19.25

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
19.25#Ex1	${\displaystyle{\displaystyle K\left(k\right)=R_{F}\left(0,{k^{\prime}}^{2},1\right)}}$ \compellintKk@{k} = \CarlsonsymellintRF@{0}{{k^{\prime}}^{2}}{1}		EllipticK(k) = 0.5*int(1/(sqrt(t+0)*sqrt(t+1 - (k)^(2))*sqrt(t+1)), t = 0..infinity)	EllipticK[(k)^2] == EllipticF[ArcCos[Sqrt[0/1]],(1-1 - (k)^(2))/(1-0)]/Sqrt[1-0]	Failure	Failure	Error	Failed [3 / 3] Result: DirectedInfinity[] Test Values: {Rule[k, 1]} Result: Complex[-0.16657773258291342, -1.0782578237498217] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.25#Ex2	${\displaystyle{\displaystyle E\left(k\right)=2R_{G}\left(0,{k^{\prime}}^{2},1\right)}}$ \compellintEk@{k} = 2\CarlsonsymellintRG@{0}{{k^{\prime}}^{2}}{1}		Error	EllipticE[(k)^2] == 2*Sqrt[1-0]*(EllipticE[ArcCos[Sqrt[0/1]],(1-1 - (k)^(2))/(1-0)]+(Cot[ArcCos[Sqrt[0/1]]])^2*EllipticF[ArcCos[Sqrt[0/1]],(1-1 - (k)^(2))/(1-0)]+Cot[ArcCos[Sqrt[0/1]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/1]]]^2])	Missing Macro Error	Failure	-	Failed [3 / 3] Result: -2.820197789027711 Test Values: {Rule[k, 1]} Result: Complex[-4.864068276731299, 1.343854231387098] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.25#Ex3	${\displaystyle{\displaystyle E\left(k\right)=\tfrac{1}{3}{k^{\prime}}^{2}\left(R_{D}\left(0,{k^{\prime}}^{2},1\right)+R_{D}\left(0,1,{k^{\prime}}^{2}\right)\right)}}$ \compellintEk@{k} = \tfrac{1}{3}{k^{\prime}}^{2}\left(\CarlsonsymellintRD@{0}{{k^{\prime}}^{2}}{1}+\CarlsonsymellintRD@{0}{1}{{k^{\prime}}^{2}}\right)		Error	EllipticE[(k)^2] == Divide[1,3]*1 - (k)^(2)*(3*(EllipticF[ArcCos[Sqrt[0/1]],(1-1 - (k)^(2))/(1-0)]-EllipticE[ArcCos[Sqrt[0/1]],(1-1 - (k)^(2))/(1-0)])/((1-1 - (k)^(2))*(1-0)^(1/2))+ 3*(EllipticF[ArcCos[Sqrt[0/1 - (k)^(2)]],(1 - (k)^(2)-1)/(1 - (k)^(2)-0)]-EllipticE[ArcCos[Sqrt[0/1 - (k)^(2)]],(1 - (k)^(2)-1)/(1 - (k)^(2)-0)])/((1 - (k)^(2)-1)*(1 - (k)^(2)-0)^(1/2)))	Missing Macro Error	Failure	-	Failed [3 / 3] Result: DirectedInfinity[] Test Values: {Rule[k, 1]} Result: Complex[7.885081986624734, -2.293856789051463] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.25#Ex4	${\displaystyle{\displaystyle K\left(k\right)-E\left(k\right)=k^{2}D\left(k\right)}}$ \compellintKk@{k}-\compellintEk@{k} = k^{2}\compellintDk@{k}		EllipticK(k)- EllipticE(k) = (k)^(2)* (EllipticK(k) - EllipticE(k))/(k)^2	EllipticK[(k)^2]- EllipticE[(k)^2] == (k)^(2)* Divide[EllipticK[(k)^2] - EllipticE[(k)^2], (k)^4]	Successful	Failure	-	Failed [3 / 3] Result: Indeterminate Test Values: {Rule[k, 1]} Result: Complex[0.3274322182097533, -1.81658404135269] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.25#Ex4	${\displaystyle{\displaystyle k^{2}D\left(k\right)=\tfrac{1}{3}k^{2}R_{D}\left(0,{k^{\prime}}^{2},1\right)}}$ k^{2}\compellintDk@{k} = \tfrac{1}{3}k^{2}\CarlsonsymellintRD@{0}{{k^{\prime}}^{2}}{1}		Error	(k)^(2)* Divide[EllipticK[(k)^2] - EllipticE[(k)^2], (k)^4] == Divide[1,3]*(k)^(2)* 3*(EllipticF[ArcCos[Sqrt[0/1]],(1-1 - (k)^(2))/(1-0)]-EllipticE[ArcCos[Sqrt[0/1]],(1-1 - (k)^(2))/(1-0)])/((1-1 - (k)^(2))*(1-0)^(1/2))	Missing Macro Error	Failure	-	Failed [3 / 3] Result: DirectedInfinity[] Test Values: {Rule[k, 1]} Result: Complex[-1.5165865988698335, -0.6055280137842299] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.25#Ex5	${\displaystyle{\displaystyle E\left(k\right)-{k^{\prime}}^{2}K\left(k\right)=\tfrac{1}{3}k^{2}{k^{\prime}}^{2}R_{D}\left(0,1,{k^{\prime}}^{2}\right)}}$ \compellintEk@{k}-{k^{\prime}}^{2}\compellintKk@{k} = \tfrac{1}{3}k^{2}{k^{\prime}}^{2}\CarlsonsymellintRD@{0}{1}{{k^{\prime}}^{2}}		Error	EllipticE[(k)^2]-1 - (k)^(2)*EllipticK[(k)^2] == Divide[1,3]*(k)^(2)*1 - (k)^(2)*3*(EllipticF[ArcCos[Sqrt[0/1 - (k)^(2)]],(1 - (k)^(2)-1)/(1 - (k)^(2)-0)]-EllipticE[ArcCos[Sqrt[0/1 - (k)^(2)]],(1 - (k)^(2)-1)/(1 - (k)^(2)-0)])/((1 - (k)^(2)-1)*(1 - (k)^(2)-0)^(1/2))	Missing Macro Error	Failure	-	Failed [3 / 3] Result: Indeterminate Test Values: {Rule[k, 1]} Result: Complex[-2.3636107378197124, 2.0191745059478237] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.25.E2	${\displaystyle{\displaystyle\Pi\left(\alpha^{2},k\right)-K\left(k\right)=\tfrac{1}{3}\alpha^{2}R_{J}\left(0,{k^{\prime}}^{2},1,1-\alpha^{2}\right)}}$ \compellintPik@{\alpha^{2}}{k}-\compellintKk@{k} = \tfrac{1}{3}\alpha^{2}\CarlsonsymellintRJ@{0}{{k^{\prime}}^{2}}{1}{1-\alpha^{2}}		Error	EllipticPi[\[Alpha]^(2), (k)^2]- EllipticK[(k)^2] == Divide[1,3]*\[Alpha]^(2)* 3*(1-0)/(1-1 - \[Alpha]^(2))*(EllipticPi[(1-1 - \[Alpha]^(2))/(1-0),ArcCos[Sqrt[0/1]],(1-1 - (k)^(2))/(1-0)]-EllipticF[ArcCos[Sqrt[0/1]],(1-1 - (k)^(2))/(1-0)])/Sqrt[1-0]	Missing Macro Error	Failure	-	Failed [9 / 9] Result: Indeterminate Test Values: {Rule[k, 1], Rule[α, 1.5]} Result: Complex[-1.5241433161083033, 0.5547659663605348] Test Values: {Rule[k, 2], Rule[α, 1.5]} ... skip entries to safe data
19.25.E4	${\displaystyle{\displaystyle\Pi\left(\alpha^{2},k\right)=-\tfrac{1}{3}(k^{2}/\alpha^{2})R_{J}\left(0,1-k^{2},1,1-(k^{2}/\alpha^{2})\right)}}$ \compellintPik@{\alpha^{2}}{k} = -\tfrac{1}{3}(k^{2}/\alpha^{2})\CarlsonsymellintRJ@{0}{1-k^{2}}{1}{1-(k^{2}/\alpha^{2})}	${\displaystyle{\displaystyle-\infty<k^{2},k^{2}<1,1<\alpha^{2}}}$	Error	EllipticPi[\[Alpha]^(2), (k)^2] == -Divide[1,3]*((k)^(2)/\[Alpha]^(2))*3*(1-0)/(1-1 -((k)^(2)/\[Alpha]^(2)))*(EllipticPi[(1-1 -((k)^(2)/\[Alpha]^(2)))/(1-0),ArcCos[Sqrt[0/1]],(1-1 - (k)^(2))/(1-0)]-EllipticF[ArcCos[Sqrt[0/1]],(1-1 - (k)^(2))/(1-0)])/Sqrt[1-0]	Missing Macro Error	Failure	-	Skip - No test values generated
19.25.E5	${\displaystyle{\displaystyle F\left(\phi,k\right)=R_{F}\left(c-1,c-k^{2},c\right)}}$ \incellintFk@{\phi}{k} = \CarlsonsymellintRF@{c-1}{c-k^{2}}{c}		EllipticF(sin(phi), k) = 0.5*int(1/(sqrt(t+c - 1)*sqrt(t+c - (k)^(2))*sqrt(t+c)), t = 0..infinity)	EllipticF[\[Phi], (k)^2] == EllipticF[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)]/Sqrt[c-c - 1]	Failure	Failure	Failed [180 / 180] Result: Float(undefined)+Float(undefined)*I Test Values: {c = -3/2, phi = 1/2*3^(1/2)+1/2*I, k = 1} Result: 3.854689052+3.461698034*I