Elliptic Integrals - 19.21 Connection Formulas

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
19.21.E1 R F ⁡ ( 0 , z + 1 , z ) ⁢ R D ⁡ ( 0 , z + 1 , 1 ) + R D ⁡ ( 0 , z + 1 , z ) ⁢ R F ⁡ ( 0 , z + 1 , 1 ) = 3 ⁢ π / ( 2 ⁢ z ) Carlson-integral-RF 0 𝑧 1 𝑧 Carlson-integral-RD 0 𝑧 1 1 Carlson-integral-RD 0 𝑧 1 𝑧 Carlson-integral-RF 0 𝑧 1 1 3 𝜋 2 𝑧 {\displaystyle{\displaystyle R_{F}\left(0,z+1,z\right)R_{D}\left(0,z+1,1\right% )+R_{D}\left(0,z+1,z\right)R_{F}\left(0,z+1,1\right)=3\pi/(2z)}}
\CarlsonsymellintRF@{0}{z+1}{z}\CarlsonsymellintRD@{0}{z+1}{1}+\CarlsonsymellintRD@{0}{z+1}{z}\CarlsonsymellintRF@{0}{z+1}{1} = 3\pi/(2z)		 

Error

EllipticF[ArcCos[Sqrt[0/z]],(z-z + 1)/(z-0)]/Sqrt[z-0]*3*(EllipticF[ArcCos[Sqrt[0/1]],(1-z + 1)/(1-0)]-EllipticE[ArcCos[Sqrt[0/1]],(1-z + 1)/(1-0)])/((1-z + 1)*(1-0)^(1/2))+ 3*(EllipticF[ArcCos[Sqrt[0/z]],(z-z + 1)/(z-0)]-EllipticE[ArcCos[Sqrt[0/z]],(z-z + 1)/(z-0)])/((z-z + 1)*(z-0)^(1/2))*EllipticF[ArcCos[Sqrt[0/1]],(1-z + 1)/(1-0)]/Sqrt[1-0] == 3*Pi/(2*z)
Missing Macro Error Failure -
Failed [7 / 7]
Result: Complex[-18.895019118218656, -13.266297761785948]
Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-2.405668177707024, 11.584123712813607]
Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.21.E2 3 ⁢ R F ⁡ ( 0 , y , z ) = z ⁢ R D ⁡ ( 0 , y , z ) + y ⁢ R D ⁡ ( 0 , z , y ) 3 Carlson-integral-RF 0 𝑦 𝑧 𝑧 Carlson-integral-RD 0 𝑦 𝑧 𝑦 Carlson-integral-RD 0 𝑧 𝑦 {\displaystyle{\displaystyle 3R_{F}\left(0,y,z\right)=zR_{D}\left(0,y,z\right)% +yR_{D}\left(0,z,y\right)}}
3\CarlsonsymellintRF@{0}{y}{z} = z\CarlsonsymellintRD@{0}{y}{z}+y\CarlsonsymellintRD@{0}{z}{y}		 

Error

3*EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]/Sqrt[x + y*I-0] == (x + y*I)*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))+ y*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 Failure -
Failed [18 / 18]
Result: Complex[-0.11482200178525792, 3.077769310376559]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}


Result: Complex[0.9930498831204495, -3.2293137034341144]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
19.21.E3 6 ⁢ R G ⁡ ( 0 , y , z ) = y ⁢ z ⁢ ( R D ⁡ ( 0 , y , z ) + R D ⁡ ( 0 , z , y ) ) 6 Carlson-integral-RG 0 𝑦 𝑧 𝑦 𝑧 Carlson-integral-RD 0 𝑦 𝑧 Carlson-integral-RD 0 𝑧 𝑦 {\displaystyle{\displaystyle 6R_{G}\left(0,y,z\right)=yz(R_{D}\left(0,y,z% \right)+R_{D}\left(0,z,y\right))}}
6\CarlsonsymellintRG@{0}{y}{z} = yz(\CarlsonsymellintRD@{0}{y}{z}+\CarlsonsymellintRD@{0}{z}{y})		 

