19.8

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
19.8#Ex1	${\displaystyle{\displaystyle a_{n+1}=\frac{a_{n}+g_{n}}{2}}}$ a_{n+1} = \frac{a_{n}+g_{n}}{2}		a[n + 1] = (a[n]+ g[n])/(2)	Subscript[a, n + 1] == Divide[Subscript[a, n]+ Subscript[g, n],2]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.8#Ex2	${\displaystyle{\displaystyle g_{n+1}=\sqrt{a_{n}g_{n}}}}$ g_{n+1} = \sqrt{a_{n}g_{n}}		g[n + 1] = sqrt(a[n]*g[n])	Subscript[g, n + 1] == Sqrt[Subscript[a, n]*Subscript[g, n]]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.8.E2	${\displaystyle{\displaystyle c_{n}=\sqrt{a_{n}^{2}-g_{n}^{2}}}}$ c_{n} = \sqrt{a_{n}^{2}-g_{n}^{2}}		c[n] = sqrt((a[n])^(2)- (g[n])^(2))	Subscript[c, n] == Sqrt[(Subscript[a, n])^(2)- (Subscript[g, n])^(2)]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.8.E3	${\displaystyle{\displaystyle c_{n+1}=\frac{a_{n}-g_{n}}{2}}}$ c_{n+1} = \frac{a_{n}-g_{n}}{2}		c[n + 1] = (a[n]- g[n])/(2)	Subscript[c, n + 1] == Divide[Subscript[a, n]- Subscript[g, n],2]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.8.E4	${\displaystyle{\displaystyle\frac{1}{M\left(a_{0},g_{0}\right)}=\frac{2}{\pi}\int_{0}^{\pi/2}\frac{\mathrm{d}\theta}{\sqrt{a_{0}^{2}{\cos^{2}}\theta+g_{0}^{2}{\sin^{2}}\theta}}}}$ \frac{1}{\AGM@{a_{0}}{g_{0}}} = \frac{2}{\pi}\int_{0}^{\pi/2}\frac{\diff{\theta}}{\sqrt{a_{0}^{2}\cos^{2}@@{\theta}+g_{0}^{2}\sin^{2}@@{\theta}}}		(1)/(GaussAGM(a[0], g[0])) = (2)/(Pi)*int((1)/(sqrt((a[0])^(2)*(cos(theta))^(2)+ (g[0])^(2)*(sin(theta))^(2))), theta = 0..Pi/2)	Error	Failure	Missing Macro Error	Error	Skip - symbolical successful subtest
19.8.E4	${\displaystyle{\displaystyle\frac{2}{\pi}\int_{0}^{\pi/2}\frac{\mathrm{d}\theta}{\sqrt{a_{0}^{2}{\cos^{2}}\theta+g_{0}^{2}{\sin^{2}}\theta}}=\frac{1}{\pi}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t(t+a_{0}^{2})(t+g_{0}^{2})}}}}$ \frac{2}{\pi}\int_{0}^{\pi/2}\frac{\diff{\theta}}{\sqrt{a_{0}^{2}\cos^{2}@@{\theta}+g_{0}^{2}\sin^{2}@@{\theta}}} = \frac{1}{\pi}\int_{0}^{\infty}\frac{\diff{t}}{\sqrt{t(t+a_{0}^{2})(t+g_{0}^{2})}}		(2)/(Pi)*int((1)/(sqrt((a[0])^(2)*(cos(theta))^(2)+ (g[0])^(2)*(sin(theta))^(2))), theta = 0..Pi/2) = (1)/(Pi)*int((1)/(sqrt(t*(t + (a[0])^(2))*(t + (g[0])^(2)))), t = 0..infinity)	Divide[2,Pi]*Integrate[Divide[1,Sqrt[(Subscript[a, 0])^(2)*(Cos[\[Theta]])^(2)+ (Subscript[g, 0])^(2)*(Sin[\[Theta]])^(2)]], {\[Theta], 0, Pi/2}, GenerateConditions->None] == Divide[1,Pi]*Integrate[Divide[1,Sqrt[t*(t + (Subscript[a, 0])^(2))*(t + (Subscript[g, 0])^(2))]], {t, 0, Infinity}, GenerateConditions->None]	Aborted	Aborted	Skipped - Because timed out	Skipped - Because timed out