Test Values: {c = -3/2, phi = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [180 / 180] Result: Complex[2.0026000841930385, 1.2187088711714384] Test Values: {Rule[c, -1.5], Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[1.4748265293714395, 0.7583435972865697] Test Values: {Rule[c, -1.5], Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.25.E6	${\displaystyle{\displaystyle\frac{\partial F\left(\phi,k\right)}{\partial k}=\tfrac{1}{3}kR_{D}\left(c-1,c,c-k^{2}\right)}}$ \pderiv{\incellintFk@{\phi}{k}}{k} = \tfrac{1}{3}k\CarlsonsymellintRD@{c-1}{c}{c-k^{2}}		Error	D[EllipticF[\[Phi], (k)^2], k] == Divide[1,3]*k*3*(EllipticF[ArcCos[Sqrt[c - 1/c - (k)^(2)]],(c - (k)^(2)-c)/(c - (k)^(2)-c - 1)]-EllipticE[ArcCos[Sqrt[c - 1/c - (k)^(2)]],(c - (k)^(2)-c)/(c - (k)^(2)-c - 1)])/((c - (k)^(2)-c)*(c - (k)^(2)-c - 1)^(1/2))	Missing Macro Error	Failure	-	Failed [180 / 180] Result: Indeterminate Test Values: {Rule[c, -1.5], Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.4045300788217367, 0.4404710702025501] Test Values: {Rule[c, -1.5], Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.25.E7	${\displaystyle{\displaystyle E\left(\phi,k\right)=2R_{G}\left(c-1,c-k^{2},c\right)-(c-1)R_{F}\left(c-1,c-k^{2},c\right)-\sqrt{(c-1)(c-k^{2})/c}}}$ \incellintEk@{\phi}{k} = 2\CarlsonsymellintRG@{c-1}{c-k^{2}}{c}-(c-1)\CarlsonsymellintRF@{c-1}{c-k^{2}}{c}-\sqrt{(c-1)(c-k^{2})/c}		Error	EllipticE[\[Phi], (k)^2] == 2*Sqrt[c-c - 1]*(EllipticE[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)]+(Cot[ArcCos[Sqrt[c - 1/c]]])^2*EllipticF[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)]+Cot[ArcCos[Sqrt[c - 1/c]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[c - 1/c]]]^2])-(c - 1)*EllipticF[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)]/Sqrt[c-c - 1]-Sqrt[(c - 1)*(c - (k)^(2))/c]	Missing Macro Error	Failure	-	Failed [180 / 180] Result: Complex[5.787775994567906, 4.022803158659452] Test Values: {Rule[c, -1.5], Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[6.805668366738806, 3.968311704298834] Test Values: {Rule[c, -1.5], Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.25.E9	${\displaystyle{\displaystyle E\left(\phi,k\right)=R_{F}\left(c-1,c-k^{2},c\right)-\tfrac{1}{3}k^{2}R_{D}\left(c-1,c-k^{2},c\right)}}$ \incellintEk@{\phi}{k} = \CarlsonsymellintRF@{c-1}{c-k^{2}}{c}-\tfrac{1}{3}k^{2}\CarlsonsymellintRD@{c-1}{c-k^{2}}{c}		Error	EllipticE[\[Phi], (k)^2] == EllipticF[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)]/Sqrt[c-c - 1]-Divide[1,3]*(k)^(2)* 3*(EllipticF[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)]-EllipticE[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)])/((c-c - (k)^(2))*(c-c - 1)^(1/2))	Missing Macro Error	Failure	-	Failed [180 / 180] Result: Complex[3.5743811704478246, 0.7698502565730785] Test Values: {Rule[c, -1.5], Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[3.9424508382496875, -1.017653751864599] Test Values: {Rule[c, -1.5], Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.25.E10	