8.18

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
8.18.E2	${\displaystyle{\displaystyle\xi=-\ln x}}$ \xi = -\ln@@{x}		xi = - ln(x)	\[Xi] == - Log[x]	Failure	Failure	Failed [30 / 30] Result: 1.271490512+.5000000000*I Test Values: {x = 1.5, xi = 1/2*3^(1/2)+1/2*I} Result: -.945348919e-1+.8660254040*I Test Values: {x = 1.5, xi = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [30 / 30] Result: Complex[1.271490511892603, 0.49999999999999994] Test Values: {Rule[x, 1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.0945348918918354, 0.8660254037844387] Test Values: {Rule[x, 1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
8.18.E4	${\displaystyle{\displaystyle aF_{k+1}=(k+b-a\xi)F_{k}+k\xi F_{k-1}}}$ aF_{k+1} = (k+b-a\xi)F_{k}+k\xi F_{k-1}		a*F[k + 1] = (k + b - a*xi)*F[k]+ k*xi*F[k - 1]	a*Subscript[F, k + 1] == (k + b - a*\[Xi])*Subscript[F, k]+ k*\[Xi]*Subscript[F, k - 1]	Skipped - no semantic math	Skipped - no semantic math	-	-
8.18#Ex1	${\displaystyle{\displaystyle F_{0}=a^{-b}Q\left(b,a\xi\right)}}$ F_{0} = a^{-b}\normincGammaQ@{b}{a\xi}	${\displaystyle{\displaystyle\Re b>0}}$	F[0] = (a)^(- b)* GAMMA(b, a*xi)/GAMMA(b)	Subscript[F, 0] == (a)^(- b)* GammaRegularized[b, a*\[Xi]]	Failure	Failure	Failed [300 / 300] Result: 1.253924788+1.407498490*I Test Values: {a = -1.5, b = -1.5, xi = 1/2*3^(1/2)+1/2*I, F[0] = 1/2*3^(1/2)+1/2*I} Result: -.1121006157+1.773523894*I Test Values: {a = -1.5, b = -1.5, xi = 1/2*3^(1/2)+1/2*I, F[0] = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[1.2539247882576399, 1.4074984905445393] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[F, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.11210061552679867, 1.7735238943289782] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[F, 0], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
8.18#Ex2	${\displaystyle{\displaystyle F_{1}=\frac{b-a\xi}{a}F_{0}+\frac{\xi^{b}e^{-a\xi}}{a\Gamma\left(b\right)}}}$ F_{1} = \frac{b-a\xi}{a}F_{0}+\frac{\xi^{b}e^{-a\xi}}{a\EulerGamma@{b}}	${\displaystyle{\displaystyle\Re b>0}}$	F[1] = (b - a*xi)/(a)*F[0]+((xi)^(b)* exp(- a*xi))/(a*GAMMA(b))	Subscript[F, 1] == Divide[b - a*\[Xi],a]*Subscript[F, 0]+Divide[\[Xi]^(b)* Exp[- a*\[Xi]],a*Gamma[b]]	Failure	Failure	Failed [300 / 300] Result: 2.329643864+4.621882749*I Test Values: {a = -1.5, b = 1.5, xi = 1/2*3^(1/2)+1/2*I, F[0] = 1/2*3^(1/2)+1/2*I, F[1] = 1/2*3^(1/2)+1/2*I} Result: .9636184598+4.987908153*I Test Values: {a = -1.5, b = 1.5, xi = 1/2*3^(1/2)+1/2*I, F[0] = 1/2*3^(1/2)+1/2*I, F[1] = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[2.32964386182885, 4.621882746395113] Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[F, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[F, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.9636184580444114, 4.9879081501795515] Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[F, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[F, 1], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