19.8.E5	${\displaystyle{\displaystyle K\left(k\right)=\frac{\pi}{2M\left(1,k^{\prime}\right)}}}$ \compellintKk@{k} = \frac{\pi}{2\AGM@{1}{k^{\prime}}}	${\displaystyle{\displaystyle-\infty<k^{2},k^{2}<1}}$	EllipticK(k) = (Pi)/(2*GaussAGM(1, sqrt(1 - (k)^(2))))	Error	Failure	Missing Macro Error	Error	-
19.8.E6	${\displaystyle{\displaystyle E\left(k\right)=\frac{\pi}{2M\left(1,k^{\prime}\right)}\left(a_{0}^{2}-\sum_{n=0}^{\infty}2^{n-1}c_{n}^{2}\right)}}$ \compellintEk@{k} = \frac{\pi}{2\AGM@{1}{k^{\prime}}}\left(a_{0}^{2}-\sum_{n=0}^{\infty}2^{n-1}c_{n}^{2}\right)	${\displaystyle{\displaystyle-\infty<k^{2},k^{2}<1,a_{0}=1,g_{0}=k^{\prime}}}$	EllipticE(k) = (Pi)/(2*GaussAGM(1, sqrt(1 - (k)^(2))))*((a[0])^(2)- sum((2)^(n - 1)* (c[n])^(2), n = 0..infinity))	Error	Failure	Missing Macro Error	Error	-
19.8.E6	${\displaystyle{\displaystyle\frac{\pi}{2M\left(1,k^{\prime}\right)}\left(a_{0}^{2}-\sum_{n=0}^{\infty}2^{n-1}c_{n}^{2}\right)=K\left(k\right)\left(a_{1}^{2}-\sum_{n=2}^{\infty}2^{n-1}c_{n}^{2}\right)}}$ \frac{\pi}{2\AGM@{1}{k^{\prime}}}\left(a_{0}^{2}-\sum_{n=0}^{\infty}2^{n-1}c_{n}^{2}\right) = \compellintKk@{k}\left(a_{1}^{2}-\sum_{n=2}^{\infty}2^{n-1}c_{n}^{2}\right)	${\displaystyle{\displaystyle-\infty<k^{2},k^{2}<1,a_{0}=1,g_{0}=k^{\prime}}}$	(Pi)/(2*GaussAGM(1, sqrt(1 - (k)^(2))))*((a[0])^(2)- sum((2)^(n - 1)* (c[n])^(2), n = 0..infinity)) = EllipticK(k)*((a[1])^(2)- sum((2)^(n - 1)* (c[n])^(2), n = 2..infinity))	Error	Failure	Missing Macro Error	Error	-
19.8.E7	${\displaystyle{\displaystyle\Pi\left(\alpha^{2},k\right)=\frac{\pi}{4M\left(1,k^{\prime}\right)}\left(2+\frac{\alpha^{2}}{1-\alpha^{2}}\sum_{n=0}^{\infty}Q_{n}\right)}}$ \compellintPik@{\alpha^{2}}{k} = \frac{\pi}{4\AGM@{1}{k^{\prime}}}\left(2+\frac{\alpha^{2}}{1-\alpha^{2}}\sum_{n=0}^{\infty}Q_{n}\right)	${\displaystyle{\displaystyle-\infty<k^{2},k^{2}<1,-\infty<\alpha^{2},\alpha^{2}<1}}$	EllipticPi((alpha)^(2), k) = (Pi)/(4*GaussAGM(1, sqrt(1 - (k)^(2))))*(2 +((alpha)^(2))/(1 - (alpha)^(2))*sum(Q[n], n = 0..infinity))	Error	Failure	Missing Macro Error	Error	-
19.8#Ex3	${\displaystyle{\displaystyle p_{n+1}=\frac{p_{n}^{2}+a_{n}g_{n}}{2p_{n}}}}$ p_{n+1} = \frac{p_{n}^{2}+a_{n}g_{n}}{2p_{n}}		p[n + 1] = ((p[n])^(2)+ a[n]*g[n])/(2*p[n])	Subscript[p, n + 1] == Divide[(Subscript[p, n])^(2)+ Subscript[a, n]*Subscript[g, n],2*Subscript[p, n]]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.8#Ex4	${\displaystyle{\displaystyle\varepsilon_{n}=\frac{p_{n}^{2}-a_{n}g_{n}}{p_{n}^{2}+a_{n}g_{n}}}}$ \varepsilon_{n} = \frac{p_{n}^{2}-a_{n}g_{n}}{p_{n}^{2}+a_{n}g_{n}}		varepsilon[n] = ((p[n])^(2)- a[n]*g[n])/((p[n])^(2)+ a[n]*g[n])	Subscript[\[CurlyEpsilon], n] == Divide[(Subscript[p, n])^(2)- Subscript[a, n]*Subscript[g, n],(Subscript[p, n])^(2)+ Subscript[a, n]*Subscript[g, n]]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.8#Ex5	${\displaystyle{\displaystyle Q_{n+1}=\tfrac{1}{2}Q_{n}\varepsilon_{n}}}$ Q_{n+1} = \tfrac{1}{2}Q_{n}\varepsilon_{n}		Q[n + 1] = (1)/(2)*Q[n]*varepsilon[n]	Subscript[Q, n + 1] == Divide[1,2]*Subscript[Q, n]*Subscript[\[CurlyEpsilon], n]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.8.E9	${\displaystyle{\displaystyle\Pi\left(\alpha^{2},k\right)=\frac{\pi}{4M\left(1,k^{\prime}\right)}\frac{k^{2}}{k^{2}-\alpha^{2}}\sum_{n=0}^{\infty}Q_{n}}}$ \compellintPik@{\alpha^{2}}{k} = \frac{\pi}{4\AGM@{1}{k^{\prime}}}\frac{k^{2}}{k^{2}-\alpha^{2}}\sum_{n=0}^{\infty}Q_{n}	${\displaystyle{\displaystyle-\infty<k^{2},k^{2}<1,1<\alpha^{2},\alpha^{2}<\infty}}$	EllipticPi((alpha)^(2), k) = (Pi)/(4*GaussAGM(1, sqrt(1 - (k)^(2))))*((k)^(2))/((k)^(2)- (alpha)^(2))*sum(Q[n], n = 0..infinity)	Error	Failure	Missing Macro Error	Error	-
19.8.E10	${\displaystyle{\displaystyle p_{0}^{2}=1-(k^{2}/\alpha^{2})}}$ p_{0}^{2} = 1-(k^{2}/\alpha^{2})		(p[0])^(2) = 1 -((k)^(2)/(alpha)^(2))	(Subscript[p, 0])^(2) == 1 -((k)^(2)/\[Alpha]^(2))	Skipped - no semantic math	Skipped - no semantic math	-	-
19.8#Ex8	${\displaystyle{\displaystyle K\left(k\right)=(1+k_{1})K\left(k_{1}\right)}}$ \compellintKk@{k} = (1+k_{1})\compellintKk@{k_{1}}		EllipticK(k) = (1 + k[1])*EllipticK(k[1])	EllipticK[(k)^2] == (1 + Subscript[k, 1])*EllipticK[(Subscript[k, 1])^2]	Failure	Failure	Error	Failed [30 / 30] Result: DirectedInfinity[] Test Values: {Rule[k, 1], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-1.44075376931664, -1.6191557371087932] Test Values: {Rule[k, 2], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.8#Ex9	${\displaystyle{\displaystyle E\left(k\right)=(1+k^{\prime})E\left(k_{1}\right)-k^{\prime}K\left(k\right)}}$ \compellintEk@{k} = (1+k^{\prime})\compellintEk@{k_{1}}-k^{\prime}\compellintKk@{k}		EllipticE(k) = (1 +sqrt(1 - (k)^(2)))*EllipticE(k[1])-sqrt(1 - (k)^(2))*EllipticK(k)	EllipticE[(k)^2] == (1 +Sqrt[1 - (k)^(2)])*EllipticE[(Subscript[k, 1])^2]-Sqrt[1 - (k)^(2)]*EllipticK[(k)^2]	Failure	Failure	Error	Failed [30 / 30] Result: Indeterminate Test Values: {Rule[k, 1], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.595329372049606, 0.2521613076710463] Test Values: {Rule[k, 2], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.8#Ex10	${\displaystyle{\displaystyle F\left(\phi,k\right)=\tfrac{1}{2}(1+k_{1})F\left(\phi_{1},k_{1}\right)}}$ \incellintFk@{\phi}{k} = \tfrac{1}{2}(1+k_{1})\incellintFk@{\phi_{1}}{k_{1}}		EllipticF(sin(phi), k) = (1)/(2)*(1 + k[1])*EllipticF(sin(phi + arctan(sqrt(1 - (k)^(2))*tan(phi))), k[1])	EllipticF[\[Phi], (k)^2] == Divide[1,2]*(1 + Subscript[k, 1])*EllipticF[\[Phi]+ ArcTan[Sqrt[1 - (k)^(2)]*Tan[\[Phi]]], (Subscript[k, 1])^2]	Failure	Failure	Failed [300 / 300] Result: .2591790565-.226164263e-1*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, k[1] = 