${\displaystyle{\displaystyle E\left(\phi,k\right)={k^{\prime}}^{2}R_{F}\left(c-1,c-k^{2},c\right)+\tfrac{1}{3}k^{2}{k^{\prime}}^{2}R_{D}\left(c-1,c,c-k^{2}\right)+k^{2}\sqrt{(c-1)/(c(c-k^{2}))}}}$ \incellintEk@{\phi}{k} = {k^{\prime}}^{2}\CarlsonsymellintRF@{c-1}{c-k^{2}}{c}+\tfrac{1}{3}k^{2}{k^{\prime}}^{2}\CarlsonsymellintRD@{c-1}{c}{c-k^{2}}+k^{2}\sqrt{(c-1)/(c(c-k^{2}))}	${\displaystyle{\displaystyle c>k^{2}}}$	Error	EllipticE[\[Phi], (k)^2] == 1 - (k)^(2)*EllipticF[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)]/Sqrt[c-c - 1]+Divide[1,3]*(k)^(2)*1 - (k)^(2)*3*(EllipticF[ArcCos[Sqrt[c - 1/c - (k)^(2)]],(c - (k)^(2)-c)/(c - (k)^(2)-c - 1)]-EllipticE[ArcCos[Sqrt[c - 1/c - (k)^(2)]],(c - (k)^(2)-c)/(c - (k)^(2)-c - 1)])/((c - (k)^(2)-c)*(c - (k)^(2)-c - 1)^(1/2))+ (k)^(2)*Sqrt[(c - 1)/(c*(c - (k)^(2)))]	Missing Macro Error	Failure	-	Failed [20 / 20] Result: Complex[-1.0687219916023158, 0.8637282710955538] Test Values: {Rule[c, 1.5], Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-1.7724732696890155, 1.0672164584507502] Test Values: {Rule[c, 1.5], Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.25.E11	${\displaystyle{\displaystyle E\left(\phi,k\right)=-\tfrac{1}{3}{k^{\prime}}^{2}R_{D}\left(c-k^{2},c,c-1\right)+\sqrt{(c-k^{2})/(c(c-1))}}}$ \incellintEk@{\phi}{k} = -\tfrac{1}{3}{k^{\prime}}^{2}\CarlsonsymellintRD@{c-k^{2}}{c}{c-1}+\sqrt{(c-k^{2})/(c(c-1))}	${\displaystyle{\displaystyle\phi\neq\tfrac{1}{2}\pi}}$	Error	EllipticE[\[Phi], (k)^2] == -Divide[1,3]*1 - (k)^(2)*3*(EllipticF[ArcCos[Sqrt[c - (k)^(2)/c - 1]],(c - 1-c)/(c - 1-c - (k)^(2))]-EllipticE[ArcCos[Sqrt[c - (k)^(2)/c - 1]],(c - 1-c)/(c - 1-c - (k)^(2))])/((c - 1-c)*(c - 1-c - (k)^(2))^(1/2))+Sqrt[(c - (k)^(2))/(c*(c - 1))]	Missing Macro Error	Failure	-	Failed [180 / 180] Result: Complex[3.6312701919621486, -1.3602272606820804] Test Values: {Rule[c, -1.5], Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.7754142926962797, -0.6029933704091625] Test Values: {Rule[c, -1.5], Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.25.E12	${\displaystyle{\displaystyle\frac{\partial E\left(\phi,k\right)}{\partial k}=-\tfrac{1}{3}kR_{D}\left(c-1,c-k^{2},c\right)}}$ \pderiv{\incellintEk@{\phi}{k}}{k} = -\tfrac{1}{3}k\CarlsonsymellintRD@{c-1}{c-k^{2}}{c}		Error	D[EllipticE[\[Phi], (k)^2], k] == -Divide[1,3]*k*3*(EllipticF[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)]-EllipticE[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)])/((c-c - (k)^(2))*(c-c - 1)^(1/2))	Missing Macro Error	Failure	-	Failed [180 / 180] Result: Complex[1.571781086254786, -0.44885861459835996] Test Values: {Rule[c, -1.5], Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[1.233812154439124, -0.8879986745755843] Test Values: {Rule[c, -1.5], Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.25.E13	${\displaystyle{\displaystyle D\left(\phi,k\right)=\tfrac{1}{3}R_{D}\left(c-1,c-k^{2},c\right)}}$ \incellintDk@{\phi}{k} = \tfrac{1}{3}\CarlsonsymellintRD@{c-1}{c-k^{2}}{c}		Error	