Elliptic Integrals - 19.20 Special Cases

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
19.20#Ex1 R F ⁑ ( x , x , x ) = x - 1 / 2 Carlson-integral-RF π‘₯ π‘₯ π‘₯ superscript π‘₯ 1 2 {\displaystyle{\displaystyle R_{F}\left(x,x,x\right)=x^{-1/2}}}
\CarlsonsymellintRF@{x}{x}{x} = x^{-1/2}		 

0.5*int(1/(sqrt(t+x)*sqrt(t+x)*sqrt(t+x)), t = 0..infinity) = (x)^(- 1/2)

EllipticF[ArcCos[Sqrt[x/x]],(x-x)/(x-x)]/Sqrt[x-x] == (x)^(- 1/2)
Failure Failure Successful [Tested: 3]
Failed [3 / 3]
Result: Indeterminate
Test Values: {Rule[x, 1.5]}


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

... skip entries to safe data
19.20#Ex2 R F ⁑ ( Ξ» ⁒ x , Ξ» ⁒ y , Ξ» ⁒ z ) = Ξ» - 1 / 2 ⁒ R F ⁑ ( x , y , z ) Carlson-integral-RF πœ† π‘₯ πœ† 𝑦 πœ† 𝑧 superscript πœ† 1 2 Carlson-integral-RF π‘₯ 𝑦 𝑧 {\displaystyle{\displaystyle R_{F}\left(\lambda x,\lambda y,\lambda z\right)=% \lambda^{-1/2}R_{F}\left(x,y,z\right)}}
\CarlsonsymellintRF@{\lambda x}{\lambda y}{\lambda z} = \lambda^{-1/2}\CarlsonsymellintRF@{x}{y}{z}		 

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

EllipticF[ArcCos[Sqrt[\[Lambda]*x/\[Lambda]*(x + y*I)]],(\[Lambda]*(x + y*I)-\[Lambda]*y)/(\[Lambda]*(x + y*I)-\[Lambda]*x)]/Sqrt[\[Lambda]*(x + y*I)-\[Lambda]*x] == \[Lambda]^(- 1/2)* 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 [180 / 180]
Result: Complex[-0.15259412278903736, 0.06775202977854555]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[Ξ», Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-0.05999241929777854, 0.15580825868890358]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[Ξ», Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.20#Ex3 R F ⁑ ( x , y , y ) = R C ⁑ ( x , y ) Carlson-integral-RF π‘₯ 𝑦 𝑦 Carlson-integral-RC π‘₯ 𝑦 {\displaystyle{\displaystyle R_{F}\left(x,y,y\right)=R_{C}\left(x,y\right)}}
\CarlsonsymellintRF@{x}{y}{y} = \CarlsonellintRC@{x}{y}		 

Error

EllipticF[ArcCos[Sqrt[x/y]],(y-y)/(y-x)]/Sqrt[y-x] == 1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)]
Missing Macro Error Failure -
Failed [3 / 18]
Result: Indeterminate
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}


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

... skip entries to safe data
19.20#Ex4 R F ⁑ ( 0 , y , y ) = 1 2 ⁒ Ο€ ⁒ y - 1 / 2 Carlson-integral-RF 0 𝑦 𝑦 1 2 πœ‹ superscript 𝑦 1 2 {\displaystyle{\displaystyle R_{F}\left(0,y,y\right)=\tfrac{1}{2}\pi y^{-1/2}}}
\CarlsonsymellintRF@{0}{y}{y} = \tfrac{1}{2}\pi y^{-1/2}		 

0.5*int(1/(sqrt(t+0)*sqrt(t+y)*sqrt(t+y)), t = 0..infinity) = (1)/(2)*Pi*(y)^(- 1/2)

EllipticF[ArcCos[Sqrt[0/y]],(y-y)/(y-0)]/Sqrt[y-0] == Divide[1,2]*Pi*(y)^(- 1/2)
Failure Successful
Failed [3 / 6]
Result: 2.565099660*I
Test Values: {y = -3/2}


Result: 4.442882938*I
Test Values: {y = -1/2}

... skip entries to safe data
 Successful [Tested: 6]
19.20#Ex5 R F ⁑ ( 0 , 0 , z ) = ∞ Carlson-integral-RF 0 0 𝑧 {\displaystyle{\displaystyle R_{F}\left(0,0,z\right)=\infty}}
\CarlsonsymellintRF@{0}{0}{z} = \infty		 

0.5*int(1/(sqrt(t+0)*sqrt(t+0)*sqrt(t+z)), t = 0..infinity) = infinity

EllipticF[ArcCos[Sqrt[0/z]],(z-0)/(z-0)]/Sqrt[z-0] == Infinity
Failure Failure Skipped - Because timed out
Failed [7 / 7]
Result: Indeterminate
Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Indeterminate
Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.20.E2 ∫ 0 1 d t 1 - t 4 = R F ⁑ ( 0 , 1 , 2 ) superscript subscript 0 1 𝑑 1 superscript 𝑑 4 Carlson-integral-RF 0 1 2 {\displaystyle{\displaystyle\int_{0}^{1}\frac{\mathrm{d}t}{\sqrt{1-t^{4}}}=R_{% F}\left(0,1,2\right)}}
\int_{0}^{1}\frac{\diff{t}}{\sqrt{1-t^{4}}} = \CarlsonsymellintRF@{0}{1}{2}		 

int((1)/(sqrt(1 - (t)^(4))), t = 0..1) = 0.5*int(1/(sqrt(t+0)*sqrt(t+1)*sqrt(t+2)), t = 0..infinity)

Integrate[Divide[1,Sqrt[1 - (t)^(4)]], {t, 0, 1}, GenerateConditions->None] == EllipticF[ArcCos[Sqrt[0/2]],(2-1)/(2-0)]/Sqrt[2-0]
Failure Successful Successful [Tested: 0] Successful [Tested: 1]
19.20.E2 R F ⁑ ( 0 , 1 , 2 ) = ( Ξ“ ⁑ ( 1 4 ) ) 2 4 ⁒ ( 2 ⁒ Ο€ ) 1 / 2 Carlson-integral-RF 0 1 2 superscript Euler-Gamma 1 4 2 4 superscript 2 πœ‹ 1 2 {\displaystyle{\displaystyle R_{F}\left(0,1,2\right)=\frac{\left(\Gamma\left(% \frac{1}{4}\right)\right)^{2}}{4(2\pi)^{1/2}}}}
\CarlsonsymellintRF@{0}{1}{2} = \frac{\left(\EulerGamma@{\frac{1}{4}}\right)^{2}}{4(2\pi)^{1/2}}		 

0.5*int(1/(sqrt(t+0)*sqrt(t+1)*sqrt(t+2)), t = 0..infinity) = ((GAMMA((1)/(4)))^(2))/(4*(2*Pi)^(1/2))

