG 𝑥 𝜇 𝑦 𝜇 𝑧 𝜇 2 Carlson-integral-RG 𝑥 𝑦 𝑧 𝜆 Carlson-integral-RF 𝑥 𝜆 𝑦 𝜆 𝑧 𝜆 𝜇 Carlson-integral-RF 𝑥 𝜇 𝑦 𝜇 𝑧 𝜇 𝜆 𝜇 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle 2R_{G}\left(x+\lambda,y+\lambda,z+\lambda\right)+% 2R_{G}\left(x+\mu,y+\mu,z+\mu\right)=2R_{G}\left(x,y,z\right)+\lambda R_{F}% \left(x+\lambda,y+\lambda,z+\lambda\right)+\mu R_{F}\left(x+\mu,y+\mu,z+\mu% \right)+\sqrt{\lambda+\mu+x+y+z}}}
2\CarlsonsymellintRG@{x+\lambda}{y+\lambda}{z+\lambda}+2\CarlsonsymellintRG@{x+\mu}{y+\mu}{z+\mu} = 2\CarlsonsymellintRG@{x}{y}{z}+\lambda\CarlsonsymellintRF@{x+\lambda}{y+\lambda}{z+\lambda}+\mu\CarlsonsymellintRF@{x+\mu}{y+\mu}{z+\mu}+\sqrt{\lambda+\mu+x+y+z}		 

Error

2*Sqrt[(x + y*I)+ \[Lambda]-x + \[Lambda]]*(EllipticE[ArcCos[Sqrt[x + \[Lambda]/(x + y*I)+ \[Lambda]]],((x + y*I)+ \[Lambda]-y + \[Lambda])/((x + y*I)+ \[Lambda]-x + \[Lambda])]+(Cot[ArcCos[Sqrt[x + \[Lambda]/(x + y*I)+ \[Lambda]]]])^2*EllipticF[ArcCos[Sqrt[x + \[Lambda]/(x + y*I)+ \[Lambda]]],((x + y*I)+ \[Lambda]-y + \[Lambda])/((x + y*I)+ \[Lambda]-x + \[Lambda])]+Cot[ArcCos[Sqrt[x + \[Lambda]/(x + y*I)+ \[Lambda]]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[x + \[Lambda]/(x + y*I)+ \[Lambda]]]]^2])+ 2*Sqrt[(x + y*I)+ \[Mu]-x + \[Mu]]*(EllipticE[ArcCos[Sqrt[x + \[Mu]/(x + y*I)+ \[Mu]]],((x + y*I)+ \[Mu]-y + \[Mu])/((x + y*I)+ \[Mu]-x + \[Mu])]+(Cot[ArcCos[Sqrt[x + \[Mu]/(x + y*I)+ \[Mu]]]])^2*EllipticF[ArcCos[Sqrt[x + \[Mu]/(x + y*I)+ \[Mu]]],((x + y*I)+ \[Mu]-y + \[Mu])/((x + y*I)+ \[Mu]-x + \[Mu])]+Cot[ArcCos[Sqrt[x + \[Mu]/(x + y*I)+ \[Mu]]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[x + \[Mu]/(x + y*I)+ \[Mu]]]]^2]) == 2*Sqrt[x + y*I-x]*(EllipticE[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]+(Cot[ArcCos[Sqrt[x/x + y*I]]])^2*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]+Cot[ArcCos[Sqrt[x/x + y*I]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[x/x + y*I]]]^2])+ \[Lambda]*EllipticF[ArcCos[Sqrt[x + \[Lambda]/(x + y*I)+ \[Lambda]]],((x + y*I)+ \[Lambda]-y + \[Lambda])/((x + y*I)+ \[Lambda]-x + \[Lambda])]/Sqrt[(x + y*I)+ \[Lambda]-x + \[Lambda]]+ \[Mu]*EllipticF[ArcCos[Sqrt[x + \[Mu]/(x + y*I)+ \[Mu]]],((x + y*I)+ \[Mu]-y + \[Mu])/((x + y*I)+ \[Mu]-x + \[Mu])]/Sqrt[(x + y*I)+ \[Mu]-x + \[Mu]]+Sqrt[\[Lambda]+ \[Mu]+ x + y +(x + y*I)]
Missing Macro Error Aborted -
Failed [300 / 300]
Result: Plus[Complex[-2.0898920996046204, 0.6803615706262403], Times[Complex[-1.7320508075688772, 1.732050807568877], Plus[Complex[0.9985512968581824, 0.2012315241723115], Times[Complex[0.3176872874027722, -1.049249833251038], Power[Plus[1.0, Times[Complex[0.0, -1.5], Power[k, 2]]], Rational[1, 2]]]]], Times[Complex[4.184639587172815, -1.9117536488739475], Plus[Complex[0.7424137617640161, 0.220635885032481], Times[Complex[0.14483575015411373, 1.3558262394954135], Power[Plus[1.0, Times[Complex[0.9940169358562925, 0.4776709006307397], Power[k, 2]]], Rational[1, 2]]]]]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[μ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Plus[Complex[-1.182728387586514, 0.2705509888970101], Times[Complex[-1.7320508075688772, 1.732050807568877], Plus[Complex[0.9985512968581824, 0.2012315241723115], Times[Complex[0.3176872874027722, -1.049249833251038], Power[Plus[1.0, Times[Complex[0.0, -1.5], Power[k, 2]]], Rational[1, 2]]]]], Times[Complex[0.7841147574434748, -1.6170454393246465], Plus[Complex[0.3473840661116648, 1.4426085854555293], Times[Complex[0.7761183239980944, 1.3014092542459557], Power[Plus[1.0, Times[Complex[0.02232909936926042, 0.49401693585629247], Power[k, 2]]], Rational[1, 2]]]]], Times[Complex[2.0923197935864075, -0.9558768244369737], Plus[Complex[0.7424137617640161, 0.220635885032481], Times[Complex[0.14483575015411373, 1.3558262394954135], Power[Plus[1.0, Times[Complex[0.9940169358562925, 0.4776709006307397], Power[k, 2]]], Rational[1, 2]]]]]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[μ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.26.E9 R J ⁡ ( x + λ , y + λ , z + λ , p + λ ) + R J ⁡ ( x + μ , y + μ , z + μ , p + μ ) = R J ⁡ ( x , y , z , p ) - 3 ⁢ R C ⁡ ( γ - δ , γ ) Carlson-integral-RJ 𝑥 𝜆 𝑦 𝜆 𝑧 𝜆 𝑝 𝜆 Carlson-integral-RJ 𝑥 𝜇 𝑦 𝜇 𝑧 𝜇 𝑝 𝜇 Carlson-integral-RJ 𝑥 𝑦 𝑧 𝑝 3 Carlson-integral-RC 𝛾 𝛿 𝛾 {\displaystyle{\displaystyle R_{J}\left(x+\lambda,y+\lambda,z+\lambda,p+% \lambda\right)+R_{J}\left(x+\mu,y+\mu,z+\mu,p+\mu\right)=R_{J}\left(x,y,z,p% \right)-3R_{C}\left(\gamma-\delta,\gamma\right)}}
\CarlsonsymellintRJ@{x+\lambda}{y+\lambda}{z+\lambda}{p+\lambda}+\CarlsonsymellintRJ@{x+\mu}{y+\mu}{z+\mu}{p+\mu} = \CarlsonsymellintRJ@{x}{y}{z}{p}-3\CarlsonellintRC@{\gamma-\delta}{\gamma}		 