Divide[EllipticF[\[Phi], (k)^2] - EllipticE[\[Phi], (k)^2], (k)^4] == Divide[1,3]*3*(EllipticF[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)]-EllipticE[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)])/((c-c - (k)^(2))*(c-c - 1)^(1/2))	Missing Macro Error	Failure	-	Failed [180 / 180] Result: Complex[-1.571781086254786, 0.44885861459835996] Test Values: {Rule[c, -1.5], Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.6083725296430629, 0.41279951787826946] Test Values: {Rule[c, -1.5], Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.25.E14	${\displaystyle{\displaystyle\Pi\left(\phi,\alpha^{2},k\right)-F\left(\phi,k\right)=\tfrac{1}{3}\alpha^{2}R_{J}\left(c-1,c-k^{2},c,c-\alpha^{2}\right)}}$ \incellintPik@{\phi}{\alpha^{2}}{k}-\incellintFk@{\phi}{k} = \tfrac{1}{3}\alpha^{2}\CarlsonsymellintRJ@{c-1}{c-k^{2}}{c}{c-\alpha^{2}}		Error	EllipticPi[\[Alpha]^(2), \[Phi],(k)^2]- EllipticF[\[Phi], (k)^2] == Divide[1,3]*\[Alpha]^(2)* 3*(c-c - 1)/(c-c - \[Alpha]^(2))*(EllipticPi[(c-c - \[Alpha]^(2))/(c-c - 1),ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)]-EllipticF[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)])/Sqrt[c-c - 1]	Missing Macro Error	Failure	-	Failed [300 / 300] Result: Complex[-0.9803588804354156, -0.9579910370435353] Test Values: {Rule[c, -1.5], Rule[k, 1], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.6164275583611891, -0.384238714210872] Test Values: {Rule[c, -1.5], Rule[k, 2], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.25.E16	${\displaystyle{\displaystyle\Pi\left(\phi,\alpha^{2},k\right)=-\tfrac{1}{3}\omega^{2}R_{J}\left(c-1,c-k^{2},c,c-\omega^{2}\right)+\sqrt{\frac{(c-1)(c-k^{2})}{(\alpha^{2}-1)(1-\omega^{2})}}\*R_{C}\left(c(\alpha^{2}-1)(1-\omega^{2}),(\alpha^{2}-c)(c-\omega^{2})\right)}}$ \incellintPik@{\phi}{\alpha^{2}}{k} = -\tfrac{1}{3}\omega^{2}\CarlsonsymellintRJ@{c-1}{c-k^{2}}{c}{c-\omega^{2}}+\sqrt{\frac{(c-1)(c-k^{2})}{(\alpha^{2}-1)(1-\omega^{2})}}\*\CarlsonellintRC@{c(\alpha^{2}-1)(1-\omega^{2})}{(\alpha^{2}-c)(c-\omega^{2})}	${\displaystyle{\displaystyle\omega^{2}=k^{2}/\alpha^{2}}}$	Error	EllipticPi[\[Alpha]^(2), \[Phi],(k)^2] == -Divide[1,3]*\[Omega]^(2)* 3*(c-c - 1)/(c-c - \[Omega]^(2))*(EllipticPi[(c-c - \[Omega]^(2))/(c-c - 1),ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)]-EllipticF[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)])/Sqrt[c-c - 1]+Sqrt[Divide[(c - 1)*(c - (k)^(2)),(\[Alpha]^(2)- 1)*(1 - \[Omega]^(2))]]* 1/Sqrt[(\[Alpha]^(2)- c)*(c - \[Omega]^(2))]*Hypergeometric2F1[1/2,1/2,3/2,1-(c*(\[Alpha]^(2)- 1)*(1 - \[Omega]^(2)))/((\[Alpha]^(2)- c)*(c - \[Omega]^(2)))]	Missing Macro Error	Aborted	-	Failed [300 / 300] Result: Complex[-0.11631142199526823, 0.9703799109463437] Test Values: {Rule[c, -1.5], Rule[k, 3], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ω, -2]} Result: Complex[-0.11631142199526823, 0.9703799109463437] Test Values: {Rule[c, -1.5], Rule[k, 3], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ω, 2]} ... skip entries to safe data