1/2*3^(1/2)+1/2*I, k = 1} Result: .8581261265-.11942686e-2*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, k[1] = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.15619877563526813, 0.03685530383845256] Test Values: {Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.6672885103059906, -0.24203301849204312] Test Values: {Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.8#Ex11	${\displaystyle{\displaystyle E\left(\phi,k\right)=\tfrac{1}{2}(1+k^{\prime})E\left(\phi_{1},k_{1}\right)-k^{\prime}F\left(\phi,k\right)+\tfrac{1}{2}(1-k^{\prime})\sin\phi_{1}}}$ \incellintEk@{\phi}{k} = \tfrac{1}{2}(1+k^{\prime})\incellintEk@{\phi_{1}}{k_{1}}-k^{\prime}\incellintFk@{\phi}{k}+\tfrac{1}{2}(1-k^{\prime})\sin@@{\phi_{1}}		EllipticE(sin(phi), k) = (1)/(2)*(1 +sqrt(1 - (k)^(2)))*EllipticE(sin(phi + arctan(sqrt(1 - (k)^(2))*tan(phi))), k[1])-sqrt(1 - (k)^(2))*EllipticF(sin(phi), k)+(1)/(2)*(1 -sqrt(1 - (k)^(2)))*sin(phi + arctan(sqrt(1 - (k)^(2))*tan(phi)))	EllipticE[\[Phi], (k)^2] == Divide[1,2]*(1 +Sqrt[1 - (k)^(2)])*EllipticE[\[Phi]+ ArcTan[Sqrt[1 - (k)^(2)]*Tan[\[Phi]]], (Subscript[k, 1])^2]-Sqrt[1 - (k)^(2)]*EllipticF[\[Phi], (k)^2]+Divide[1,2]*(1 -Sqrt[1 - (k)^(2)])*Sin[\[Phi]+ ArcTan[Sqrt[1 - (k)^(2)]*Tan[\[Phi]]]]	Failure	Failure	Failed [300 / 300] Result: -.627821156e-1-.413169945e-1*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, k[1] = 1/2*3^(1/2)+1/2*I, k = 1} Result: .886069620e-1-.4575597e-3*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, k[1] = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-0.0022565574667213206, -0.009009769525654576] Test Values: {Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.11756483394447081, -0.05872123913100852] Test Values: {Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.8.E14	${\displaystyle{\displaystyle 2(k^{2}-\alpha^{2})\Pi\left(\phi,\alpha^{2},k\right)=\frac{\omega^{2}-\alpha^{2}}{1+k^{\prime}}\Pi\left(\phi_{1},\alpha_{1}^{2},k_{1}\right)+k^{2}F\left(\phi,k\right)-{(1+k^{\prime})\alpha_{1}^{2}R_{C}\left(c_{1},c_{1}-\alpha_{1}^{2}\right)}}}$ 2(k^{2}-\alpha^{2})\incellintPik@{\phi}{\alpha^{2}}{k} = \frac{\omega^{2}-\alpha^{2}}{1+k^{\prime}}\incellintPik@{\phi_{1}}{\alpha_{1}^{2}}{k_{1}}+k^{2}\incellintFk@{\phi}{k}-{(1+k^{\prime})\alpha_{1}^{2}\CarlsonellintRC@{c_{1}}{c_{1}-\alpha_{1}^{2}}}		Error	2*((k)^(2)- \[Alpha]^(2))*EllipticPi[\[Alpha]^(2), \[Phi],(k)^2] == Divide[\[Omega]^(2)- \[Alpha]^(2),1 +Sqrt[1 - (k)^(2)]]*EllipticPi[(Subscript[\[Alpha], 1])^(2), \[Phi]+ ArcTan[Sqrt[1 - (k)^(2)]*Tan[\[Phi]]],(Subscript[k, 1])^2]+ (k)^(2)* EllipticF[\[Phi], (k)^2]-(1 +Sqrt[1 - (k)^(2)])*(Subscript[\[Alpha], 1])^(2)*1/Sqrt[((Csc[Subscript[\[Phi], 1]])^(2))- (Subscript[\[Alpha], 1])^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-((Csc[Subscript[\[Phi], 