Error

3*((x + y*I)+ \[Lambda]-x + \[Lambda])/((x + y*I)+ \[Lambda]-p + \[Lambda])*(EllipticPi[((x + y*I)+ \[Lambda]-p + \[Lambda])/((x + y*I)+ \[Lambda]-x + \[Lambda]),ArcCos[Sqrt[x + \[Lambda]/(x + y*I)+ \[Lambda]]],((x + y*I)+ \[Lambda]-y + \[Lambda])/((x + y*I)+ \[Lambda]-x + \[Lambda])]-EllipticF[ArcCos[Sqrt[x + \[Lambda]/(x + y*I)+ \[Lambda]]],((x + y*I)+ \[Lambda]-y + \[Lambda])/((x + y*I)+ \[Lambda]-x + \[Lambda])])/Sqrt[(x + y*I)+ \[Lambda]-x + \[Lambda]]+ 3*((x + y*I)+ \[Mu]-x + \[Mu])/((x + y*I)+ \[Mu]-p + \[Mu])*(EllipticPi[((x + y*I)+ \[Mu]-p + \[Mu])/((x + y*I)+ \[Mu]-x + \[Mu]),ArcCos[Sqrt[x + \[Mu]/(x + y*I)+ \[Mu]]],((x + y*I)+ \[Mu]-y + \[Mu])/((x + y*I)+ \[Mu]-x + \[Mu])]-EllipticF[ArcCos[Sqrt[x + \[Mu]/(x + y*I)+ \[Mu]]],((x + y*I)+ \[Mu]-y + \[Mu])/((x + y*I)+ \[Mu]-x + \[Mu])])/Sqrt[(x + y*I)+ \[Mu]-x + \[Mu]] == 3*(x + y*I-x)/(x + y*I-p)*(EllipticPi[(x + y*I-p)/(x + y*I-x),ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/Sqrt[x + y*I-x]- 3*1/Sqrt[\[Gamma]]*Hypergeometric2F1[1/2,1/2,3/2,1-(\[Gamma]- \[Delta])/(\[Gamma])]
Missing Macro Error Failure -
Failed [300 / 300]
Result: Complex[6.482970499990588, -0.8807575715831795]
Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, -1.5], Rule[γ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[δ, 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]]]], Rule[μ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[7.020988185402777, -1.8389880807014276]
Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, -1.5], Rule[γ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[δ, 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]]]], Rule[μ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.26#Ex3 γ = p ⁢ ( p + λ ) ⁢ ( p + μ ) 𝛾 𝑝 𝑝 𝜆 𝑝 𝜇 {\displaystyle{\displaystyle\gamma=p(p+\lambda)(p+\mu)}}
\gamma = p(p+\lambda)(p+\mu)		 

gamma = p*(p + lambda)*(p + mu)
 
\[Gamma] == p*(p + \[Lambda])*(p + \[Mu])
 Skipped - no semantic math Skipped - no semantic math - -
19.26#Ex4 δ = ( p - x ) ⁢ ( p - y ) ⁢ ( p - z ) 𝛿 𝑝 𝑥 𝑝 𝑦 𝑝 𝑧 {\displaystyle{\displaystyle\delta=(p-x)(p-y)(p-z)}}
\delta = (p-x)(p-y)(p-z)		 

delta = (p - x)*(p - y)*(p -(x + y*I))
 
\[Delta] == (p - x)*(p - y)*(p -(x + y*I))
 Skipped - no semantic math Skipped - no semantic math - -
19.26.E11 R C ⁡ ( x + λ , y + λ ) + R C ⁡ ( x + μ , y + μ ) = R C ⁡ ( x , y ) Carlson-integral-RC 𝑥 𝜆 𝑦 𝜆 Carlson-integral-RC 𝑥 𝜇 𝑦 𝜇 Carlson-integral-RC 𝑥 𝑦 {\displaystyle{\displaystyle R_{C}\left(x+\lambda,y+\lambda\right)+R_{C}\left(% x+\mu,y+\mu\right)=R_{C}\left(x,y\right)}}
\CarlsonellintRC@{x+\lambda}{y+\lambda}+\CarlsonellintRC@{x+\mu}{y+\mu} = \CarlsonellintRC@{x}{y}		 