Error

6*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]) == y*(x + y*I)*(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 Failure -
Failed [18 / 18]
Result: Plus[Complex[-2.3418687704988255, 4.458096439149204], Times[Complex[8.07364639949469, -3.344213836475408], 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[1.800571487249528, -2.4291108001544095], Times[Complex[8.07364639949469, 3.344213836475408], 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
19.21.E3 y ⁢ z ⁢ ( R D ⁡ ( 0 , y , z ) + R D ⁡ ( 0 , z , y ) ) = 3 ⁢ z ⁢ R F ⁡ ( 0 , y , z ) + z ⁢ ( y - z ) ⁢ R D ⁡ ( 0 , y , z ) 𝑦 𝑧 Carlson-integral-RD 0 𝑦 𝑧 Carlson-integral-RD 0 𝑧 𝑦 3 𝑧 Carlson-integral-RF 0 𝑦 𝑧 𝑧 𝑦 𝑧 Carlson-integral-RD 0 𝑦 𝑧 {\displaystyle{\displaystyle yz(R_{D}\left(0,y,z\right)+R_{D}\left(0,z,y\right% ))=3zR_{F}\left(0,y,z\right)+z(y-z)R_{D}\left(0,y,z\right)}}
yz(\CarlsonsymellintRD@{0}{y}{z}+\CarlsonsymellintRD@{0}{z}{y}) = 3z\CarlsonsymellintRF@{0}{y}{z}+z(y-z)\CarlsonsymellintRD@{0}{y}{z}		 

Error

y*(x + y*I)*(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))) == 3*(x + y*I)*EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]/Sqrt[x + y*I-0]+(x + y*I)*(y -(x + y*I))*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))
Missing Macro Error Failure -
Failed [18 / 18]
Result: Complex[-4.444420962886951, -4.788886968242726]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}


Result: Complex[-6.333545379831845, 3.3543957304704977]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
19.21.E4 R F ⁡ ( 0 , z - 1 , z ) = R F ⁡ ( 0 , 1 - z , 1 ) - i ⁢ R F ⁡ ( 0 , z , 1 ) Carlson-integral-RF 0 𝑧 1 𝑧 Carlson-integral-RF 0 1 𝑧 1 imaginary-unit Carlson-integral-RF 0 𝑧 1 {\displaystyle{\displaystyle R_{F}\left(0,z-1,z\right)=R_{F}\left(0,1-z,1% \right)-\mathrm{i}R_{F}\left(0,z,1\right)}}
\CarlsonsymellintRF@{0}{z-1}{z} = \CarlsonsymellintRF@{0}{1-z}{1}-\iunit\CarlsonsymellintRF@{0}{z}{1}		 

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

EllipticF[ArcCos[Sqrt[0/z]],(z-z - 1)/(z-0)]/Sqrt[z-0] == EllipticF[ArcCos[Sqrt[0/1]],(1-1 - z)/(1-0)]/Sqrt[1-0]- I*EllipticF[ArcCos[Sqrt[0/1]],(1-z)/(1-0)]/Sqrt[1-0]
Failure Failure
Failed [7 / 7]
Result: 3.197606220-2.012137137*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}


Result: 1.024722154-2.538160454*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [7 / 7]
Result: Complex[0.447882135735306, 1.6422203572966838]
Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[0.9112982419758283, 0.8007121244739206]
Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.21.E4 R F ⁡ ( 0 , z - 1 , z ) = R F ⁡ ( 0 , 1 - z , 1 ) + i ⁢ R F ⁡ ( 0 , z , 1 ) Carlson-integral-RF 0 𝑧 1 𝑧 Carlson-integral-RF 0 1 𝑧 1 imaginary-unit Carlson-integral-RF 0 𝑧 1 {\displaystyle{\displaystyle R_{F}\left(0,z-1,z\right)=R_{F}\left(0,1-z,1% \right)+\mathrm{i}R_{F}\left(0,z,1\right)}}
\CarlsonsymellintRF@{0}{z-1}{z} = \CarlsonsymellintRF@{0}{1-z}{1}+\iunit\CarlsonsymellintRF@{0}{z}{1}		 