19.25.E17	${\displaystyle{\displaystyle F\left(\phi,k\right)=R_{F}\left(x,y,z\right)}}$ \incellintFk@{\phi}{k} = \CarlsonsymellintRF@{x}{y}{z}		EllipticF(sin(phi), k) = 0.5*int(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+x + y*I)), t = 0..infinity)	EllipticF[\[Phi], (k)^2] == EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]	Aborted	Failure	Failed [300 / 300] Result: 2.547570015-.6488873983*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2, k = 1} Result: 2.209888328-.6080126261*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2, k = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.5939484671297026, -0.40701440305540804] Test Values: {Rule[k, 1], Rule[x, 1.5], Rule[y, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.5587134153531784, -0.34669285510288844] Test Values: {Rule[k, 2], Rule[x, 1.5], Rule[y, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.25.E18	${\displaystyle{\displaystyle(x,y,z)=(c-1,c-k^{2},c)}}$ (x,y,z) = (c-1,c-k^{2},c)		(x , y ,(x + y*I)) = (c - 1 , c - (k)^(2), c)	(x , y ,(x + y*I)) == (c - 1 , c - (k)^(2), c)	Skipped - no semantic math	Skipped - no semantic math	-	-
19.25#Ex6	${\displaystyle{\displaystyle\phi=\operatorname{arccos}\sqrt{\ifrac{x}{z}}}}$ \phi = \acos@@{\sqrt{\ifrac{x}{z}}}		phi = arccos(sqrt((x)/(x + y*I)))	\[Phi] == ArcCos[Sqrt[Divide[x,x + y*I]]]	Failure	Failure	Failed [180 / 180] Result: .806272406e-1+.9406867936*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2} Result: .806272406e-1+.593132064e-1*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, y = 3/2} ... skip entries to safe data	Failed [180 / 180] Result: Complex[-0.35238546150522904, 0.6906867935097715] Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-1.0353981633974483, 0.8736994954019909] Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.25#Ex6	${\displaystyle{\displaystyle\operatorname{arccos}\sqrt{\ifrac{x}{z}}=\operatorname{arcsin}\sqrt{\ifrac{(z-x)}{z}}}}$ \acos@@{\sqrt{\ifrac{x}{z}}} = \asin@@{\sqrt{\ifrac{(z-x)}{z}}}		arccos(sqrt((x)/(x + y*I))) = arcsin(sqrt(((x + y*I)- x)/(x + y*I)))	ArcCos[Sqrt[Divide[x,x + y*I]]] == ArcSin[Sqrt[Divide[(x + y*I)- x,x + y*I]]]	Failure	Failure	Successful [Tested: 18]	Successful [Tested: 18]
19.25#Ex7	${\displaystyle{\displaystyle k=\sqrt{\frac{z-y}{z-x}}}}$ k = \sqrt{\frac{z-y}{z-x}}		k = sqrt(((x + y*I)- y)/((x + y*I)- x))	k == Sqrt[Divide[(x + y*I)- y,(x + y*I)- x]]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.25#Ex8	${\displaystyle{\displaystyle\alpha^{2}=\frac{z-p}{z-x}}}$ \alpha^{2} = \frac{z-p}{z-x}		(alpha)^(2) = ((x + y*I)- p)/((x + y*I)- x)	\[Alpha]^(2) == Divide[(x + y*I)- p,(x + y*I)- x]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.25.E24	${\displaystyle{\displaystyle(z-x)^{1/2}R_{F}\left(x,y,z\right)=F\left(\phi,k\right)}}$ (z-x)^{1/2}\CarlsonsymellintRF@{x}{y}{z} = \incellintFk@{\phi}{k}		((x + y*I)- x)^(1/2)* 0.5*int(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+x + y*I)), t = 0..infinity) = EllipticF(sin(phi), k)	((x + y*I)- x)^(1/2)* EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x] == EllipticF[\[Phi], (k)^2]	