19.16

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
19.16.E1	${\displaystyle{\displaystyle R_{F}\left(x,y,z\right)=\frac{1}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{s(t)}}}$ \CarlsonsymellintRF@{x}{y}{z} = \frac{1}{2}\int_{0}^{\infty}\frac{\diff{t}}{s(t)}		0.5*int(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+x + y*I)), t = 0..infinity) = (1)/(2)*int((1)/(s(t)), t = 0..infinity)	EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x] == Divide[1,2]*Integrate[Divide[1,s[t]], {t, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Failed [108 / 108] Result: Float(infinity)+1.326265449*I Test Values: {s = -3/2, x = 3/2, y = -3/2} Result: Float(infinity)+Float(infinity)*I Test Values: {s = -3/2, x = 3/2, y = 3/2} ... skip entries to safe data	Failed [108 / 108] Result: Complex[52.57956240437182, 0.6784437678906974] Test Values: {Rule[s, -1.5], Rule[x, 1.5], Rule[y, -1.5]} Result: Complex[52.453473067488765, -0.7809212115368181] Test Values: {Rule[s, -1.5], Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.16.E2	${\displaystyle{\displaystyle R_{J}\left(x,y,z,p\right)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{s(t)(t+p)}}}$ \CarlsonsymellintRJ@{x}{y}{z}{p} = \frac{3}{2}\int_{0}^{\infty}\frac{\diff{t}}{s(t)(t+p)}		Error	3*(x + y*I-x)/(x + y*I-p)*(EllipticPi[(x + y*I-p)/(x + y*I-x),ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/Sqrt[x + y*I-x] == Divide[3,2]*Integrate[Divide[1,s[t]*(t + p)], {t, 0, Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Skipped - Because timed out
19.16.E3	${\displaystyle{\displaystyle R_{G}\left(x,y,z\right)=\frac{1}{4\pi}\int_{0}^{2\pi}\!\!\!\!\int_{0}^{\pi}\left(x{\sin^{2}}\theta{\cos^{2}}\phi+y{\sin^{2}}\theta{\sin^{2}}\phi+z{\cos^{2}}\theta\right)^{\frac{1}{2}}\sin\theta\mathrm{d}\theta\mathrm{d}\phi}}$ \CarlsonsymellintRG@{x}{y}{z} = \frac{1}{4\pi}\int_{0}^{2\pi}\!\!\!\!\int_{0}^{\pi}\left(x\sin^{2}@@{\theta}\cos^{2}@@{\phi}+y\sin^{2}@@{\theta}\sin^{2}@@{\phi}+z\cos^{2}@@{\theta}\right)^{\frac{1}{2}}\sin@@{\theta}\diff{\theta}\diff{\phi}		Error	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]) == Divide[1,4*Pi]*Integrate[Integrate[(x*(Sin[\[Theta]])^(2)* (Cos[\[Phi]])^(2)+ y*(Sin[\[Theta]])^(2)* (Sin[\[Phi]])^(2)+(x + y*I)*(Cos[\[Theta]])^(2))^(Divide[1,2])* Sin[\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None], {\[Phi], 0, 2*Pi}, GenerateConditions->None]	Missing Macro Error	Aborted	-	Skipped - Because timed out
19.16.E4	${\displaystyle{\displaystyle s(t)=\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}}}$ s(t) = \sqrt{t+x}\sqrt{t+y}\sqrt{t+z}		s(t) = sqrt(t + x)*sqrt(t + y)*sqrt(t +(x + y*I))	s[t] == Sqrt[t + x]*Sqrt[t + y]*Sqrt[t +(x + y*I)]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.16.E5	${\displaystyle{\displaystyle R_{D}\left(x,y,z\right)=R_{J}\left(x,y,z,z\right)}}$ \CarlsonsymellintRD@{x}{y}{z} = \CarlsonsymellintRJ@{x}{y}{z}{z}		Error	3*(EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticE[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/((x + y*I-y)*(x + y*I-x)^(1/2)) == 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]	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
19.16.E5	${\displaystyle{\displaystyle R_{J}\left(x,y,z,z\right)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{s(t)(t+z)}}}$ \CarlsonsymellintRJ@{x}{y}{z}{z} = \frac{3}{2}\int_{0}^{\infty}\frac{\diff{t}}{s(t)(t+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] == Divide[3,2]*Integrate[Divide[1,s[t]*(t +(x + y*I))], {t, 0, Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Skipped - Because timed out
19.16.E6	${\displaystyle{\displaystyle R_{C}\left(x,y\right)=R_{F}\left(x,y,y\right)}}$ \CarlsonellintRC@{x}{y} = \CarlsonsymellintRF@{x}{y}{y}		Error	1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)] == EllipticF[ArcCos[Sqrt[x/y]],(y-y)/(y-x)]/Sqrt[y-x]	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
19.16#Ex3	${\displaystyle{\displaystyle c={\csc^{2}}\phi}}$ c = \csc^{2}@@{\phi}		c = (csc(phi))^(2)	c == (Csc[\[Phi]])^(2)	Failure	Failure	Failed [60 / 60] Result: -2.359812877+.7993130071*I Test Values: {c = -3/2, phi = 1/2*3^(1/2)+1/2*I} Result: -1.296085040-.8173084059*I Test Values: {c = -3/2, phi = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [60 / 60] Result: Complex[-3.841312467237177, 3.4490957612740374] Test Values: {Rule[c, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.17530792640393877, -3.4502399957777015] Test Values: {Rule[c, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data