EllipticF[ArcCos[Sqrt[0/2]],(2-1)/(2-0)]/Sqrt[2-0] == Divide[(Gamma[Divide[1,4]])^(2),4*(2*Pi)^(1/2)]
Successful Failure Skip - symbolical successful subtest Successful [Tested: 1]
19.20.E2 ( Ξ“ ⁑ ( 1 4 ) ) 2 4 ⁒ ( 2 ⁒ Ο€ ) 1 / 2 = 1.31102 87771 46059 90523 ⁒ … superscript Euler-Gamma 1 4 2 4 superscript 2 πœ‹ 1 2 1.31102 87771 46059 90523 … {\displaystyle{\displaystyle\frac{\left(\Gamma\left(\frac{1}{4}\right)\right)^% {2}}{4(2\pi)^{1/2}}=1.31102\;87771\;46059\;90523\;\dots}}
\frac{\left(\EulerGamma@{\frac{1}{4}}\right)^{2}}{4(2\pi)^{1/2}} = 1.31102\;87771\;46059\;90523\;\dots		 

((GAMMA((1)/(4)))^(2))/(4*(2*Pi)^(1/2)) = 1.31102877714605990523

Divide[(Gamma[Divide[1,4]])^(2),4*(2*Pi)^(1/2)] == 1.31102877714605990523
Failure Successful Successful [Tested: 0] Successful [Tested: 1]
19.20#Ex6 R G ⁑ ( x , x , x ) = x 1 / 2 Carlson-integral-RG π‘₯ π‘₯ π‘₯ superscript π‘₯ 1 2 {\displaystyle{\displaystyle R_{G}\left(x,x,x\right)=x^{1/2}}}
\CarlsonsymellintRG@{x}{x}{x} = x^{1/2}		 

Error

Sqrt[x-x]*(EllipticE[ArcCos[Sqrt[x/x]],(x-x)/(x-x)]+(Cot[ArcCos[Sqrt[x/x]]])^2*EllipticF[ArcCos[Sqrt[x/x]],(x-x)/(x-x)]+Cot[ArcCos[Sqrt[x/x]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[x/x]]]^2]) == (x)^(1/2)
Missing Macro Error Failure -
Failed [3 / 3]
Result: Indeterminate
Test Values: {Rule[x, 1.5]}


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

... skip entries to safe data
19.20#Ex7 R G ⁑ ( Ξ» ⁒ x , Ξ» ⁒ y , Ξ» ⁒ z ) = Ξ» 1 / 2 ⁒ R G ⁑ ( x , y , z ) Carlson-integral-RG πœ† π‘₯ πœ† 𝑦 πœ† 𝑧 superscript πœ† 1 2 Carlson-integral-RG π‘₯ 𝑦 𝑧 {\displaystyle{\displaystyle R_{G}\left(\lambda x,\lambda y,\lambda z\right)=% \lambda^{1/2}R_{G}\left(x,y,z\right)}}
\CarlsonsymellintRG@{\lambda x}{\lambda y}{\lambda z} = \lambda^{1/2}\CarlsonsymellintRG@{x}{y}{z}		 

Error

Sqrt[\[Lambda]*(x + y*I)-\[Lambda]*x]*(EllipticE[ArcCos[Sqrt[\[Lambda]*x/\[Lambda]*(x + y*I)]],(\[Lambda]*(x + y*I)-\[Lambda]*y)/(\[Lambda]*(x + y*I)-\[Lambda]*x)]+(Cot[ArcCos[Sqrt[\[Lambda]*x/\[Lambda]*(x + y*I)]]])^2*EllipticF[ArcCos[Sqrt[\[Lambda]*x/\[Lambda]*(x + y*I)]],(\[Lambda]*(x + y*I)-\[Lambda]*y)/(\[Lambda]*(x + y*I)-\[Lambda]*x)]+Cot[ArcCos[Sqrt[\[Lambda]*x/\[Lambda]*(x + y*I)]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[\[Lambda]*x/\[Lambda]*(x + y*I)]]]^2]) == \[Lambda]^(1/2)* 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])
Missing Macro Error Aborted -
Failed [180 / 180]
Result: Plus[Times[Complex[-0.75, 0.4330127018922193], 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[0.75, -0.43301270189221935], Plus[Complex[0.469094970899074, 0.7900882534928779], Times[Complex[0.1542171038749957, -1.1011185950707625], Power[Plus[1.0, Times[Complex[1.25, -2.25], Power[k, 2]]], Rational[1, 2]]]]]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[Ξ», Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Plus[Times[Complex[-0.8365163037378078, -0.22414386804201336], 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[0.8365163037378078, 0.22414386804201325], Plus[Complex[0.46909497089907387, 0.7900882534928779], Times[Complex[0.1542171038749957, -1.1011185950707625], Power[Plus[1.0, Times[Complex[1.25, -2.25], Power[k, 2]]], Rational[1, 2]]]]]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[Ξ», Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.20#Ex8 R G ⁑ ( 0 , y , y ) = 1 4 ⁒ Ο€ ⁒ y 1 / 2 Carlson-integral-RG 0 𝑦 𝑦 1 4 πœ‹ superscript 𝑦 1 2 {\displaystyle{\displaystyle R_{G}\left(0,y,y\right)=\tfrac{1}{4}\pi y^{1/2}}}
\CarlsonsymellintRG@{0}{y}{y} = \tfrac{1}{4}\pi y^{1/2}		 

Error

Sqrt[y-0]*(EllipticE[ArcCos[Sqrt[0/y]],(y-y)/(y-0)]+(Cot[ArcCos[Sqrt[0/y]]])^2*EllipticF[ArcCos[Sqrt[0/y]],(y-y)/(y-0)]+Cot[ArcCos[Sqrt[0/y]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/y]]]^2]) == Divide[1,4]*Pi*(y)^(1/2)
Missing Macro Error Failure -
Failed [6 / 6]
Result: Complex[0.0, 0.961912372621398]
Test Values: {Rule[y, -1.5]}


Result: 0.961912372621398
Test Values: {Rule[y, 1.5]}

... skip entries to safe data
19.20#Ex9 R G ⁑ ( 0 , 0 , z ) = 1 2 ⁒ z 1 / 2 Carlson-integral-RG 0 0 𝑧 1 2 superscript 𝑧 1 2 {\displaystyle{\displaystyle R_{G}\left(0,0,z\right)=\tfrac{1}{2}z^{1/2}}}
\CarlsonsymellintRG@{0}{0}{z} = \tfrac{1}{2}z^{1/2}		 