Error

1/Sqrt[y + \[Lambda]]*Hypergeometric2F1[1/2,1/2,3/2,1-(x + \[Lambda])/(y + \[Lambda])]+ 1/Sqrt[y + \[Mu]]*Hypergeometric2F1[1/2,1/2,3/2,1-(x + \[Mu])/(y + \[Mu])] == 1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)]
Missing Macro Error Failure -
Failed [300 / 300]
Result: Complex[1.7722794006718585, -0.740880873447254]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[μ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[1.579678795390187, -0.7154745309495683]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[μ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.26#Ex5 x + μ = λ - 2 ⁢ ( x + λ ⁢ y + x ⁢ ( y + λ ) ) 2 𝑥 𝜇 superscript 𝜆 2 superscript 𝑥 𝜆 𝑦 𝑥 𝑦 𝜆 2 {\displaystyle{\displaystyle x+\mu=\lambda^{-2}(\sqrt{x+\lambda}y+\sqrt{x}(y+% \lambda))^{2}}}
x+\mu = \lambda^{-2}(\sqrt{x+\lambda}y+\sqrt{x}(y+\lambda))^{2}		 

x + mu = (lambda)^(- 2)*(sqrt(x + lambda)*y +sqrt(x)*(y + lambda))^(2)
 
x + \[Mu] == \[Lambda]^(- 2)*(Sqrt[x + \[Lambda]]*y +Sqrt[x]*(y + \[Lambda]))^(2)
 Skipped - no semantic math Skipped - no semantic math - -
19.26#Ex6 y + μ = ( y ⁢ ( y + λ ) / λ 2 ) ⁢ ( x + x + λ ) 2 𝑦 𝜇 𝑦 𝑦 𝜆 superscript 𝜆 2 superscript 𝑥 𝑥 𝜆 2 {\displaystyle{\displaystyle y+\mu=(y(y+\lambda)/\lambda^{2})(\sqrt{x}+\sqrt{x% +\lambda})^{2}}}
y+\mu = (y(y+\lambda)/\lambda^{2})(\sqrt{x}+\sqrt{x+\lambda})^{2}		 

y + mu = (y*(y + lambda)/(lambda)^(2))*(sqrt(x)+sqrt(x + lambda))^(2)
 
y + \[Mu] == (y*(y + \[Lambda])/\[Lambda]^(2))*(Sqrt[x]+Sqrt[x + \[Lambda]])^(2)
 Skipped - no semantic math Skipped - no semantic math - -
19.26.E13 R C ⁡ ( α 2 , α 2 - θ ) + R C ⁡ ( β 2 , β 2 - θ ) = R C ⁡ ( σ 2 , σ 2 - θ ) Carlson-integral-RC superscript 𝛼 2 superscript 𝛼 2 𝜃 Carlson-integral-RC superscript 𝛽 2 superscript 𝛽 2 𝜃 Carlson-integral-RC superscript 𝜎 2 superscript 𝜎 2 𝜃 {\displaystyle{\displaystyle R_{C}\left(\alpha^{2},\alpha^{2}-\theta\right)+R_% {C}\left(\beta^{2},\beta^{2}-\theta\right)=R_{C}\left(\sigma^{2},\sigma^{2}-% \theta\right)}}
\CarlsonellintRC@{\alpha^{2}}{\alpha^{2}-\theta}+\CarlsonellintRC@{\beta^{2}}{\beta^{2}-\theta} = \CarlsonellintRC@{\sigma^{2}}{\sigma^{2}-\theta}		 σ = ( α ⁢ β + θ ) / ( α + β ) 𝜎 𝛼 𝛽 𝜃 𝛼 𝛽 {\displaystyle{\displaystyle\sigma=(\alpha\beta+\theta)/(\alpha+\beta)}} 
Error

1/Sqrt[\[Alpha]^(2)- \[Theta]]*Hypergeometric2F1[1/2,1/2,3/2,1-(\[Alpha]^(2))/(\[Alpha]^(2)- \[Theta])]+ 1/Sqrt[\[Beta]^(2)- \[Theta]]*Hypergeometric2F1[1/2,1/2,3/2,1-(\[Beta]^(2))/(\[Beta]^(2)- \[Theta])] == 1/Sqrt[\[Sigma]^(2)- \[Theta]]*Hypergeometric2F1[1/2,1/2,3/2,1-(\[Sigma]^(2))/(\[Sigma]^(2)- \[Theta])]
Missing Macro Error Aborted - Successful [Tested: 2]
19.26.E14 ( p - y ) ⁢ R C ⁡ ( x , p ) + ( q - y ) ⁢ R C ⁡ ( x , q ) = ( η - ξ ) ⁢ R C ⁡ ( ξ , η ) 𝑝 𝑦 Carlson-integral-RC 𝑥 𝑝 𝑞 𝑦 Carlson-integral-RC 𝑥 𝑞 𝜂 𝜉 Carlson-integral-RC 𝜉 𝜂 {\displaystyle{\displaystyle(p-y)R_{C}\left(x,p\right)+(q-y)R_{C}\left(x,q% \right)=(\eta-\xi)R_{C}\left(\xi,\eta\right)}}
(p-y)\CarlsonellintRC@{x}{p}+(q-y)\CarlsonellintRC@{x}{q} = (\eta-\xi)\CarlsonellintRC@{\xi}{\eta}		 x ≥ 0 , y ≥ 0 formulae-sequence 𝑥 0 𝑦 0 {\displaystyle{\displaystyle x\geq 0,y\geq 0}} 
Error