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

EllipticF[ArcCos[Sqrt[0/z]],(z-z - 1)/(z-0)]/Sqrt[z-0] == EllipticF[ArcCos[Sqrt[0/1]],(1-1 - z)/(1-0)]/Sqrt[1-0]+ I*EllipticF[ArcCos[Sqrt[0/1]],(1-z)/(1-0)]/Sqrt[1-0]
Failure Failure
Failed [7 / 7]
Result: 3.609429842+1.115973839*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}


Result: 2.710472508+.381644808*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [7 / 7]
Result: Complex[0.0036174998504115152, -2.054982714938571]
Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-0.9086726238549093, -2.7316000638683375]
Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.21.E5 2 ⁢ R G ⁡ ( 0 , z - 1 , z ) = 2 ⁢ R G ⁡ ( 0 , 1 - z , 1 ) + i ⁢ 2 ⁢ R G ⁡ ( 0 , z , 1 ) + ( z - 1 ) ⁢ R F ⁡ ( 0 , 1 - z , 1 ) - i ⁢ z ⁢ R F ⁡ ( 0 , z , 1 ) 2 Carlson-integral-RG 0 𝑧 1 𝑧 2 Carlson-integral-RG 0 1 𝑧 1 imaginary-unit 2 Carlson-integral-RG 0 𝑧 1 𝑧 1 Carlson-integral-RF 0 1 𝑧 1 imaginary-unit 𝑧 Carlson-integral-RF 0 𝑧 1 {\displaystyle{\displaystyle 2R_{G}\left(0,z-1,z\right)=2R_{G}\left(0,1-z,1% \right)+\mathrm{i}2R_{G}\left(0,z,1\right)+(z-1)R_{F}\left(0,1-z,1\right)-% \mathrm{i}zR_{F}\left(0,z,1\right)}}
2\CarlsonsymellintRG@{0}{z-1}{z} = 2\CarlsonsymellintRG@{0}{1-z}{1}+\iunit 2\CarlsonsymellintRG@{0}{z}{1}+(z-1)\CarlsonsymellintRF@{0}{1-z}{1}-\iunit z\CarlsonsymellintRF@{0}{z}{1}		 

Error

2*Sqrt[z-0]*(EllipticE[ArcCos[Sqrt[0/z]],(z-z - 1)/(z-0)]+(Cot[ArcCos[Sqrt[0/z]]])^2*EllipticF[ArcCos[Sqrt[0/z]],(z-z - 1)/(z-0)]+Cot[ArcCos[Sqrt[0/z]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/z]]]^2]) == 2*Sqrt[1-0]*(EllipticE[ArcCos[Sqrt[0/1]],(1-1 - z)/(1-0)]+(Cot[ArcCos[Sqrt[0/1]]])^2*EllipticF[ArcCos[Sqrt[0/1]],(1-1 - z)/(1-0)]+Cot[ArcCos[Sqrt[0/1]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/1]]]^2])+ I*2*Sqrt[1-0]*(EllipticE[ArcCos[Sqrt[0/1]],(1-z)/(1-0)]+(Cot[ArcCos[Sqrt[0/1]]])^2*EllipticF[ArcCos[Sqrt[0/1]],(1-z)/(1-0)]+Cot[ArcCos[Sqrt[0/1]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/1]]]^2])+(z - 1)*EllipticF[ArcCos[Sqrt[0/1]],(1-1 - z)/(1-0)]/Sqrt[1-0]- I*z*EllipticF[ArcCos[Sqrt[0/1]],(1-z)/(1-0)]/Sqrt[1-0]
Missing Macro Error Failure -
Failed [7 / 7]
Result: Complex[0.23313173408598564, -1.9381268036446178]
Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[0.6842698833888152, -2.1985132995849304]
Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.21.E5 2 ⁢ R G ⁡ ( 0 , z - 1 , z ) = 2 ⁢ R G ⁡ ( 0 , 1 - z , 1 ) - i ⁢ 2 ⁢ R G ⁡ ( 0 , z , 1 ) + ( z - 1 ) ⁢ R F ⁡ ( 0 , 1 - z , 1 ) + i ⁢ z ⁢ R F ⁡ ( 0 , z , 1 ) 2 Carlson-integral-RG 0 𝑧 1 𝑧 2 Carlson-integral-RG 0 1 𝑧 1 imaginary-unit 2 Carlson-integral-RG 0 𝑧 1 𝑧 1 Carlson-integral-RF 0 1 𝑧 1 imaginary-unit 𝑧 Carlson-integral-RF 0 𝑧 1 {\displaystyle{\displaystyle 2R_{G}\left(0,z-1,z\right)=2R_{G}\left(0,1-z,1% \right)-\mathrm{i}2R_{G}\left(0,z,1\right)+(z-1)R_{F}\left(0,1-z,1\right)+% \mathrm{i}zR_{F}\left(0,z,1\right)}}
2\CarlsonsymellintRG@{0}{z-1}{z} = 2\CarlsonsymellintRG@{0}{1-z}{1}-\iunit 2\CarlsonsymellintRG@{0}{z}{1}+(z-1)\CarlsonsymellintRF@{0}{1-z}{1}+\iunit z\CarlsonsymellintRF@{0}{z}{1}		 