Error

Sqrt[z-0]*(EllipticE[ArcCos[Sqrt[0/z]],(z-0)/(z-0)]+(Cot[ArcCos[Sqrt[0/z]]])^2*EllipticF[ArcCos[Sqrt[0/z]],(z-0)/(z-0)]+Cot[ArcCos[Sqrt[0/z]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/z]]]^2]) == Divide[1,2]*(z)^(1/2)
Missing Macro Error Failure -
Failed [7 / 7]
Result: Indeterminate
Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Indeterminate
Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.20.E5 2 ⁒ R G ⁑ ( x , y , y ) = y ⁒ R C ⁑ ( x , y ) + x 2 Carlson-integral-RG π‘₯ 𝑦 𝑦 𝑦 Carlson-integral-RC π‘₯ 𝑦 π‘₯ {\displaystyle{\displaystyle 2R_{G}\left(x,y,y\right)=yR_{C}\left(x,y\right)+% \sqrt{x}}}
2\CarlsonsymellintRG@{x}{y}{y} = y\CarlsonellintRC@{x}{y}+\sqrt{x}		 

Error

2*Sqrt[y-x]*(EllipticE[ArcCos[Sqrt[x/y]],(y-y)/(y-x)]+(Cot[ArcCos[Sqrt[x/y]]])^2*EllipticF[ArcCos[Sqrt[x/y]],(y-y)/(y-x)]+Cot[ArcCos[Sqrt[x/y]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[x/y]]]^2]) == y*1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)]+Sqrt[x]
Missing Macro Error Failure -
Failed [18 / 18]
Result: Plus[Complex[-1.988036787975128, -1.360349523175663], Times[Complex[0.0, 3.4641016151377544], Plus[Complex[0.7853981633974483, -0.44068679350977147], Times[Complex[0.0, 0.7071067811865475], Power[Plus[1.0, Times[-2.0, Power[k, 2]]], Rational[1, 2]]]]]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}


Result: Indeterminate
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
19.20#Ex10 R J ⁑ ( x , x , x , x ) = x - 3 / 2 Carlson-integral-RJ π‘₯ π‘₯ π‘₯ π‘₯ superscript π‘₯ 3 2 {\displaystyle{\displaystyle R_{J}\left(x,x,x,x\right)=x^{-3/2}}}
\CarlsonsymellintRJ@{x}{x}{x}{x} = x^{-3/2}		 

Error

3*(x-x)/(x-x)*(EllipticPi[(x-x)/(x-x),ArcCos[Sqrt[x/x]],(x-x)/(x-x)]-EllipticF[ArcCos[Sqrt[x/x]],(x-x)/(x-x)])/Sqrt[x-x] == (x)^(- 3/2)
Missing Macro Error Failure -
Failed [3 / 3]
Result: Indeterminate
Test Values: {Rule[x, 1.5]}


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

... skip entries to safe data
19.20#Ex11 R J ⁑ ( Ξ» ⁒ x , Ξ» ⁒ y , Ξ» ⁒ z , Ξ» ⁒ p ) = Ξ» - 3 / 2 ⁒ R J ⁑ ( x , y , z , p ) Carlson-integral-RJ πœ† π‘₯ πœ† 𝑦 πœ† 𝑧 πœ† 𝑝 superscript πœ† 3 2 Carlson-integral-RJ π‘₯ 𝑦 𝑧 𝑝 {\displaystyle{\displaystyle R_{J}\left(\lambda x,\lambda y,\lambda z,\lambda p% \right)=\lambda^{-3/2}R_{J}\left(x,y,z,p\right)}}
\CarlsonsymellintRJ@{\lambda x}{\lambda y}{\lambda z}{\lambda p} = \lambda^{-3/2}\CarlsonsymellintRJ@{x}{y}{z}{p}		 

Error

3*(\[Lambda]*(x + y*I)-\[Lambda]*x)/(\[Lambda]*(x + y*I)-\[Lambda]*p)*(EllipticPi[(\[Lambda]*(x + y*I)-\[Lambda]*p)/(\[Lambda]*(x + y*I)-\[Lambda]*x),ArcCos[Sqrt[\[Lambda]*x/\[Lambda]*(x + y*I)]],(\[Lambda]*(x + y*I)-\[Lambda]*y)/(\[Lambda]*(x + y*I)-\[Lambda]*x)]-EllipticF[ArcCos[Sqrt[\[Lambda]*x/\[Lambda]*(x + y*I)]],(\[Lambda]*(x + y*I)-\[Lambda]*y)/(\[Lambda]*(x + y*I)-\[Lambda]*x)])/Sqrt[\[Lambda]*(x + y*I)-\[Lambda]*x] == \[Lambda]^(- 3/2)* 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]
Missing Macro Error Failure -
Failed [300 / 300]
Result: Complex[0.8261798979421457, -0.5239696989052641]
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[Ξ», Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-1.5256524914787406, -1.066611458671583]
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[Ξ», Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.20#Ex12 R J ⁑ ( x , y , z , z ) = R D ⁑ ( x , y , z ) Carlson-integral-RJ π‘₯ 𝑦 𝑧 𝑧 Carlson-integral-RD π‘₯ 𝑦 𝑧 {\displaystyle{\displaystyle R_{J}\left(x,y,z,z\right)=R_{D}\left(x,y,z\right)}}
\CarlsonsymellintRJ@{x}{y}{z}{z} = \CarlsonsymellintRD@{x}{y}{z}		 

Error

3*(x + y*I-x)/(x + y*I-x + y*I)*(EllipticPi[(x + y*I-x + y*I)/(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*(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))
Missing Macro Error Failure -
Failed [18 / 18]
Result: Complex[-0.37100270206594405, 0.09129381935817127]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}


Result: Complex[-0.5182279531589904, -0.0513630200054771]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
19.20#Ex13 R J ⁑ ( 0 , 0 , z , p ) = ∞ Carlson-integral-RJ 0 0 𝑧 𝑝 {\displaystyle{\displaystyle R_{J}\left(0,0,z,p\right)=\infty}}
\CarlsonsymellintRJ@{0}{0}{z}{p} = \infty		 

Error

3*(z-0)/(z-p)*(EllipticPi[(z-p)/(z-0),ArcCos[Sqrt[0/z]],(z-0)/(z-0)]-EllipticF[ArcCos[Sqrt[0/z]],(z-0)/(z-0)])/Sqrt[z-0] == Infinity
Missing Macro Error Failure -
Failed [70 / 70]
Result: Indeterminate
Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[z, 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[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.20#Ex14 R J ⁑ ( x , x , x , p ) = R D ⁑ ( p , p , x ) Carlson-integral-RJ π‘₯ π‘₯ π‘₯ 𝑝 Carlson-integral-RD 𝑝 𝑝 π‘₯ {\displaystyle{\displaystyle R_{J}\left(x,x,x,p\right)=R_{D}\left(p,p,x\right)}}
\CarlsonsymellintRJ@{x}{x}{x}{p} = \CarlsonsymellintRD@{p}{p}{x}		 x β‰  p , x ⁒ p β‰  0 formulae-sequence π‘₯ 𝑝 π‘₯ 𝑝 0 {\displaystyle{\displaystyle x\neq p,xp\neq 0}} 
Error