8.18.E6	${\displaystyle{\displaystyle\left(\frac{1-e^{-t}}{t}\right)^{b-1}=\sum_{k=0}^{\infty}d_{k}(t-\xi)^{k}}}$ \left(\frac{1-e^{-t}}{t}\right)^{b-1} = \sum_{k=0}^{\infty}d_{k}(t-\xi)^{k}		((1 - exp(- t))/(t))^(b - 1) = sum(d[k]*(t - xi)^(k), k = 0..infinity)	(Divide[1 - Exp[- t],t])^(b - 1) == Sum[Subscript[d, k]*(t - \[Xi])^(k), {k, 0, Infinity}, GenerateConditions->None]	Skipped - no semantic math	Skipped - no semantic math	-	-
8.18#Ex3	${\displaystyle{\displaystyle d_{0}=\left(\frac{1-x}{\xi}\right)^{b-1}}}$ d_{0} = \left(\frac{1-x}{\xi}\right)^{b-1}		d[0] = ((1 - x)/(xi))^(b - 1)	Subscript[d, 0] == (Divide[1 - x,\[Xi]])^(b - 1)	Skipped - no semantic math	Skipped - no semantic math	-	-
8.18#Ex4	${\displaystyle{\displaystyle d_{1}=\frac{x\xi+x-1}{(1-x)\xi}(b-1)d_{0}}}$ d_{1} = \frac{x\xi+x-1}{(1-x)\xi}(b-1)d_{0}		d[1] = (x*xi + x - 1)/((1 - x)*xi)*(b - 1)*d[0]	Subscript[d, 1] == Divide[x*\[Xi]+ x - 1,(1 - x)*\[Xi]]*(b - 1)*Subscript[d, 0]	Skipped - no semantic math	Skipped - no semantic math	-	-
8.18.E8	${\displaystyle{\displaystyle x_{0}=a/(a+b)}}$ x_{0} = a/(a+b)		x[0] = a/(a + b)	Subscript[x, 0] == a/(a + b)	Skipped - no semantic math	Skipped - no semantic math	-	-
8.18.E10	${\displaystyle{\displaystyle-\tfrac{1}{2}\eta^{2}=x_{0}\ln\left(\frac{x}{x_{0}}\right)+(1-x_{0})\ln\left(\frac{1-x}{1-x_{0}}\right)}}$ -\tfrac{1}{2}\eta^{2} = x_{0}\ln@{\frac{x}{x_{0}}}+(1-x_{0})\ln@{\frac{1-x}{1-x_{0}}}		-(1)/(2)*(eta)^(2) = x[0]*ln((x)/(x[0]))+(1 - x[0])*ln((1 - x)/(1 - x[0]))	-Divide[1,2]*\[Eta]^(2) == Subscript[x, 0]*Log[Divide[x,Subscript[x, 0]]]+(1 - Subscript[x, 0])*Log[Divide[1 - x,1 - Subscript[x, 0]]]	Failure	Failure	Failed [300 / 300] Result: .580000474e-1+.458917392e-1*I Test Values: {eta = 1/2*3^(1/2)+1/2*I, x = 1.5, x[0] = 1/2*3^(1/2)+1/2*I} Result: 2.269862383+1.019641337*I Test Values: {eta = 1/2*3^(1/2)+1/2*I, x = 1.5, x[0] = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.058000047924774145, 0.04589173995258988] Test Values: {Rule[x, 1.5], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[2.2698623824536366, 1.0196413375539057] Test Values: {Rule[x, 1.5], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, 0], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
8.18.E11	${\displaystyle{\displaystyle c_{0}(\eta)=\frac{1}{\eta}-\frac{\sqrt{x_{0}(1-x_{0})}}{x-x_{0}}}}$ c_{0}(\eta) = \frac{1}{\eta}-\frac{\sqrt{x_{0}(1-x_{0})}}{x-x_{0}}		c[0](eta) = (1)/(eta)-(sqrt(x[0]*(1 - x[0])))/(x - x[0])	Subscript[c, 0][\[Eta]] == Divide[1,\[Eta]]-Divide[Sqrt[Subscript[x, 0]*(1 - Subscript[x, 0])],x - Subscript[x, 0]]	Skipped - no semantic math	Skipped - no semantic math	-	-