Error

2*Sqrt[z-0]*(EllipticE[ArcCos[Sqrt[0/z]],(z-z - 1)/(z-0)]+(Cot[ArcCos[Sqrt[0/z]]])^2*EllipticF[ArcCos[Sqrt[0/z]],(z-z - 1)/(z-0)]+Cot[ArcCos[Sqrt[0/z]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/z]]]^2]) == 2*Sqrt[1-0]*(EllipticE[ArcCos[Sqrt[0/1]],(1-1 - z)/(1-0)]+(Cot[ArcCos[Sqrt[0/1]]])^2*EllipticF[ArcCos[Sqrt[0/1]],(1-1 - z)/(1-0)]+Cot[ArcCos[Sqrt[0/1]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/1]]]^2])- I*2*Sqrt[1-0]*(EllipticE[ArcCos[Sqrt[0/1]],(1-z)/(1-0)]+(Cot[ArcCos[Sqrt[0/1]]])^2*EllipticF[ArcCos[Sqrt[0/1]],(1-z)/(1-0)]+Cot[ArcCos[Sqrt[0/1]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/1]]]^2])+(z - 1)*EllipticF[ArcCos[Sqrt[0/1]],(1-1 - z)/(1-0)]/Sqrt[1-0]+ I*z*EllipticF[ArcCos[Sqrt[0/1]],(1-z)/(1-0)]/Sqrt[1-0]
Missing Macro Error Failure -
Failed [7 / 7]
Result: Complex[0.44709928924442033, 1.6621495887309192]
Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[1.1273665829731985, 1.9939163092606038]
Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.21.E6 ( r ⁢ p / z ) ⁢ R J ⁡ ( 0 , y , z , p ) = ( r - 1 ) ⁢ R F ⁡ ( 0 , y , z ) ⁢ R D ⁡ ( p , r ⁢ z , z ) + R D ⁡ ( 0 , y , z ) ⁢ R F ⁡ ( p , r ⁢ z , z ) 𝑟 𝑝 𝑧 Carlson-integral-RJ 0 𝑦 𝑧 𝑝 𝑟 1 Carlson-integral-RF 0 𝑦 𝑧 Carlson-integral-RD 𝑝 𝑟 𝑧 𝑧 Carlson-integral-RD 0 𝑦 𝑧 Carlson-integral-RF 𝑝 𝑟 𝑧 𝑧 {\displaystyle{\displaystyle(\sqrt{rp}/z)R_{J}\left(0,y,z,p\right)={(r-1)}R_{F% }\left(0,y,z\right)R_{D}\left(p,rz,z\right)+R_{D}\left(0,y,z\right)R_{F}\left(% p,rz,z\right)}}
(\sqrt{rp}/z)\CarlsonsymellintRJ@{0}{y}{z}{p} = {(r-1)}\CarlsonsymellintRF@{0}{y}{z}\CarlsonsymellintRD@{p}{rz}{z}+\CarlsonsymellintRD@{0}{y}{z}\CarlsonsymellintRF@{p}{rz}{z}		 r = ( y - p ) / ( y - z ) , ( y - p ) / ( y - z ) > 0 formulae-sequence 𝑟 𝑦 𝑝 𝑦 𝑧 𝑦 𝑝 𝑦 𝑧 0 {\displaystyle{\displaystyle r=(y-p)/(y-z),(y-p)/(y-z)>0}} 
Error