(p - y)*1/Sqrt[p]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(p)]+(q - y)*1/Sqrt[q]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(q)] == (\[Eta]- \[Xi])*1/Sqrt[\[Eta]]*Hypergeometric2F1[1/2,1/2,3/2,1-(\[Xi])/(\[Eta])]
Missing Macro Error Failure -
Failed [300 / 300]
Result: Indeterminate
Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[q, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, 1.5], Rule[η, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ξ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-3.0971074607887266, 1.6817857583573725]
Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[q, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, 1.5], Rule[η, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ξ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.26#Ex7 ( p - x ) ⁢ ( q - x ) = ( y - x ) 2 𝑝 𝑥 𝑞 𝑥 superscript 𝑦 𝑥 2 {\displaystyle{\displaystyle(p-x)(q-x)=(y-x)^{2}}}
(p-x)(q-x) = (y-x)^{2}		 

(p - x)*(q - x) = (y - x)^(2)
 
(p - x)*(q - x) == (y - x)^(2)
 Skipped - no semantic math Skipped - no semantic math - -
19.26#Ex8 ξ = y 2 / x 𝜉 superscript 𝑦 2 𝑥 {\displaystyle{\displaystyle\xi=y^{2}/x}}
\xi = y^{2}/x		 

xi = (y)^(2)/x
 
\[Xi] == (y)^(2)/x
 Skipped - no semantic math Skipped - no semantic math - -
19.26#Ex9 η = p ⁢ q / x 𝜂 𝑝 𝑞 𝑥 {\displaystyle{\displaystyle\eta=pq/x}}
\eta = pq/x		 

eta = p*q/x
 
\[Eta] == p*q/x
 Skipped - no semantic math Skipped - no semantic math - -
19.26#Ex10 η - ξ = p + q - 2 ⁢ y 𝜂 𝜉 𝑝 𝑞 2 𝑦 {\displaystyle{\displaystyle\eta-\xi=p+q-2y}}
\eta-\xi = p+q-2y		 

eta - xi = p + q - 2*y
 
\[Eta]- \[Xi] == p + q - 2*y
 Skipped - no semantic math Skipped - no semantic math - -
19.26.E16 R F ⁡ ( λ , y + λ , z + λ ) = R F ⁡ ( 0 , y , z ) - R F ⁡ ( μ , y + μ , z + μ ) , Carlson-integral-RF 𝜆 𝑦 𝜆 𝑧 𝜆 Carlson-integral-RF 0 𝑦 𝑧 Carlson-integral-RF 𝜇 𝑦 𝜇 𝑧 𝜇 {\displaystyle{\displaystyle R_{F}\left(\lambda,y+\lambda,z+\lambda\right)={R_% {F}\left(0,y,z\right)-R_{F}\left(\mu,y+\mu,z+\mu\right),}}}
\CarlsonsymellintRF@{\lambda}{y+\lambda}{z+\lambda} = {\CarlsonsymellintRF@{0}{y}{z}-\CarlsonsymellintRF@{\mu}{y+\mu}{z+\mu},}		 λ ⁢ μ = y ⁢ z 𝜆 𝜇 𝑦 𝑧 {\displaystyle{\displaystyle\lambda\mu=yz}} 
0.5*int(1/(sqrt(t+lambda)*sqrt(t+y + lambda)*sqrt(t+(x + y*I)+ lambda)), t = 0..infinity) = 0.5*int(1/(sqrt(t+0)*sqrt(t+y)*sqrt(t+x + y*I)), t = 0..infinity)- 0.5*int(1/(sqrt(t+mu)*sqrt(t+y + mu)*sqrt(t+(x + y*I)+ mu)), t = 0..infinity)

EllipticF[ArcCos[Sqrt[\[Lambda]/(x + y*I)+ \[Lambda]]],((x + y*I)+ \[Lambda]-y + \[Lambda])/((x + y*I)+ \[Lambda]-\[Lambda])]/Sqrt[(x + y*I)+ \[Lambda]-\[Lambda]] == EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]/Sqrt[x + y*I-0]- EllipticF[ArcCos[Sqrt[\[Mu]/(x + y*I)+ \[Mu]]],((x + y*I)+ \[Mu]-y + \[Mu])/((x + y*I)+ \[Mu]-\[Mu])]/Sqrt[(x + y*I)+ \[Mu]-\[Mu]]
Error Failure - Error
19.26.E17 α ⁢ R C ⁡ ( β , α + β ) + β ⁢ R C ⁡ ( α , α + β ) = π / 2 𝛼 Carlson-integral-RC 𝛽 𝛼 𝛽 𝛽 Carlson-integral-RC 𝛼 𝛼 𝛽 𝜋 2 {\displaystyle{\displaystyle\sqrt{\alpha}R_{C}\left(\beta,\alpha+\beta\right)+% \sqrt{\beta}R_{C}\left(\alpha,\alpha+\beta\right)=\pi/2}}
\sqrt{\alpha}\CarlsonellintRC@{\beta}{\alpha+\beta}+\sqrt{\beta}\CarlsonellintRC@{\alpha}{\alpha+\beta} = \pi/2		 α + β > 0 𝛼 𝛽 0 {\displaystyle{\displaystyle\alpha+\beta>0}} 
Error