Aborted	Failure	Failed [300 / 300] Result: -1.167656510+1.966567574*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2, k = 1} Result: -.8299748231+1.925692802*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2, k = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.015324342917649614, 0.4565416109140732] Test Values: {Rule[k, 1], Rule[x, 1.5], Rule[y, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.050559394694173865, 0.3962200629615536] Test Values: {Rule[k, 2], Rule[x, 1.5], Rule[y, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.25.E25	${\displaystyle{\displaystyle(z-x)^{3/2}R_{D}\left(x,y,z\right)=(3/k^{2})(F\left(\phi,k\right)-E\left(\phi,k\right))}}$ (z-x)^{3/2}\CarlsonsymellintRD@{x}{y}{z} = (3/k^{2})(\incellintFk@{\phi}{k}-\incellintEk@{\phi}{k})		Error	((x + y*I)- x)^(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)) == (3/(k)^(2))*(EllipticF[\[Phi], (k)^2]- EllipticE[\[Phi], (k)^2])	Missing Macro Error	Failure	-	Failed [300 / 300] Result: Complex[-0.9041684186949032, 0.18989946051507803] Test Values: {Rule[k, 1], Rule[x, 1.5], Rule[y, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.8729885067685752, 0.19149534336253457] Test Values: {Rule[k, 2], Rule[x, 1.5], Rule[y, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.25.E26	${\displaystyle{\displaystyle(z-x)^{3/2}R_{J}\left(x,y,z,p\right)=(3/\alpha^{2}){(\Pi\left(\phi,\alpha^{2},k\right)-F\left(\phi,k\right))}}}$ (z-x)^{3/2}\CarlsonsymellintRJ@{x}{y}{z}{p} = (3/\alpha^{2}){(\incellintPik@{\phi}{\alpha^{2}}{k}-\incellintFk@{\phi}{k})}		Error	((x + y*I)- x)^(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] == (3/\[Alpha]^(2))*(EllipticPi[\[Alpha]^(2), \[Phi],(k)^2]- EllipticF[\[Phi], (k)^2])	Missing Macro Error	Failure	-	Failed [300 / 300] Result: Complex[-8.905365206673954*^-4, 0.6653826564189609] Test Values: {Rule[k, 1], 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[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.030816807002235325, 0.6810951786851601] Test Values: {Rule[k, 2], 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[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.25.E27	${\displaystyle{\displaystyle 2(z-x)^{-1/2}R_{G}\left(x,y,z\right)=E\left(\phi,k\right)+(\cot\phi)^{2}F\left(\phi,k\right)+(\cot\phi)\sqrt{1-k^{2}{\sin^{2}}\phi}}}$ 2(z-x)^{-1/2}\CarlsonsymellintRG@{x}{y}{z} = \incellintEk@{\phi}{k}+(\cot@@{\phi})^{2}\incellintFk@{\phi}{k}+(\cot@@{\phi})\sqrt{1-k^{2}\sin^{2}@@{\phi}}		Error	2*((x + y*I)- x)^(- 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]) == EllipticE[\[Phi], (k)^2]+(Cot[\[Phi]])^(2)* EllipticF[\[Phi], (k)^2]+(Cot[\[Phi]])*Sqrt[1 - (k)^(2)* (Sin[\[Phi]])^(2)]	Missing Macro Error	Failure	-	Failed [300 / 300] Result: Complex[-1.8997799949200251, -0.4031557744461449] Test Values: {Rule[k, 1], Rule[x, 1.5], Rule[y, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-3.0701379688219372, -2.1411109504853227] Test Values: {Rule[k, 2], Rule[x, 1.5], Rule[y, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.25#Ex9	${\displaystyle{\displaystyle\Delta(\mathrm{n,d})=k^{2}}}$ \Delta(\mathrm{n,d}) = k^{2}		Delta(n , d) = (k)^(2)	\[CapitalDelta][n , d] == (k)^(2)	Skipped - no semantic math	Skipped - no semantic math	-	-