3*(x-x)/(x-p)*(EllipticPi[(x-p)/(x-x),ArcCos[Sqrt[x/x]],(x-x)/(x-x)]-EllipticF[ArcCos[Sqrt[x/x]],(x-x)/(x-x)])/Sqrt[x-x] == 3*(EllipticF[ArcCos[Sqrt[p/x]],(x-p)/(x-p)]-EllipticE[ArcCos[Sqrt[p/x]],(x-p)/(x-p)])/((x-p)*(x-p)^(1/2))
Missing Macro Error Failure -
Failed [27 / 27]
Result: Indeterminate
Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5]}


Result: Indeterminate
Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 0.5]}

... skip entries to safe data
19.20#Ex14 R D ⁑ ( p , p , x ) = 3 x - p ⁒ ( R C ⁑ ( x , p ) - 1 x ) Carlson-integral-RD 𝑝 𝑝 π‘₯ 3 π‘₯ 𝑝 Carlson-integral-RC π‘₯ 𝑝 1 π‘₯ {\displaystyle{\displaystyle R_{D}\left(p,p,x\right)=\frac{3}{x-p}\left(R_{C}% \left(x,p\right)-\frac{1}{\sqrt{x}}\right)}}
\CarlsonsymellintRD@{p}{p}{x} = \frac{3}{x-p}\left(\CarlsonellintRC@{x}{p}-\frac{1}{\sqrt{x}}\right)		 x β‰  p , x ⁒ p β‰  0 formulae-sequence π‘₯ 𝑝 π‘₯ 𝑝 0 {\displaystyle{\displaystyle x\neq p,xp\neq 0}} 
Error

3*(EllipticF[ArcCos[Sqrt[p/x]],(x-p)/(x-p)]-EllipticE[ArcCos[Sqrt[p/x]],(x-p)/(x-p)])/((x-p)*(x-p)^(1/2)) == Divide[3,x - p]*(1/Sqrt[p]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(p)]-Divide[1,Sqrt[x]])
Missing Macro Error Failure -
Failed [9 / 27]
Result: Complex[1.0177225554447191, 2.220446049250313*^-16]
Test Values: {Rule[p, -1.5], Rule[x, 1.5]}


Result: Complex[1.1652542988181402, 6.661338147750939*^-16]
Test Values: {Rule[p, -1.5], Rule[x, 0.5]}

... skip entries to safe data
19.20#Ex15 R J ⁑ ( 0 , y , y , p ) = 3 ⁒ Ο€ 2 ⁒ ( y ⁒ p + p ⁒ y ) Carlson-integral-RJ 0 𝑦 𝑦 𝑝 3 πœ‹ 2 𝑦 𝑝 𝑝 𝑦 {\displaystyle{\displaystyle R_{J}\left(0,y,y,p\right)=\frac{3\pi}{2(y\sqrt{p}% +p\sqrt{y})}}}
\CarlsonsymellintRJ@{0}{y}{y}{p} = \frac{3\pi}{2(y\sqrt{p}+p\sqrt{y})}		 p > 0 𝑝 0 {\displaystyle{\displaystyle p>0}} 
Error

3*(y-0)/(y-p)*(EllipticPi[(y-p)/(y-0),ArcCos[Sqrt[0/y]],(y-y)/(y-0)]-EllipticF[ArcCos[Sqrt[0/y]],(y-y)/(y-0)])/Sqrt[y-0] == Divide[3*Pi,2*(y*Sqrt[p]+ p*Sqrt[y])]
Missing Macro Error Failure -
Failed [18 / 18]
Result: Complex[-0.6412749150809316, 3.2063745754046598]
Test Values: {Rule[p, 1.5], Rule[y, -1.5]}


Result: Indeterminate
Test Values: {Rule[p, 1.5], Rule[y, 1.5]}

... skip entries to safe data
19.20#Ex16 R J ⁑ ( 0 , y , y , - q ) = - 3 ⁒ Ο€ 2 ⁒ y ⁒ ( y + q ) Carlson-integral-RJ 0 𝑦 𝑦 π‘ž 3 πœ‹ 2 𝑦 𝑦 π‘ž {\displaystyle{\displaystyle R_{J}\left(0,y,y,-q\right)=\frac{-3\pi}{2\sqrt{y}% (y+q)}}}
\CarlsonsymellintRJ@{0}{y}{y}{-q} = \frac{-3\pi}{2\sqrt{y}(y+q)}		 q > 0 π‘ž 0 {\displaystyle{\displaystyle q>0}} 
Error

3*(y-0)/(y-- q)*(EllipticPi[(y-- q)/(y-0),ArcCos[Sqrt[0/y]],(y-y)/(y-0)]-EllipticF[ArcCos[Sqrt[0/y]],(y-y)/(y-0)])/Sqrt[y-0] == Divide[- 3*Pi,2*Sqrt[y]*(y + q)]
Missing Macro Error Failure -
Failed [18 / 18]
Result: DirectedInfinity[]
Test Values: {Rule[q, 1.5], Rule[y, -1.5]}


Result: Plus[1.282549830161864, Times[2.449489742783178, Plus[-1.5707963267948966, Times[1.5707963267948966, Power[Plus[1.0, Times[-1.0, Decrement[1.5]]], Rational[-1, 2]]]], Power[Decrement[1.5], -1]]]
Test Values: {Rule[q, 1.5], Rule[y, 1.5]}

... skip entries to safe data
19.20#Ex17 R J ⁑ ( x , y , y , p ) = 3 p - y ⁒ ( R C ⁑ ( x , y ) - R C ⁑ ( x , p ) ) Carlson-integral-RJ π‘₯ 𝑦 𝑦 𝑝 3 𝑝 𝑦 Carlson-integral-RC π‘₯ 𝑦 Carlson-integral-RC π‘₯ 𝑝 {\displaystyle{\displaystyle R_{J}\left(x,y,y,p\right)=\frac{3}{p-y}(R_{C}% \left(x,y\right)-R_{C}\left(x,p\right))}}
\CarlsonsymellintRJ@{x}{y}{y}{p} = \frac{3}{p-y}(\CarlsonellintRC@{x}{y}-\CarlsonellintRC@{x}{p})		 p β‰  y 𝑝 𝑦 {\displaystyle{\displaystyle p\neq y}} 
Error
 
3*(y-x)/(y-p)*(EllipticPi[(y-p)/(y-x),ArcCos[Sqrt[x/y]],(y-y)/(y-x)]-EllipticF[ArcCos[Sqrt[x/y]],(y-y)/(y-x)])/Sqrt[y-x] == Divide[3,p - y]*(1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)]- 1/Sqrt[p]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(p)])
 Missing Macro Error Aborted -
Failed [157 / 162]
Result: Complex[0.40904124998304914, 6.107600792054881]
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]}

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]}

... skip entries to safe data
19.20#Ex18 R J ⁑ ( x , y , y , y ) = R D ⁑ ( x , y , y ) Carlson-integral-RJ π‘₯ 𝑦 𝑦 𝑦 Carlson-integral-RD π‘₯ 𝑦 𝑦 {\displaystyle{\displaystyle R_{J}\left(x,y,y,y\right)=R_{D}\left(x,y,y\right)}}
\CarlsonsymellintRJ@{x}{y}{y}{y} = \CarlsonsymellintRD@{x}{y}{y}		 