8.18.E12	${\displaystyle{\displaystyle c_{0}(0)=\frac{1-2x_{0}}{3\sqrt{x_{0}(1-x_{0})}}}}$ c_{0}(0) = \frac{1-2x_{0}}{3\sqrt{x_{0}(1-x_{0})}}		c[0](0) = (1 - 2*x[0])/(3*sqrt(x[0]*(1 - x[0])))	Subscript[c, 0][0] == Divide[1 - 2*Subscript[x, 0],3*Sqrt[Subscript[x, 0]*(1 - Subscript[x, 0])]]	Skipped - no semantic math	Skipped - no semantic math	-	-
8.18.E15	${\displaystyle{\displaystyle\mu\ln\zeta-\zeta=\ln x+\mu\ln\left(1-x\right)+(1+\mu)\ln\left(1+\mu\right)-\mu}}$ \mu\ln@@{\zeta}-\zeta = \ln@@{x}+\mu\ln@{1-x}+(1+\mu)\ln@{1+\mu}-\mu		mu*ln(zeta)- zeta = ln(x)+ mu*ln(1 - x)+(1 + mu)*ln(1 + mu)- mu	\[Mu]*Log[\[Zeta]]- \[Zeta] == Log[x]+ \[Mu]*Log[1 - x]+(1 + \[Mu])*Log[1 + \[Mu]]- \[Mu]	Failure	Failure	Failed [299 / 300] Result: .405976146-2.738439399*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, x = 1.5, zeta = 1/2*3^(1/2)+1/2*I} Result: .9866033870-1.744115280*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, x = 1.5, zeta = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [299 / 300] Result: Complex[0.4059761460255107, -2.7384393975724306] Test Values: {Rule[x, 1.5], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.0560847852373059, 1.7517066341083583] Test Values: {Rule[x, 1.5], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
8.18.E16	${\displaystyle{\displaystyle h_{0}(\zeta,\mu)=\mu\left(\frac{1}{\zeta-\mu}-\frac{(1+\mu)^{-3/2}}{x_{0}-x}\right)}}$ h_{0}(\zeta,\mu) = \mu\left(\frac{1}{\zeta-\mu}-\frac{(1+\mu)^{-3/2}}{x_{0}-x}\right)		h[0](zeta , mu) = mu*((1)/(zeta - mu)-((1 + mu)^(- 3/2))/(x[0]- x))	Subscript[h, 0][\[Zeta], \[Mu]] == \[Mu]*(Divide[1,\[Zeta]- \[Mu]]-Divide[(1 + \[Mu])^(- 3/2),Subscript[x, 0]- x])	Skipped - no semantic math	Skipped - no semantic math	-	-
8.18.E17	${\displaystyle{\displaystyle h_{0}(\mu,\mu)=\frac{1}{3}\left(\frac{1-\mu}{\sqrt{1+\mu}}-1\right)}}$ h_{0}(\mu,\mu) = \frac{1}{3}\left(\frac{1-\mu}{\sqrt{1+\mu}}-1\right)		h[0](mu , mu) = (1)/(3)*((1 - mu)/(sqrt(1 + mu))- 1)	Subscript[h, 0][\[Mu], \[Mu]] == Divide[1,3]*(Divide[1 - \[Mu],Sqrt[1 + \[Mu]]]- 1)	Skipped - no semantic math	Skipped - no semantic math	-	-
8.18.E18	${\displaystyle{\displaystyle I_{x}\left(a,b\right)=p}}$ \normincBetaI{x}@{a}{b} = p	${\displaystyle{\displaystyle 0\leq p,p\leq 1,\Re a>0,\Re b>0,\Re(a+b)>0}}$	Error	BetaRegularized[x, a, b] == p	Missing Macro Error	Failure	-	Failed [105 / 108] Result: DirectedInfinity[] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[p, 0.5], Rule[x, 1.5]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[p, 0.5], Rule[x, 0.5]} ... skip entries to safe data