19.25#Ex10	${\displaystyle{\displaystyle\Delta(\mathrm{d,c})={k^{\prime}}^{2}}}$ \Delta(\mathrm{d,c}) = {k^{\prime}}^{2}		Delta(d , c) = 1 - (k)^(2)	\[CapitalDelta][d , c] == 1 - (k)^(2)	Skipped - no semantic math	Skipped - no semantic math	-	-
19.25#Ex11	${\displaystyle{\displaystyle\Delta(\mathrm{n,c})=1}}$ \Delta(\mathrm{n,c}) = 1		Delta(n , c) = 1	\[CapitalDelta][n , c] == 1	Skipped - no semantic math	Skipped - no semantic math	-	-
19.25.E30	${\displaystyle{\displaystyle\operatorname{am}\left(u,k\right)=R_{C}\left({\operatorname{cs}^{2}}\left(u,k\right),{\operatorname{ns}^{2}}\left(u,k\right)\right)}}$ \Jacobiamk@{u}{k} = \CarlsonellintRC@{\Jacobiellcsk^{2}@{u}{k}}{\Jacobiellnsk^{2}@{u}{k}}		Error	JacobiAmplitude[u, Power[k, 2]] == 1/Sqrt[(JacobiNS[u, (k)^2])^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-((JacobiCS[u, (k)^2])^(2))/((JacobiNS[u, (k)^2])^(2))]	Missing Macro Error	Aborted	-	Failed [18 / 30] Result: Complex[-0.5428587296705786, 0.8636075147962846] Test Values: {Rule[k, 1], Rule[u, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} Result: Complex[-0.6732377468613371, 0.8494366739388763] Test Values: {Rule[k, 2], Rule[u, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.25.E31	${\displaystyle{\displaystyle u=R_{F}\left({\operatorname{ps}^{2}}\left(u,k\right),{\operatorname{qs}^{2}}\left(u,k\right),{\operatorname{rs}^{2}}\left(u,k\right)\right)}}$ u = \CarlsonsymellintRF@{\genJacobiellk{p}{s}^{2}@{u}{k}}{\genJacobiellk{q}{s}^{2}@{u}{k}}{\genJacobiellk{r}{s}^{2}@{u}{k}}		u = 0.5*int(1/(sqrt(t+genJacobiellk(p)*(s)^(2)* u*k)*sqrt(t+genJacobiellk(q)*(s)^(2)* u*k)*sqrt(t+genJacobiellk(r)*(s)^(2)* u*k)), t = 0..infinity)	u == EllipticF[ArcCos[Sqrt[genJacobiellk[p]*(s)^(2)* u*k/genJacobiellk[r]*(s)^(2)* u*k]],(genJacobiellk[r]*(s)^(2)* u*k-genJacobiellk[q]*(s)^(2)* u*k)/(genJacobiellk[r]*(s)^(2)* u*k-genJacobiellk[p]*(s)^(2)* u*k)]/Sqrt[genJacobiellk[r]*(s)^(2)* u*k-genJacobiellk[p]*(s)^(2)* u*k]	Aborted	Failure	Error	Failed [300 / 300] Result: Plus[Complex[0.43301270189221935, 0.24999999999999997], Times[Complex[-0.78471422644353, -0.9906313764027224], Power[Times[Complex[-1.7426678688862403, -1.3308892896287465], genJacobiellk], Rational[-1, 2]]]] Test Values: {Rule[k, 1], 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[r, -1.5], Rule[s, -1.5], Rule[u, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Plus[Complex[0.43301270189221935, 0.24999999999999997], Times[Complex[-0.3766936106342851, -1.225388931598258], Power[Times[Complex[-3.4853357377724805, -2.661778579257493], genJacobiellk], Rational[-1, 2]]]] Test Values: {Rule[k, 2], 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[r, -1.5], Rule[s, -1.5], Rule[u, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data