Sqrt[\[Alpha]]*1/Sqrt[\[Alpha]+ \[Beta]]*Hypergeometric2F1[1/2,1/2,3/2,1-(\[Beta])/(\[Alpha]+ \[Beta])]+Sqrt[\[Beta]]*1/Sqrt[\[Alpha]+ \[Beta]]*Hypergeometric2F1[1/2,1/2,3/2,1-(\[Alpha])/(\[Alpha]+ \[Beta])] == Pi/2
Missing Macro Error Failure - Successful [Tested: 9]
19.26.E18 R F ⁡ ( x , y , z ) = 2 ⁢ R F ⁡ ( x + λ , y + λ , z + λ ) Carlson-integral-RF 𝑥 𝑦 𝑧 2 Carlson-integral-RF 𝑥 𝜆 𝑦 𝜆 𝑧 𝜆 {\displaystyle{\displaystyle R_{F}\left(x,y,z\right)=2R_{F}\left(x+\lambda,y+% \lambda,z+\lambda\right)}}
\CarlsonsymellintRF@{x}{y}{z} = 2\CarlsonsymellintRF@{x+\lambda}{y+\lambda}{z+\lambda}		 

0.5*int(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+x + y*I)), t = 0..infinity) = 2*0.5*int(1/(sqrt(t+x + lambda)*sqrt(t+y + lambda)*sqrt(t+(x + y*I)+ lambda)), t = 0..infinity)

EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x] == 2*EllipticF[ArcCos[Sqrt[x + \[Lambda]/(x + y*I)+ \[Lambda]]],((x + y*I)+ \[Lambda]-y + \[Lambda])/((x + y*I)+ \[Lambda]-x + \[Lambda])]/Sqrt[(x + y*I)+ \[Lambda]-x + \[Lambda]]
Aborted Failure Skipped - Because timed out
Failed [180 / 180]
Result: Complex[-0.6992255245511445, 1.8246422705609677]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-1.7332476531334464, -0.3074481161267689]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.26.E18 2 ⁢ R F ⁡ ( x + λ , y + λ , z + λ ) = R F ⁡ ( x + λ 4 , y + λ 4 , z + λ 4 ) 2 Carlson-integral-RF 𝑥 𝜆 𝑦 𝜆 𝑧 𝜆 Carlson-integral-RF 𝑥 𝜆 4 𝑦 𝜆 4 𝑧 𝜆 4 {\displaystyle{\displaystyle 2R_{F}\left(x+\lambda,y+\lambda,z+\lambda\right)=% R_{F}\left(\frac{x+\lambda}{4},\frac{y+\lambda}{4},\frac{z+\lambda}{4}\right)}}
2\CarlsonsymellintRF@{x+\lambda}{y+\lambda}{z+\lambda} = \CarlsonsymellintRF@{\frac{x+\lambda}{4}}{\frac{y+\lambda}{4}}{\frac{z+\lambda}{4}}		 

2*0.5*int(1/(sqrt(t+x + lambda)*sqrt(t+y + lambda)*sqrt(t+(x + y*I)+ lambda)), t = 0..infinity) = 0.5*int(1/(sqrt(t+(x + lambda)/(4))*sqrt(t+(y + lambda)/(4))*sqrt(t+((x + y*I)+ lambda)/(4))), t = 0..infinity)

2*EllipticF[ArcCos[Sqrt[x + \[Lambda]/(x + y*I)+ \[Lambda]]],((x + y*I)+ \[Lambda]-y + \[Lambda])/((x + y*I)+ \[Lambda]-x + \[Lambda])]/Sqrt[(x + y*I)+ \[Lambda]-x + \[Lambda]] == EllipticF[ArcCos[Sqrt[Divide[x + \[Lambda],4]/Divide[(x + y*I)+ \[Lambda],4]]],(Divide[(x + y*I)+ \[Lambda],4]-Divide[y + \[Lambda],4])/(Divide[(x + y*I)+ \[Lambda],4]-Divide[x + \[Lambda],4])]/Sqrt[Divide[(x + y*I)+ \[Lambda],4]-Divide[x + \[Lambda],4]]
Failure Failure Skipped - Because timed out
Failed [180 / 180]
Result: Complex[-1.1343270456997319, -2.101834604175173]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-0.07907692856233961, -0.3004487668798371]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.26.E19 λ = x ⁢ y + y ⁢ z + z ⁢ x 𝜆 𝑥 𝑦 𝑦 𝑧 𝑧 𝑥 {\displaystyle{\displaystyle\lambda=\sqrt{x}\sqrt{y}+\sqrt{y}\sqrt{z}+\sqrt{z}% \sqrt{x}}}
\lambda = \sqrt{x}\sqrt{y}+\sqrt{y}\sqrt{z}+\sqrt{z}\sqrt{x}		 

lambda = sqrt(x)*sqrt(y)+sqrt(y)*sqrt(x + y*I)+sqrt(x + y*I)*sqrt(x)
 
\[Lambda] == Sqrt[x]*Sqrt[y]+Sqrt[y]*Sqrt[x + y*I]+Sqrt[x + y*I]*Sqrt[x]
 Skipped - no semantic math Skipped - no semantic math - -
19.26.E20 R D ⁡ ( x , y , z ) = 2 ⁢ R D ⁡ ( x + λ , y + λ , z + λ ) + 3 z ⁢ ( z + λ ) Carlson-integral-RD 𝑥 𝑦 𝑧 2 Carlson-integral-RD 𝑥 𝜆 𝑦 𝜆 𝑧 𝜆 3 𝑧 𝑧 𝜆 {\displaystyle{\displaystyle R_{D}\left(x,y,z\right)=2R_{D}\left(x+\lambda,y+% \lambda,z+\lambda\right)+\frac{3}{\sqrt{z}(z+\lambda)}}}
\CarlsonsymellintRD@{x}{y}{z} = 2\CarlsonsymellintRD@{x+\lambda}{y+\lambda}{z+\lambda}+\frac{3}{\sqrt{z}(z+\lambda)}		 