(Sqrt[r*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] == (r - 1)*EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]/Sqrt[x + y*I-0]*3*(EllipticF[ArcCos[Sqrt[p/x + y*I]],(x + y*I-r*(x + y*I))/(x + y*I-p)]-EllipticE[ArcCos[Sqrt[p/x + y*I]],(x + y*I-r*(x + y*I))/(x + y*I-p)])/((x + y*I-r*(x + y*I))*(x + y*I-p)^(1/2))+ 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))*EllipticF[ArcCos[Sqrt[p/x + y*I]],(x + y*I-r*(x + y*I))/(x + y*I-p)]/Sqrt[x + y*I-p]
Missing Macro Error Aborted -
Failed [300 / 300]
Result: Complex[-0.019107479205769995, -0.26821779662698253]
Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[r, -1.5], Rule[x, 1.5], Rule[y, -1.5]}


Result: Complex[1.626010920193221, 0.604709928225457]
Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[r, -1.5], Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
19.21.E7 ( x - y ) ⁢ R D ⁡ ( y , z , x ) + ( z - y ) ⁢ R D ⁡ ( x , y , z ) = 3 ⁢ R F ⁡ ( x , y , z ) - 3 ⁢ y / ( x ⁢ z ) 𝑥 𝑦 Carlson-integral-RD 𝑦 𝑧 𝑥 𝑧 𝑦 Carlson-integral-RD 𝑥 𝑦 𝑧 3 Carlson-integral-RF 𝑥 𝑦 𝑧 3 𝑦 𝑥 𝑧 {\displaystyle{\displaystyle(x-y)R_{D}\left(y,z,x\right)+(z-y)R_{D}\left(x,y,z% \right)=3R_{F}\left(x,y,z\right)-3\sqrt{y/(xz)}}}
(x-y)\CarlsonsymellintRD@{y}{z}{x}+(z-y)\CarlsonsymellintRD@{x}{y}{z} = 3\CarlsonsymellintRF@{x}{y}{z}-3\sqrt{y/(xz)}		 

Error

(x - y)*3*(EllipticF[ArcCos[Sqrt[y/x]],(x-x + y*I)/(x-y)]-EllipticE[ArcCos[Sqrt[y/x]],(x-x + y*I)/(x-y)])/((x-x + y*I)*(x-y)^(1/2))+((x + y*I)- y)*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)) == 3*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]- 3*Sqrt[y/(x*(x + y*I))]
Missing Macro Error Failure -
Failed [18 / 18]
Result: Complex[2.01816993619941, -7.647648832317454]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}


Result: Complex[1.9029767059950156, 2.211761239496786]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
19.21.E8 R D ⁡ ( y , z , x ) + R D ⁡ ( z , x , y ) + R D ⁡ ( x , y , z ) = 3 ⁢ ( x ⁢ y ⁢ z ) - 1 / 2 Carlson-integral-RD 𝑦 𝑧 𝑥 Carlson-integral-RD 𝑧 𝑥 𝑦 Carlson-integral-RD 𝑥 𝑦 𝑧 3 superscript 𝑥 𝑦 𝑧 1 2 {\displaystyle{\displaystyle R_{D}\left(y,z,x\right)+R_{D}\left(z,x,y\right)+R% _{D}\left(x,y,z\right)=3(xyz)^{-1/2}}}
\CarlsonsymellintRD@{y}{z}{x}+\CarlsonsymellintRD@{z}{x}{y}+\CarlsonsymellintRD@{x}{y}{z} = 3(xyz)^{-1/2}		 