Error
 
3*(y-x)/(y-y)*(EllipticPi[(y-y)/(y-x),ArcCos[Sqrt[x/y]],(y-y)/(y-x)]-EllipticF[ArcCos[Sqrt[x/y]],(y-y)/(y-x)])/Sqrt[y-x] == 3*(EllipticF[ArcCos[Sqrt[x/y]],(y-y)/(y-x)]-EllipticE[ArcCos[Sqrt[x/y]],(y-y)/(y-x)])/((y-y)*(y-x)^(1/2))
 Missing Macro Error Failure -
Failed [18 / 18]
Result: Indeterminate
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}

Result: Indeterminate
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
19.20.E9 R J ⁑ ( 0 , y , z , + y ⁒ z ) = + 3 2 ⁒ y ⁒ z ⁒ R F ⁑ ( 0 , y , z ) Carlson-integral-RJ 0 𝑦 𝑧 𝑦 𝑧 3 2 𝑦 𝑧 Carlson-integral-RF 0 𝑦 𝑧 {\displaystyle{\displaystyle R_{J}\left(0,y,z,+\sqrt{yz}\right)=+\frac{3}{2% \sqrt{yz}}R_{F}\left(0,y,z\right)}}
\CarlsonsymellintRJ@{0}{y}{z}{+\sqrt{yz}} = +\frac{3}{2\sqrt{yz}}\CarlsonsymellintRF@{0}{y}{z}		 

Error
 
3*(x + y*I-0)/(x + y*I-+Sqrt[y*(x + y*I)])*(EllipticPi[(x + y*I-+Sqrt[y*(x + y*I)])/(x + y*I-0),ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/Sqrt[x + y*I-0] == +Divide[3,2*Sqrt[y*(x + y*I)]]*EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]/Sqrt[x + y*I-0]
 Missing Macro Error Failure -
Failed [18 / 18]
Result: Complex[-0.9141259292931587, -0.9706303463287326]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}

Result: Complex[-4.407772019377616, 0.7576222483343515]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
19.20.E9 R J ⁑ ( 0 , y , z , - y ⁒ z ) = - 3 2 ⁒ y ⁒ z ⁒ R F ⁑ ( 0 , y , z ) Carlson-integral-RJ 0 𝑦 𝑧 𝑦 𝑧 3 2 𝑦 𝑧 Carlson-integral-RF 0 𝑦 𝑧 {\displaystyle{\displaystyle R_{J}\left(0,y,z,-\sqrt{yz}\right)=-\frac{3}{2% \sqrt{yz}}R_{F}\left(0,y,z\right)}}
\CarlsonsymellintRJ@{0}{y}{z}{-\sqrt{yz}} = -\frac{3}{2\sqrt{yz}}\CarlsonsymellintRF@{0}{y}{z}		 

Error
 
3*(x + y*I-0)/(x + y*I--Sqrt[y*(x + y*I)])*(EllipticPi[(x + y*I--Sqrt[y*(x + y*I)])/(x + y*I-0),ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/Sqrt[x + y*I-0] == -Divide[3,2*Sqrt[y*(x + y*I)]]*EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]/Sqrt[x + y*I-0]
 Missing Macro Error Failure -
Failed [18 / 18]
Result: Complex[0.1671030668705316, -0.09828926199489627]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}

Result: Complex[-0.7387931095854892, 1.0731895314108653]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
19.20#Ex19 lim p β†’ 0 + ⁑ p ⁒ R J ⁑ ( 0 , y , z , p ) = 3 ⁒ Ο€ 2 ⁒ y ⁒ z subscript β†’ 𝑝 limit-from 0 𝑝 Carlson-integral-RJ 0 𝑦 𝑧 𝑝 3 πœ‹ 2 𝑦 𝑧 {\displaystyle{\displaystyle\lim_{p\to 0+}\sqrt{p}R_{J}\left(0,y,z,p\right)=% \frac{3\pi}{2\sqrt{yz}}}}
\lim_{p\to 0+}\sqrt{p}\CarlsonsymellintRJ@{0}{y}{z}{p} = \frac{3\pi}{2\sqrt{yz}}		 

Error
 
Limit[Sqrt[p]*3*(x + y*I-0)/(x + y*I-p)*(EllipticPi[(x + y*I-p)/(x + y*I-0),ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/Sqrt[x + y*I-0], p -> 0, Direction -> "FromAbove", GenerateConditions->None] == Divide[3*Pi,2*Sqrt[y*(x + y*I)]]
 Missing Macro Error Aborted - Skipped - Because timed out
19.20#Ex20 lim p β†’ 0 - ⁑ R J ⁑ ( 0 , y , z , p ) = - R D ⁑ ( 0 , y , z ) - R D ⁑ ( 0 , z , y ) subscript β†’ 𝑝 limit-from 0 Carlson-integral-RJ 0 𝑦 𝑧 𝑝 Carlson-integral-RD 0 𝑦 𝑧 Carlson-integral-RD 0 𝑧 𝑦 {\displaystyle{\displaystyle\lim_{p\to 0-}R_{J}\left(0,y,z,p\right)={-R_{D}% \left(0,y,z\right)-R_{D}\left(0,z,y\right)}}}
\lim_{p\to 0-}\CarlsonsymellintRJ@{0}{y}{z}{p} = {-\CarlsonsymellintRD@{0}{y}{z}-\CarlsonsymellintRD@{0}{z}{y}}		 

Error
 
Limit[3*(x + y*I-0)/(x + y*I-p)*(EllipticPi[(x + y*I-p)/(x + y*I-0),ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/Sqrt[x + y*I-0], p -> 0, Direction -> "FromBelow", GenerateConditions->None] == - 3*(EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticE[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/((x + y*I-y)*(x + y*I-0)^(1/2))- 3*(EllipticF[ArcCos[Sqrt[0/y]],(y-x + y*I)/(y-0)]-EllipticE[ArcCos[Sqrt[0/y]],(y-x + y*I)/(y-0)])/((y-x + y*I)*(y-0)^(1/2))
 Missing Macro Error Aborted - Skipped - Because timed out
19.20#Ex20 - R D ⁑ ( 0 , y , z ) - R D ⁑ ( 0 , z , y ) = - 6 y ⁒ z ⁒ R G ⁑ ( 0 , y , z ) Carlson-integral-RD 0 𝑦 𝑧 Carlson-integral-RD 0 𝑧 𝑦 6 𝑦 𝑧 Carlson-integral-RG 0 𝑦 𝑧 {\displaystyle{\displaystyle{-R_{D}\left(0,y,z\right)-R_{D}\left(0,z,y\right)}% =\frac{-6}{yz}R_{G}\left(0,y,z\right)}}
{-\CarlsonsymellintRD@{0}{y}{z}-\CarlsonsymellintRD@{0}{z}{y}} = \frac{-6}{yz}\CarlsonsymellintRG@{0}{y}{z}		 