Error

3*(EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticE[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/((x + y*I-y)*(x + y*I-x)^(1/2)) == 2*3*(EllipticF[ArcCos[Sqrt[x + \[Lambda]/(x + y*I)+ \[Lambda]]],((x + y*I)+ \[Lambda]-y + \[Lambda])/((x + y*I)+ \[Lambda]-x + \[Lambda])]-EllipticE[ArcCos[Sqrt[x + \[Lambda]/(x + y*I)+ \[Lambda]]],((x + y*I)+ \[Lambda]-y + \[Lambda])/((x + y*I)+ \[Lambda]-x + \[Lambda])])/(((x + y*I)+ \[Lambda]-y + \[Lambda])*((x + y*I)+ \[Lambda]-x + \[Lambda])^(1/2))+Divide[3,Sqrt[x + y*I]*((x + y*I)+ \[Lambda])]
Missing Macro Error Failure -
Failed [180 / 180]
Result: Complex[0.4984590390126629, -1.2092907867192135]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-0.5295690158190058, -2.8195127867822802]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.26.E21 2 ⁢ R G ⁡ ( x , y , z ) = 4 ⁢ R G ⁡ ( x + λ , y + λ , z + λ ) - λ ⁢ R F ⁡ ( x , y , z ) - x - y - z 2 Carlson-integral-RG 𝑥 𝑦 𝑧 4 Carlson-integral-RG 𝑥 𝜆 𝑦 𝜆 𝑧 𝜆 𝜆 Carlson-integral-RF 𝑥 𝑦 𝑧 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle 2R_{G}\left(x,y,z\right)=4R_{G}\left(x+\lambda,y+% \lambda,z+\lambda\right)-\lambda R_{F}\left(x,y,z\right)-\sqrt{x}-\sqrt{y}-% \sqrt{z}}}
2\CarlsonsymellintRG@{x}{y}{z} = 4\CarlsonsymellintRG@{x+\lambda}{y+\lambda}{z+\lambda}-\lambda\CarlsonsymellintRF@{x}{y}{z}-\sqrt{x}-\sqrt{y}-\sqrt{z}		 

Error

2*Sqrt[x + y*I-x]*(EllipticE[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]+(Cot[ArcCos[Sqrt[x/x + y*I]]])^2*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]+Cot[ArcCos[Sqrt[x/x + y*I]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[x/x + y*I]]]^2]) == 4*Sqrt[(x + y*I)+ \[Lambda]-x + \[Lambda]]*(EllipticE[ArcCos[Sqrt[x + \[Lambda]/(x + y*I)+ \[Lambda]]],((x + y*I)+ \[Lambda]-y + \[Lambda])/((x + y*I)+ \[Lambda]-x + \[Lambda])]+(Cot[ArcCos[Sqrt[x + \[Lambda]/(x + y*I)+ \[Lambda]]]])^2*EllipticF[ArcCos[Sqrt[x + \[Lambda]/(x + y*I)+ \[Lambda]]],((x + y*I)+ \[Lambda]-y + \[Lambda])/((x + y*I)+ \[Lambda]-x + \[Lambda])]+Cot[ArcCos[Sqrt[x + \[Lambda]/(x + y*I)+ \[Lambda]]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[x + \[Lambda]/(x + y*I)+ \[Lambda]]]]^2])- \[Lambda]*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]-Sqrt[x]-Sqrt[y]-Sqrt[x + y*I]
Missing Macro Error Aborted -
Failed [180 / 180]
Result: Plus[Complex[2.330530943809637, 0.9206144902290859], Times[Complex[1.7320508075688772, -1.732050807568877], Plus[Complex[0.9985512968581824, 0.2012315241723115], Times[Complex[0.3176872874027722, -1.049249833251038], Power[Plus[1.0, Times[Complex[0.0, -1.5], Power[k, 2]]], Rational[1, 2]]]]], Times[Complex[-4.184639587172815, 1.9117536488739475], Plus[Complex[0.7424137617640161, 0.220635885032481], Times[Complex[0.14483575015411373, 1.3558262394954135], Power[Plus[1.0, Times[Complex[0.9940169358562925, 0.4776709006307397], Power[k, 2]]], Rational[1, 2]]]]]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Plus[Complex[2.3171140130573056, 0.42755423781462054], Times[Complex[1.7320508075688772, -1.732050807568877], Plus[Complex[0.9985512968581824, 0.2012315241723115], Times[Complex[0.3176872874027722, -1.049249833251038], Power[Plus[1.0, Times[Complex[0.0, -1.5], Power[k, 2]]], Rational[1, 2]]]]], Times[Complex[-1.5682295148869496, 3.234090878649293], Plus[Complex[0.3473840661116648, 1.4426085854555293], Times[Complex[0.7761183239980944, 1.3014092542459557], Power[Plus[1.0, Times[Complex[0.02232909936926042, 0.49401693585629247], Power[k, 2]]], Rational[1, 2]]]]]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.26.E22 R J ⁡ ( x , y , z , p ) = 2 ⁢ R J ⁡ ( x + λ , y + λ , z + λ , p + λ ) + 3 ⁢ R C ⁡ ( α 2 , β 2 ) Carlson-integral-RJ 𝑥 𝑦 𝑧 𝑝 2 Carlson-integral-RJ 𝑥 𝜆 𝑦 𝜆 𝑧 𝜆 𝑝 𝜆 3 Carlson-integral-RC superscript 𝛼 2 superscript 𝛽 2 {\displaystyle{\displaystyle R_{J}\left(x,y,z,p\right)=2R_{J}\left(x+\lambda,y% +\lambda,z+\lambda,p+\lambda\right)+3R_{C}\left(\alpha^{2},\beta^{2}\right)}}
\CarlsonsymellintRJ@{x}{y}{z}{p} = 2\CarlsonsymellintRJ@{x+\lambda}{y+\lambda}{z+\lambda}{p+\lambda}+3\CarlsonellintRC@{\alpha^{2}}{\beta^{2}}		 

