19.21

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
19.21.E1	${\displaystyle{\displaystyle R_{F}\left(0,z+1,z\right)R_{D}\left(0,z+1,1\right)+R_{D}\left(0,z+1,z\right)R_{F}\left(0,z+1,1\right)=3\pi/(2z)}}$ \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	${\displaystyle{\displaystyle 3R_{F}\left(0,y,z\right)=zR_{D}\left(0,y,z\right)+yR_{D}\left(0,z,y\right)}}$ 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	${\displaystyle{\displaystyle 6R_{G}\left(0,y,z\right)=yz(R_{D}\left(0,y,z\right)+R_{D}\left(0,z,y\right))}}$ 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	${\displaystyle{\displaystyle yz(R_{D}\left(0,y,z\right)+R_{D}\left(0,z,y\right))=3zR_{F}\left(0,y,z\right)+z(y-z)R_{D}\left(0,y,z\right)}}$ 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	${\displaystyle{\displaystyle R_{F}\left(0,z-1,z\right)=R_{F}\left(0,1-z,1\right)-\mathrm{i}R_{F}\left(0,z,1\right)}}$ \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	${\displaystyle{\displaystyle R_{F}\left(0,z-1,z\right)=R_{F}\left(0,1-z,1\right)+\mathrm{i}R_{F}\left(0,z,1\right)}}$ \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	${\displaystyle{\displaystyle 2R_{G}\left(0,z-1,z\right)=2R_{G}\left(0,1-z,1\right)+\mathrm{i}2R_{G}\left(0,z,1\right)+(z-1)R_{F}\left(0,1-z,1\right)-\mathrm{i}zR_{F}\left(0,z,1\right)}}$ 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	${\displaystyle{\displaystyle 2R_{G}\left(0,z-1,z\right)=2R_{G}\left(0,1-z,1\right)-\mathrm{i}2R_{G}\left(0,z,1\right)+(z-1)R_{F}\left(0,1-z,1\right)+\mathrm{i}zR_{F}\left(0,z,1\right)}}$ 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	${\displaystyle{\displaystyle(\sqrt{rp}/z)R_{J}\left(0,y,z,p\right)={(r-1)}R_{F}\left(0,y,z\right)R_{D}\left(p,rz,z\right)+R_{D}\left(0,y,z\right)R_{F}\left(p,rz,z\right)}}$ (\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}	${\displaystyle{\displaystyle r=(y-p)/(y-z),(y-p)/(y-z)>0}}$	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	${\displaystyle{\displaystyle(x-y)R_{D}\left(y,z,x\right)+(z-y)R_{D}\left(x,y,z\right)=3R_{F}\left(x,y,z\right)-3\sqrt{y/(xz)}}}$ (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	${\displaystyle{\displaystyle R_{D}\left(y,z,x\right)+R_{D}\left(z,x,y\right)+R_{D}\left(x,y,z\right)=3(xyz)^{-1/2}}}$ \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	${\displaystyle{\displaystyle xR_{D}\left(y,z,x\right)+yR_{D}\left(z,x,y\right)+zR_{D}\left(x,y,z\right)=3R_{F}\left(x,y,z\right)}}$ 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	${\displaystyle{\displaystyle 2R_{G}\left(x,y,z\right)=zR_{F}\left(x,y,z\right)-\tfrac{1}{3}(x-z)(y-z)R_{D}\left(x,y,z\right)+\sqrt{xy/z}}}$ 2\CarlsonsymellintRG@{x}{y}{z} = z\CarlsonsymellintRF@{x}{y}{z}-\tfrac{1}{3}(x-z)(y-z)\CarlsonsymellintRD@{x}{y}{z}+\sqrt{xy/z}	${\displaystyle{\displaystyle z\neq 0}}$	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	${\displaystyle{\displaystyle(p-x)R_{J}\left(x,y,z,p\right)+(q-x)R_{J}\left(x,y,z,q\right)=3R_{F}\left(x,y,z\right)-3R_{C}\left(\xi,\eta\right)}}$ (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	${\displaystyle{\displaystyle(p-x)(q-x)=(y-x)(z-x)}}$ (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	${\displaystyle{\displaystyle\xi=yz/x}}$ \xi = yz/x		xi = y*z/x	\[Xi] == y*z/x	Skipped - no semantic math	Skipped - no semantic math	-	-
19.21#Ex3	${\displaystyle{\displaystyle\eta=pq/x}}$ \eta = pq/x		eta = p*q/x	\[Eta] == p*q/x	Skipped - no semantic math	Skipped - no semantic math	-	-
19.21.E14	${\displaystyle{\displaystyle\eta-\xi=p+q-y-z}}$ \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	${\displaystyle{\displaystyle pR_{J}\left(0,y,z,p\right)+qR_{J}\left(0,y,z,q\right)=3R_{F}\left(0,y,z\right)}}$ p\CarlsonsymellintRJ@{0}{y}{z}{p}+q\CarlsonsymellintRJ@{0}{y}{z}{q} = 3\CarlsonsymellintRF@{0}{y}{z}	${\displaystyle{\displaystyle pq=yz}}$	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