Error
 
- 3*(EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticE[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/((x + y*I-y)*(x + y*I-0)^(1/2))- 3*(EllipticF[ArcCos[Sqrt[0/y]],(y-x + y*I)/(y-0)]-EllipticE[ArcCos[Sqrt[0/y]],(y-x + y*I)/(y-0)])/((y-x + y*I)*(y-0)^(1/2)) == Divide[- 6,y*(x + y*I)]*Sqrt[x + y*I-0]*(EllipticE[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]+(Cot[ArcCos[Sqrt[0/x + y*I]]])^2*EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]+Cot[ArcCos[Sqrt[0/x + y*I]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/x + y*I]]]^2])
 Missing Macro Error Failure -
Failed [18 / 18]
Result: Plus[Complex[1.5111033799217843, -0.47027281525563985], Times[Complex[-2.537302274660022, -1.050985014004285], Plus[Complex[1.465481142300126, -0.24396122198922798], Times[Complex[0.2643318009908678, -0.8730286325904596], Power[Plus[1.0, Times[Complex[-1.0, -1.5], Power[k, 2]]], Rational[1, 2]]]]]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}

Result: Plus[Complex[-0.13967540286775149, -0.9399293972008751], Times[Complex[2.537302274660022, -1.050985014004285], Plus[Complex[1.0084590214609772, 0.7147093671486319], Times[Complex[0.2643318009908678, 0.8730286325904596], Power[Plus[1.0, Times[Complex[-1.0, 1.5], Power[k, 2]]], Rational[1, 2]]]]]]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
19.20.E12 lim p β†’ + ∞ ⁑ p ⁒ R J ⁑ ( x , y , z , p ) = 3 ⁒ R F ⁑ ( x , y , z ) subscript β†’ 𝑝 𝑝 Carlson-integral-RJ π‘₯ 𝑦 𝑧 𝑝 3 Carlson-integral-RF π‘₯ 𝑦 𝑧 {\displaystyle{\displaystyle\lim_{p\to+\infty}pR_{J}\left(x,y,z,p\right)=3R_{F% }\left(x,y,z\right)}}
\lim_{p\to+\infty}p\CarlsonsymellintRJ@{x}{y}{z}{p} = 3\CarlsonsymellintRF@{x}{y}{z}		 

Error
 
Limit[p*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], p -> + Infinity, GenerateConditions->None] == 3*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]
 Missing Macro Error Aborted - Skipped - Because timed out
19.20.E12 lim p β†’ - ∞ ⁑ p ⁒ R J ⁑ ( x , y , z , p ) = 3 ⁒ R F ⁑ ( x , y , z ) subscript β†’ 𝑝 𝑝 Carlson-integral-RJ π‘₯ 𝑦 𝑧 𝑝 3 Carlson-integral-RF π‘₯ 𝑦 𝑧 {\displaystyle{\displaystyle\lim_{p\to-\infty}pR_{J}\left(x,y,z,p\right)=3R_{F% }\left(x,y,z\right)}}
\lim_{p\to-\infty}p\CarlsonsymellintRJ@{x}{y}{z}{p} = 3\CarlsonsymellintRF@{x}{y}{z}		 

Error
 
Limit[p*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], p -> - Infinity, GenerateConditions->None] == 3*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]
 Missing Macro Error Aborted - Skipped - Because timed out
19.20.E13 2 ⁒ ( p - x ) ⁒ R J ⁑ ( x , y , z , p ) = 3 ⁒ R F ⁑ ( x , y , z ) - 3 ⁒ x ⁒ R C ⁑ ( y ⁒ z , p 2 ) 2 𝑝 π‘₯ Carlson-integral-RJ π‘₯ 𝑦 𝑧 𝑝 3 Carlson-integral-RF π‘₯ 𝑦 𝑧 3 π‘₯ Carlson-integral-RC 𝑦 𝑧 superscript 𝑝 2 {\displaystyle{\displaystyle 2(p-x)R_{J}\left(x,y,z,p\right)=3R_{F}\left(x,y,z% \right)-3\sqrt{x}R_{C}\left(yz,p^{2}\right)}}
2(p-x)\CarlsonsymellintRJ@{x}{y}{z}{p} = 3\CarlsonsymellintRF@{x}{y}{z}-3\sqrt{x}\CarlsonellintRC@{yz}{p^{2}}		 

Error
 
2*(p - x)*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*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]- 3*Sqrt[x]*1/Sqrt[(p)^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-(y*(x + y*I))/((p)^(2))]
 Missing Macro Error Aborted -
Failed [180 / 180]
Result: Complex[3.989482635019833, -4.816521080718802]
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]}

Result: Complex[5.152296981249878, -0.7434346709776987]
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]}

... skip entries to safe data
19.20.E14 ( q + z ) ⁒ R J ⁑ ( x , y , z , - q ) = ( p - z ) ⁒ R J ⁑ ( x , y , z , p ) - 3 ⁒ R F ⁑ ( x , y , z ) + 3 ⁒ ( x ⁒ y ⁒ z x ⁒ y + p ⁒ q ) 1 / 2 ⁒ R C ⁑ ( x ⁒ y + p ⁒ q , p ⁒ q ) π‘ž 𝑧 Carlson-integral-RJ π‘₯ 𝑦 𝑧 π‘ž 𝑝 𝑧 Carlson-integral-RJ π‘₯ 𝑦 𝑧 𝑝 3 Carlson-integral-RF π‘₯ 𝑦 𝑧 3 superscript π‘₯ 𝑦 𝑧 π‘₯ 𝑦 𝑝 π‘ž 1 2 Carlson-integral-RC π‘₯ 𝑦 𝑝 π‘ž 𝑝 π‘ž {\displaystyle{\displaystyle(q+z)R_{J}\left(x,y,z,-q\right)=(p-z)R_{J}\left(x,% y,z,p\right)-3R_{F}\left(x,y,z\right)+3\left(\frac{xyz}{xy+pq}\right)^{1/2}R_{% C}\left(xy+pq,pq\right)}}
(q+z)\CarlsonsymellintRJ@{x}{y}{z}{-q} = (p-z)\CarlsonsymellintRJ@{x}{y}{z}{p}-3\CarlsonsymellintRF@{x}{y}{z}+3\left(\frac{xyz}{xy+pq}\right)^{1/2}\CarlsonellintRC@{xy+pq}{pq}		 

Error
 
(q +(x + y*I))*3*(x + y*I-x)/(x + y*I-- q)*(EllipticPi[(x + y*I-- q)/(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] == (p -(x + y*I))*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*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]+ 3*(Divide[x*y*(x + y*I),x*y + p*q])^(1/2)* 1/Sqrt[p*q]*Hypergeometric2F1[1/2,1/2,3/2,1-(x*y + p*q)/(p*q)]
 Missing Macro Error Aborted -
Failed [300 / 300]
Result: Complex[-3.4116287326863786, 8.252883937385896]
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]}