Error

3*(EllipticF[ArcCos[Sqrt[y/x]],(x-x + y*I)/(x-y)]-EllipticE[ArcCos[Sqrt[y/x]],(x-x + y*I)/(x-y)])/((x-x + y*I)*(x-y)^(1/2))+ 3*(EllipticF[ArcCos[Sqrt[x + y*I/y]],(y-x)/(y-x + y*I)]-EllipticE[ArcCos[Sqrt[x + y*I/y]],(y-x)/(y-x + y*I)])/((y-x)*(y-x + y*I)^(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)) == 3*(x*y*(x + y*I))^(- 1/2)
Missing Macro Error Failure -
Failed [18 / 18]
Result: Complex[0.061772053426947915, 0.22732915812456822]
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
19.21.E9 x ⁢ R D ⁡ ( y , z , x ) + y ⁢ R D ⁡ ( z , x , y ) + z ⁢ R D ⁡ ( x , y , z ) = 3 ⁢ R F ⁡ ( x , y , z ) 𝑥 Carlson-integral-RD 𝑦 𝑧 𝑥 𝑦 Carlson-integral-RD 𝑧 𝑥 𝑦 𝑧 Carlson-integral-RD 𝑥 𝑦 𝑧 3 Carlson-integral-RF 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle xR_{D}\left(y,z,x\right)+yR_{D}\left(z,x,y\right)% +zR_{D}\left(x,y,z\right)=3R_{F}\left(x,y,z\right)}}
x\CarlsonsymellintRD@{y}{z}{x}+y\CarlsonsymellintRD@{z}{x}{y}+z\CarlsonsymellintRD@{x}{y}{z} = 3\CarlsonsymellintRF@{x}{y}{z}		 

Error

x*3*(EllipticF[ArcCos[Sqrt[y/x]],(x-x + y*I)/(x-y)]-EllipticE[ArcCos[Sqrt[y/x]],(x-x + y*I)/(x-y)])/((x-x + y*I)*(x-y)^(1/2))+ y*3*(EllipticF[ArcCos[Sqrt[x + y*I/y]],(y-x)/(y-x + y*I)]-EllipticE[ArcCos[Sqrt[x + y*I/y]],(y-x)/(y-x + y*I)])/((y-x)*(y-x + y*I)^(1/2))+(x + y*I)*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)) == 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 -
Failed [18 / 18]
Result: Complex[0.3490343350525606, -4.182689157514275]
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
19.21.E10 2 ⁢ R G ⁡ ( x , y , z ) = z ⁢ R F ⁡ ( x , y , z ) - 1 3 ⁢ ( x - z ) ⁢ ( y - z ) ⁢ R D ⁡ ( x , y , z ) + x ⁢ y / z 2 Carlson-integral-RG 𝑥 𝑦 𝑧 𝑧 Carlson-integral-RF 𝑥 𝑦 𝑧 1 3 𝑥 𝑧 𝑦 𝑧 Carlson-integral-RD 𝑥 𝑦 𝑧 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle 2R_{G}\left(x,y,z\right)=zR_{F}\left(x,y,z\right)% -\tfrac{1}{3}(x-z)(y-z)R_{D}\left(x,y,z\right)+\sqrt{xy/z}}}
2\CarlsonsymellintRG@{x}{y}{z} = z\CarlsonsymellintRF@{x}{y}{z}-\tfrac{1}{3}(x-z)(y-z)\CarlsonsymellintRD@{x}{y}{z}+\sqrt{xy/z}		 z ≠ 0 𝑧 0 {\displaystyle{\displaystyle z\neq 0}} 
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]) == (x + y*I)*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]-Divide[1,3]*(x -(x + y*I))*(y -(x + y*I))*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))+Sqrt[x*y/(x + y*I)]
Missing Macro Error Failure -
Failed [18 / 18]
Result: Plus[Complex[-2.045465659795318, -0.2973389532409781], 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]]]]]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}