Error

3*(x + y*I-x)/(x + y*I-p)*(EllipticPi[(x + y*I-p)/(x + y*I-x),ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/Sqrt[x + y*I-x] == 2*3*((x + y*I)+ \[Lambda]-x + \[Lambda])/((x + y*I)+ \[Lambda]-p + \[Lambda])*(EllipticPi[((x + y*I)+ \[Lambda]-p + \[Lambda])/((x + y*I)+ \[Lambda]-x + \[Lambda]),ArcCos[Sqrt[x + \[Lambda]/(x + y*I)+ \[Lambda]]],((x + y*I)+ \[Lambda]-y + \[Lambda])/((x + y*I)+ \[Lambda]-x + \[Lambda])]-EllipticF[ArcCos[Sqrt[x + \[Lambda]/(x + y*I)+ \[Lambda]]],((x + y*I)+ \[Lambda]-y + \[Lambda])/((x + y*I)+ \[Lambda]-x + \[Lambda])])/Sqrt[(x + y*I)+ \[Lambda]-x + \[Lambda]]+ 3*1/Sqrt[\[Beta]^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-(\[Alpha]^(2))/(\[Beta]^(2))]
Missing Macro Error Failure -
Failed [300 / 300]
Result: Indeterminate
Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, -1.5], Rule[α, 1.5], Rule[β, 1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Indeterminate
Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, -1.5], Rule[α, 1.5], Rule[β, 1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.26#Ex11 α = p ⁢ ( x + y + z ) + x ⁢ y ⁢ z 𝛼 𝑝 𝑥 𝑦 𝑧 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle\alpha=p(\sqrt{x}+\sqrt{y}+\sqrt{z})+\sqrt{x}\sqrt% {y}\sqrt{z}}}
\alpha = p(\sqrt{x}+\sqrt{y}+\sqrt{z})+\sqrt{x}\sqrt{y}\sqrt{z}		 

alpha = p*(sqrt(x)+sqrt(y)+sqrt(x + y*I))+sqrt(x)*sqrt(y)*sqrt(x + y*I)
 
\[Alpha] == p*(Sqrt[x]+Sqrt[y]+Sqrt[x + y*I])+Sqrt[x]*Sqrt[y]*Sqrt[x + y*I]
 Skipped - no semantic math Skipped - no semantic math - -
19.26#Ex12 β = p ⁢ ( p + λ ) 𝛽 𝑝 𝑝 𝜆 {\displaystyle{\displaystyle\beta=\sqrt{p}(p+\lambda)}}
\beta = \sqrt{p}(p+\lambda)		 

beta = sqrt(p)*(p + lambda)
 
\[Beta] == Sqrt[p]*(p + \[Lambda])
 Skipped - no semantic math Skipped - no semantic math - -
19.26#Ex13 β + α = ( p + x ) ⁢ ( p + y ) ⁢ ( p + z ) 𝛽 𝛼 𝑝 𝑥 𝑝 𝑦 𝑝 𝑧 {\displaystyle{\displaystyle\beta+\alpha=(\sqrt{p}+\sqrt{x})(\sqrt{p}+\sqrt{y}% )(\sqrt{p}+\sqrt{z})}}
\beta+\alpha = (\sqrt{p}+\sqrt{x})(\sqrt{p}+\sqrt{y})(\sqrt{p}+\sqrt{z})		 

beta + alpha = (sqrt(p)+sqrt(x))*(sqrt(p)+sqrt(y))*(sqrt(p)+sqrt(x + y*I))
 
\[Beta]+ \[Alpha] == (Sqrt[p]+Sqrt[x])*(Sqrt[p]+Sqrt[y])*(Sqrt[p]+Sqrt[x + y*I])
 Skipped - no semantic math Skipped - no semantic math - -
19.26#Ex14 β 2 - α 2 = ( p - x ) ⁢ ( p - y ) ⁢ ( p - z ) superscript 𝛽 2 superscript 𝛼 2 𝑝 𝑥 𝑝 𝑦 𝑝 𝑧 {\displaystyle{\displaystyle\beta^{2}-\alpha^{2}=(p-x)(p-y)(p-z)}}
\beta^{2}-\alpha^{2} = (p-x)(p-y)(p-z)		 