Result: Complex[-8.900891250450524, -2.579723477019983]
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]}

... skip entries to safe data
19.20#Ex21 q > 0 π‘ž 0 {\displaystyle{\displaystyle q>0}}
q > 0		 

q > 0
 
q > 0
 Skipped - no semantic math Skipped - no semantic math - -
19.20#Ex22 p = z ⁒ ( x + y + q ) - x ⁒ y z + q 𝑝 𝑧 π‘₯ 𝑦 π‘ž π‘₯ 𝑦 𝑧 π‘ž {\displaystyle{\displaystyle p=\frac{z(x+y+q)-xy}{z+q}}}
p = \frac{z(x+y+q)-xy}{z+q}		 

p = ((x + y*I)*(x + y + q)- x*y)/((x + y*I)+ q)
 
p == Divide[(x + y*I)*(x + y + q)- x*y,(x + y*I)+ q]
 Skipped - no semantic math Skipped - no semantic math - -
19.20#Ex23 p = w ⁒ y + ( 1 - w ) ⁒ z 𝑝 𝑀 𝑦 1 𝑀 𝑧 {\displaystyle{\displaystyle p=wy+(1-w)z}}
p = wy+(1-w)z		 

p = w*y +(1 - w)*(x + y*I)
 
p == w*y +(1 - w)*(x + y*I)
 Skipped - no semantic math Skipped - no semantic math - -
19.20#Ex24 w = z - x z + q 𝑀 𝑧 π‘₯ 𝑧 π‘ž {\displaystyle{\displaystyle w=\frac{z-x}{z+q}}}
w = \frac{z-x}{z+q}		 

w = ((x + y*I)- x)/((x + y*I)+ q)
 
w == Divide[(x + y*I)- x,(x + y*I)+ q]
 Skipped - no semantic math Skipped - no semantic math - -
19.20#Ex25 0 < w 0 𝑀 {\displaystyle{\displaystyle 0<w}}
0 < w		 

0 < w
 
0 < w
 Skipped - no semantic math Skipped - no semantic math - -
19.20.E17 ( q + z ) ⁒ R J ⁑ ( 0 , y , z , - q ) = ( p - z ) ⁒ R J ⁑ ( 0 , y , z , p ) - 3 ⁒ R F ⁑ ( 0 , y , z ) π‘ž 𝑧 Carlson-integral-RJ 0 𝑦 𝑧 π‘ž 𝑝 𝑧 Carlson-integral-RJ 0 𝑦 𝑧 𝑝 3 Carlson-integral-RF 0 𝑦 𝑧 {\displaystyle{\displaystyle(q+z)R_{J}\left(0,y,z,-q\right)=(p-z)R_{J}\left(0,% y,z,p\right)-3R_{F}\left(0,y,z\right)}}
(q+z)\CarlsonsymellintRJ@{0}{y}{z}{-q} = (p-z)\CarlsonsymellintRJ@{0}{y}{z}{p}-3\CarlsonsymellintRF@{0}{y}{z}		 p = z ⁒ ( y + q ) / ( z + q ) , w = z / ( z + q ) formulae-sequence 𝑝 𝑧 𝑦 π‘ž 𝑧 π‘ž 𝑀 𝑧 𝑧 π‘ž {\displaystyle{\displaystyle p=z(y+q)/(z+q),w=z/(z+q)}} 
Error
 
(q +(x + y*I))*3*(x + y*I-0)/(x + y*I-- q)*(EllipticPi[(x + y*I-- q)/(x + y*I-0),ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/Sqrt[x + y*I-0] == (p -(x + y*I))*3*(x + y*I-0)/(x + y*I-p)*(EllipticPi[(x + y*I-p)/(x + y*I-0),ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/Sqrt[x + y*I-0]- 3*EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]/Sqrt[x + y*I-0]
 Missing Macro Error Failure -
Failed [300 / 300]
Result: Complex[-3.556352843352318, 3.1308549992075583]
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]}

Result: Complex[-7.694083210877473, -5.44447388199589]
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]}

... skip entries to safe data
19.20#Ex26 R D ⁑ ( x , x , x ) = x - 3 / 2 Carlson-integral-RD π‘₯ π‘₯ π‘₯ superscript π‘₯ 3 2 {\displaystyle{\displaystyle R_{D}\left(x,x,x\right)=x^{-3/2}}}
\CarlsonsymellintRD@{x}{x}{x} = x^{-3/2}		 

Error
 
3*(EllipticF[ArcCos[Sqrt[x/x]],(x-x)/(x-x)]-EllipticE[ArcCos[Sqrt[x/x]],(x-x)/(x-x)])/((x-x)*(x-x)^(1/2)) == (x)^(- 3/2)
 Missing Macro Error Failure -
Failed [3 / 3]
Result: Indeterminate
Test Values: {Rule[x, 1.5]}

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

... skip entries to safe data
19.20#Ex27 R D ⁑ ( Ξ» ⁒ x , Ξ» ⁒ y , Ξ» ⁒ z ) = Ξ» - 3 / 2 ⁒ R D ⁑ ( x , y , z ) Carlson-integral-RD πœ† π‘₯ πœ† 𝑦 πœ† 𝑧 superscript πœ† 3 2 Carlson-integral-RD π‘₯ 𝑦 𝑧 {\displaystyle{\displaystyle R_{D}\left(\lambda x,\lambda y,\lambda z\right)=% \lambda^{-3/2}R_{D}\left(x,y,z\right)}}
\CarlsonsymellintRD@{\lambda x}{\lambda y}{\lambda z} = \lambda^{-3/2}\CarlsonsymellintRD@{x}{y}{z}		 

Error
 
3*(EllipticF[ArcCos[Sqrt[\[Lambda]*x/\[Lambda]*(x + y*I)]],(\[Lambda]*(x + y*I)-\[Lambda]*y)/(\[Lambda]*(x + y*I)-\[Lambda]*x)]-EllipticE[ArcCos[Sqrt[\[Lambda]*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)^(1/2)) == \[Lambda]^(- 3/2)* 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))
 Missing Macro Error Failure -
Failed [180 / 180]
Result: Complex[1.0149076549010991, -0.8161311339182895]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[Ξ», Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

Result: Complex[-1.2947399441897933, -0.14055622592761496]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[Ξ», Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.20#Ex29 R D ⁑ ( 0 , 0 , z ) = ∞ Carlson-integral-RD 0 0 𝑧 {\displaystyle{\displaystyle R_{D}\left(0,0,z\right)=\infty}}
\CarlsonsymellintRD@{0}{0}{z} = \infty		 