Result: Plus[Complex[-2.0191372830755783, 1.5655011975568338], Times[Complex[1.7320508075688772, 1.732050807568877], Plus[Complex[1.0566228789425183, 0.3443432776585209], Times[Complex[0.3176872874027722, 1.049249833251038], Power[Plus[1.0, Times[Complex[0.0, 1.5], Power[k, 2]]], Rational[1, 2]]]]]]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
19.21.E12 ( p - x ) ⁢ R J ⁡ ( x , y , z , p ) + ( q - x ) ⁢ R J ⁡ ( x , y , z , q ) = 3 ⁢ R F ⁡ ( x , y , z ) - 3 ⁢ R C ⁡ ( ξ , η ) 𝑝 𝑥 Carlson-integral-RJ 𝑥 𝑦 𝑧 𝑝 𝑞 𝑥 Carlson-integral-RJ 𝑥 𝑦 𝑧 𝑞 3 Carlson-integral-RF 𝑥 𝑦 𝑧 3 Carlson-integral-RC 𝜉 𝜂 {\displaystyle{\displaystyle(p-x)R_{J}\left(x,y,z,p\right)+(q-x)R_{J}\left(x,y% ,z,q\right)=3R_{F}\left(x,y,z\right)-3R_{C}\left(\xi,\eta\right)}}
(p-x)\CarlsonsymellintRJ@{x}{y}{z}{p}+(q-x)\CarlsonsymellintRJ@{x}{y}{z}{q} = 3\CarlsonsymellintRF@{x}{y}{z}-3\CarlsonellintRC@{\xi}{\eta}		 

Error

(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]+(q - x)*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] == 3*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]- 3*1/Sqrt[\[Eta]]*Hypergeometric2F1[1/2,1/2,3/2,1-(\[Xi])/(\[Eta])]
Missing Macro Error Aborted -
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[5.153237655786464, -3.718995107844719]
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
19.21#Ex1 ( p - x ) ⁢ ( q - x ) = ( y - x ) ⁢ ( z - x ) 𝑝 𝑥 𝑞 𝑥 𝑦 𝑥 𝑧 𝑥 {\displaystyle{\displaystyle(p-x)(q-x)=(y-x)(z-x)}}
(p-x)(q-x) = (y-x)(z-x)		 

(p - x)*(q - x) = (y - x)*((x + y*I)- x)
 
(p - x)*(q - x) == (y - x)*((x + y*I)- x)
 Skipped - no semantic math Skipped - no semantic math - -
19.21#Ex2 ξ = y ⁢ z / x 𝜉 𝑦 𝑧 𝑥 {\displaystyle{\displaystyle\xi=yz/x}}
\xi = yz/x		 

xi = y*z/x
 
\[Xi] == y*z/x
 Skipped - no semantic math Skipped - no semantic math - -
19.21#Ex3 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 - -
19.21.E14 η - ξ = p + q - y - z 𝜂 𝜉 𝑝 𝑞 𝑦 𝑧 {\displaystyle{\displaystyle\eta-\xi=p+q-y-z}}
\eta-\xi = p+q-y-z		 

eta - xi = p + q - y -(x + y*I)
 
\[Eta]- \[Xi] == p + q - y -(x + y*I)
 Skipped - no semantic math Skipped - no semantic math - -
19.21.E15 p ⁢ R J ⁡ ( 0 , y , z , p ) + q ⁢ R J ⁡ ( 0 , y , z , q ) = 3 ⁢ R F ⁡ ( 0 , y , z ) 𝑝 Carlson-integral-RJ 0 𝑦 𝑧 𝑝 𝑞 Carlson-integral-RJ 0 𝑦 𝑧 𝑞 3 Carlson-integral-RF 0 𝑦 𝑧 {\displaystyle{\displaystyle pR_{J}\left(0,y,z,p\right)+qR_{J}\left(0,y,z,q% \right)=3R_{F}\left(0,y,z\right)}}
p\CarlsonsymellintRJ@{0}{y}{z}{p}+q\CarlsonsymellintRJ@{0}{y}{z}{q} = 3\CarlsonsymellintRF@{0}{y}{z}		 p ⁢ q = y ⁢ z 𝑝 𝑞 𝑦 𝑧 {\displaystyle{\displaystyle pq=yz}} 
Error

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]+ q*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] == 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[-0.5878632565330948, -3.2355968294614907]
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[-2.320767562800481, 3.5603464097743847]
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
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