(beta)^(2)- (alpha)^(2) = (p - x)*(p - y)*(p -(x + y*I))
 
\[Beta]^(2)- \[Alpha]^(2) == (p - x)*(p - y)*(p -(x + y*I))
 Skipped - no semantic math Skipped - no semantic math - -
19.26.E24 z = ( ξ ⁢ ζ + η ⁢ ζ - ξ ⁢ η ) 2 / ( 4 ⁢ ξ ⁢ η ⁢ ζ ) 𝑧 superscript 𝜉 𝜁 𝜂 𝜁 𝜉 𝜂 2 4 𝜉 𝜂 𝜁 {\displaystyle{\displaystyle z=(\xi\zeta+\eta\zeta-\xi\eta)^{2}/(4\xi\eta\zeta% )}}
z = (\xi\zeta+\eta\zeta-\xi\eta)^{2}/(4\xi\eta\zeta)		 ( ξ = ( x + λ , η = ( x + λ , ζ ) = ( x + λ fragments ( ξ fragments ( x λ , η fragments ( x λ , ζ ) fragments ( x λ {\displaystyle{\displaystyle(\xi=(x+\lambda,\eta=(x+\lambda,\zeta)=(x+\lambda}} 
z = (xi*zeta + eta*zeta - xi*eta)^(2)/(4*xi*eta*zeta)
 
z == (\[Xi]*\[Zeta]+ \[Eta]*\[Zeta]- \[Xi]*\[Eta])^(2)/(4*\[Xi]*\[Eta]*\[Zeta])
 Skipped - no semantic math Skipped - no semantic math - -
19.26.E25 R C ⁡ ( x , y ) = 2 ⁢ R C ⁡ ( x + λ , y + λ ) Carlson-integral-RC 𝑥 𝑦 2 Carlson-integral-RC 𝑥 𝜆 𝑦 𝜆 {\displaystyle{\displaystyle R_{C}\left(x,y\right)=2R_{C}\left(x+\lambda,y+% \lambda\right)}}
\CarlsonellintRC@{x}{y} = 2\CarlsonellintRC@{x+\lambda}{y+\lambda}		 λ = y + 2 ⁢ x ⁢ y 𝜆 𝑦 2 𝑥 𝑦 {\displaystyle{\displaystyle\lambda=y+2\sqrt{x}\sqrt{y}}} 
Error

1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)] == 2*1/Sqrt[y + \[Lambda]]*Hypergeometric2F1[1/2,1/2,3/2,1-(x + \[Lambda])/(y + \[Lambda])]
Missing Macro Error Failure -
Failed [1 / 1]
Result: Indeterminate
Test Values: {Rule[x, 0.5], Rule[y, 0.5], Rule[λ, 1.5]}

19.26.E26 R C ⁡ ( x 2 , y 2 ) = R C ⁡ ( a 2 , a ⁢ y ) Carlson-integral-RC superscript 𝑥 2 superscript 𝑦 2 Carlson-integral-RC superscript 𝑎 2 𝑎 𝑦 {\displaystyle{\displaystyle R_{C}\left(x^{2},y^{2}\right)=R_{C}\left(a^{2},ay% \right)}}
\CarlsonellintRC@{x^{2}}{y^{2}} = \CarlsonellintRC@{a^{2}}{ay}		 a = ( x + y ) / 2 , ℜ ⁡ x ≥ 0 , ℜ ⁡ y > 0 formulae-sequence 𝑎 𝑥 𝑦 2 formulae-sequence 𝑥 0 𝑦 0 {\displaystyle{\displaystyle a=(x+y)/2,\Re x\geq 0,\Re y>0}} 
Error

1/Sqrt[(y)^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-((x)^(2))/((y)^(2))] == 1/Sqrt[a*y]*Hypergeometric2F1[1/2,1/2,3/2,1-((a)^(2))/(a*y)]
Missing Macro Error Aborted -
Failed [3 / 3]
Result: Indeterminate
Test Values: {Rule[a, 1.5], Rule[x, 1.5], Rule[y, 1.5]}


Result: Indeterminate
Test Values: {Rule[a, 0.5], Rule[x, 0.5], Rule[y, 0.5]}

... skip entries to safe data
19.26.E27 R C ⁡ ( x 2 , x 2 - θ ) = 2 ⁢ R C ⁡ ( s 2 , s 2 - θ ) Carlson-integral-RC superscript 𝑥 2 superscript 𝑥 2 𝜃 2 Carlson-integral-RC superscript 𝑠 2 superscript 𝑠 2 𝜃 {\displaystyle{\displaystyle R_{C}\left(x^{2},x^{2}-\theta\right)=2R_{C}\left(% s^{2},s^{2}-\theta\right)}}
\CarlsonellintRC@{x^{2}}{x^{2}-\theta} = 2\CarlsonellintRC@{s^{2}}{s^{2}-\theta}		 s = x + x 2 - θ , θ ≠ x 2 formulae-sequence 𝑠 𝑥 superscript 𝑥 2 𝜃 𝜃 superscript 𝑥 2 {\displaystyle{\displaystyle s=x+\sqrt{x^{2}-\theta},\theta\neq x^{2}}} 
Error

1/Sqrt[(x)^(2)- \[Theta]]*Hypergeometric2F1[1/2,1/2,3/2,1-((x)^(2))/((x)^(2)- \[Theta])] == 2*1/Sqrt[(s)^(2)- \[Theta]]*Hypergeometric2F1[1/2,1/2,3/2,1-((s)^(2))/((s)^(2)- \[Theta])]
Missing Macro Error Failure - Successful [Tested: 2]
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