Error
 
3*(EllipticF[ArcCos[Sqrt[0/z]],(z-0)/(z-0)]-EllipticE[ArcCos[Sqrt[0/z]],(z-0)/(z-0)])/((z-0)*(z-0)^(1/2)) == Infinity
 Missing Macro Error Failure -
Failed [7 / 7]
Result: Indeterminate
Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

Result: Indeterminate
Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.20.E20 R D ⁑ ( x , y , y ) = 3 2 ⁒ ( y - x ) ⁒ ( R C ⁑ ( x , y ) - x y ) Carlson-integral-RD π‘₯ 𝑦 𝑦 3 2 𝑦 π‘₯ Carlson-integral-RC π‘₯ 𝑦 π‘₯ 𝑦 {\displaystyle{\displaystyle R_{D}\left(x,y,y\right)=\frac{3}{2(y-x)}\left(R_{% C}\left(x,y\right)-\frac{\sqrt{x}}{y}\right)}}
\CarlsonsymellintRD@{x}{y}{y} = \frac{3}{2(y-x)}\left(\CarlsonellintRC@{x}{y}-\frac{\sqrt{x}}{y}\right)		 x β‰  y , y β‰  0 formulae-sequence π‘₯ 𝑦 𝑦 0 {\displaystyle{\displaystyle x\neq y,y\neq 0}} 
Error
 
3*(EllipticF[ArcCos[Sqrt[x/y]],(y-y)/(y-x)]-EllipticE[ArcCos[Sqrt[x/y]],(y-y)/(y-x)])/((y-y)*(y-x)^(1/2)) == Divide[3,2*(y - x)]*(1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)]-Divide[Sqrt[x],y])
 Missing Macro Error Failure -
Failed [15 / 15]
Result: Indeterminate
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}

Result: Indeterminate
Test Values: {Rule[x, 1.5], Rule[y, -0.5]}

... skip entries to safe data
19.20.E21 R D ⁑ ( x , x , z ) = 3 z - x ⁒ ( R C ⁑ ( z , x ) - 1 z ) Carlson-integral-RD π‘₯ π‘₯ 𝑧 3 𝑧 π‘₯ Carlson-integral-RC 𝑧 π‘₯ 1 𝑧 {\displaystyle{\displaystyle R_{D}\left(x,x,z\right)=\frac{3}{z-x}\left(R_{C}% \left(z,x\right)-\frac{1}{\sqrt{z}}\right)}}
\CarlsonsymellintRD@{x}{x}{z} = \frac{3}{z-x}\left(\CarlsonellintRC@{z}{x}-\frac{1}{\sqrt{z}}\right)		 x β‰  z , x ⁒ z β‰  0 formulae-sequence π‘₯ 𝑧 π‘₯ 𝑧 0 {\displaystyle{\displaystyle x\neq z,xz\neq 0}} 
Error
 
3*(EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-x)/(x + y*I-x)]-EllipticE[ArcCos[Sqrt[x/x + y*I]],(x + y*I-x)/(x + y*I-x)])/((x + y*I-x)*(x + y*I-x)^(1/2)) == Divide[3,(x + y*I)- x]*(1/Sqrt[x]*Hypergeometric2F1[1/2,1/2,3/2,1-(x + y*I)/(x)]-Divide[1,Sqrt[x + y*I]])
 Missing Macro Error Failure -
Failed [18 / 18]
Result: Complex[0.13486015646372063, -0.8506635330353051]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}

Result: Complex[0.13486015646372096, 0.8506635330353054]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
19.20.E22 ∫ 0 1 t 2 ⁒ d t 1 - t 4 = 1 3 ⁒ R D ⁑ ( 0 , 2 , 1 ) superscript subscript 0 1 superscript 𝑑 2 𝑑 1 superscript 𝑑 4 1 3 Carlson-integral-RD 0 2 1 {\displaystyle{\displaystyle\int_{0}^{1}\frac{t^{2}\mathrm{d}t}{\sqrt{1-t^{4}}% }=\tfrac{1}{3}R_{D}\left(0,2,1\right)}}
\int_{0}^{1}\frac{t^{2}\diff{t}}{\sqrt{1-t^{4}}} = \tfrac{1}{3}\CarlsonsymellintRD@{0}{2}{1}		 

Error
 
Integrate[Divide[(t)^(2),Sqrt[1 - (t)^(4)]], {t, 0, 1}, GenerateConditions->None] == Divide[1,3]*3*(EllipticF[ArcCos[Sqrt[0/1]],(1-2)/(1-0)]-EllipticE[ArcCos[Sqrt[0/1]],(1-2)/(1-0)])/((1-2)*(1-0)^(1/2))
 Missing Macro Error Successful Skip - symbolical successful subtest Successful [Tested: 1]
19.20.E22 1 3 ⁒ R D ⁑ ( 0 , 2 , 1 ) = ( Ξ“ ⁑ ( 3 4 ) ) 2 ( 2 ⁒ Ο€ ) 1 / 2 1 3 Carlson-integral-RD 0 2 1 superscript Euler-Gamma 3 4 2 superscript 2 πœ‹ 1 2 {\displaystyle{\displaystyle\tfrac{1}{3}R_{D}\left(0,2,1\right)=\frac{\left(% \Gamma\left(\frac{3}{4}\right)\right)^{2}}{(2\pi)^{1/2}}}}
\tfrac{1}{3}\CarlsonsymellintRD@{0}{2}{1} = \frac{\left(\EulerGamma@{\frac{3}{4}}\right)^{2}}{(2\pi)^{1/2}}		 

Error
 
Divide[1,3]*3*(EllipticF[ArcCos[Sqrt[0/1]],(1-2)/(1-0)]-EllipticE[ArcCos[Sqrt[0/1]],(1-2)/(1-0)])/((1-2)*(1-0)^(1/2)) == Divide[(Gamma[Divide[3,4]])^(2),(2*Pi)^(1/2)]
 Missing Macro Error Successful Skip - symbolical successful subtest Successful [Tested: 1]
19.20.E22 ( Ξ“ ⁑ ( 3 4 ) ) 2 ( 2 ⁒ Ο€ ) 1 / 2 = 0.59907 01173 67796 10371 ⁒ … superscript Euler-Gamma 3 4 2 superscript 2 πœ‹ 1 2 0.59907 01173 67796 10371 … {\displaystyle{\displaystyle\frac{\left(\Gamma\left(\frac{3}{4}\right)\right)^% {2}}{(2\pi)^{1/2}}=0.59907\;01173\;67796\;10371\dots}}
\frac{\left(\EulerGamma@{\frac{3}{4}}\right)^{2}}{(2\pi)^{1/2}} = 0.59907\;01173\;67796\;10371\dots		 

((GAMMA((3)/(4)))^(2))/((2*Pi)^(1/2)) = 0.59907011736779610371
 
Divide[(Gamma[Divide[3,4]])^(2),(2*Pi)^(1/2)] == 0.59907011736779610371
 Failure Successful Successful [Tested: 0] Successful [Tested: 1]
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