1]])^(2))/(((Csc[Subscript[\[Phi], 1]])^(2))- (Subscript[\[Alpha], 1])^(2))]	Missing Macro Error	Aborted	-	Failed [300 / 300] Result: Complex[-1.4115811709537147, -1.2227387134851169] Test Values: {Rule[k, 1], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ω, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[α, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[1.5976966939439394, -1.230515427208163] Test Values: {Rule[k, 2], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ω, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[α, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.8#Ex17	${\displaystyle{\displaystyle F\left(\phi,k\right)=\frac{2}{1+k}F\left(\phi_{2},k_{2}\right)}}$ \incellintFk@{\phi}{k} = \frac{2}{1+k}\incellintFk@{\phi_{2}}{k_{2}}		EllipticF(sin(phi), k) = (2)/(1 + k)*EllipticF(sin(phi[2]), k[2])	EllipticF[\[Phi], (k)^2] == Divide[2,1 + k]*EllipticF[Subscript[\[Phi], 2], (Subscript[k, 2])^2]	Failure	Failure	Failed [300 / 300] Result: .716161018e-1+.1278882161*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, k[2] = 1/2*3^(1/2)+1/2*I, phi[2] = 1/2*3^(1/2)+1/2*I, k = 1} Result: -.163142760e-1+.3519262665*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, k[2] = 1/2*3^(1/2)+1/2*I, phi[2] = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.0030858847214221274, 0.01883659064247678] Test Values: {Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[ϕ, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.11075679050380455, 0.16335572999260056] Test Values: {Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[ϕ, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.8#Ex18	${\displaystyle{\displaystyle E\left(\phi,k\right)=(1+k)E\left(\phi_{2},k_{2}\right)+(1-k)F\left(\phi_{2},k_{2}\right)-k\sin\phi}}$ \incellintEk@{\phi}{k} = (1+k)\incellintEk@{\phi_{2}}{k_{2}}+(1-k)\incellintFk@{\phi_{2}}{k_{2}}-k\sin@@{\phi}		EllipticE(sin(phi), k) = (1 + k)*EllipticE(sin(phi[2]), k[2])+(1 - k)*EllipticF(sin(phi[2]), k[2])- k*sin(phi)	EllipticE[\[Phi], (k)^2] == (1 + k)*EllipticE[Subscript[\[Phi], 2], (Subscript[k, 2])^2]+(1 - k)*EllipticF[Subscript[\[Phi], 2], (Subscript[k, 2])^2]- k*Sin[\[Phi]]	Failure	Failure	Failed [300 / 300] Result: -.251128463-.1652679776*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, k[2] = 1/2*3^(1/2)+1/2*I, phi[2] = 1/2*3^(1/2)+1/2*I, k = 1} Result: .549972877-.903450862e-1*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, k[2] = 1/2*3^(1/2)+1/2*I, phi[2] = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-0.009026229866885283, -0.03603907810261833] Test Values: {Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[ϕ, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.42447097038130677, 0.1345883883024661] Test Values: {Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[ϕ, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.8#Ex22	${\displaystyle{\displaystyle F\left(\phi,k\right)=(1+k_{1})F\left(\psi_{1},k_{1}\right)}}$ \incellintFk@{\phi}{k} = (1+k_{1})\incellintFk@{\psi_{1}}{k_{1}}		EllipticF(sin(phi), k) = (1 + k[1])*EllipticF(sin(psi[1]), k[1])	EllipticF[\[Phi], (k)^2] == (1 + Subscript[k, 1])*EllipticF[Subscript[\[Psi], 1], (Subscript[k, 1])^2]	Failure	Failure	Failed [299 / 300] Result: -.3025119160-.7226109033*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k[1] = 1/2*3^(1/2)+1/2*I, psi[1] = 1/2*3^(1/2)+1/2*I, k = 1} Result: -.6401936029-.6817361311*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k[1] = 1/2*3^(1/2)+1/2*I, psi[1] = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [299 / 300] Result: Complex[-0.11940620612760577, -0.19771875715838422] Test Values: {Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[ψ, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.15464125790413003, -0.13739720920586462] Test Values: {Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[ψ, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.8#Ex23	${\displaystyle{\displaystyle E\left(\phi,k\right)=(1+k^{\prime})E\left(\psi_{1},k_{1}\right)-k^{\prime}F\left(\phi,k\right)+(1-\Delta)\cot\phi}}$ \incellintEk@{\phi}{k} = (1+k^{\prime})\incellintEk@{\psi_{1}}{k_{1}}-k^{\prime}\incellintFk@{\phi}{k}+(1-\Delta)\cot@@{\phi}		EllipticE(sin(phi), k) = (1 +sqrt(1 - (k)^(2)))*EllipticE(sin(psi[1]), k[1])-sqrt(1 - (k)^(2))*EllipticF(sin(phi), k)+(1 -(sqrt(1 - (k)^(2)* (sin(phi))^(2))))*cot(phi)	EllipticE[\[Phi], (k)^2] == (1 +Sqrt[1 - (k)^(2)])*EllipticE[Subscript[\[Psi], 1], (Subscript[k, 1])^2]-Sqrt[1 - (k)^(2)]*EllipticF[\[Phi], (k)^2]+(1 -(Sqrt[1 - (k)^(2)* (Sin[\[Phi]])^(2)]))*Cot[\[Phi]]	Failure	Failure	Failed [300 / 300] Result: -.5555013192-.1267358774*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k[1] = 1/2*3^(1/2)+1/2*I, psi[1] = 1/2*3^(1/2)+1/2*I, k = 1} Result: -1.589246368-2.046785663*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k[1] = 1/2*3^(1/2)+1/2*I, psi[1] = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-0.22091089534718378, -0.1170454776590783] Test Values: {Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[ψ, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.9299237807056446, -0.7272990802320405] Test Values: {Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[ψ, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.8.E20	${\displaystyle{\displaystyle\rho\Pi\left(\phi,\alpha^{2},k\right)=\frac{4}{1+k^{\prime}}\Pi\left(\psi_{1},\alpha_{1}^{2},k_{1}\right)+(\rho-1)F\left(\phi,k\right)-R_{C}\left(c-1,c-\alpha^{2}\right)}}$ \rho\incellintPik@{\phi}{\alpha^{2}}{k} = \frac{4}{1+k^{\prime}}\incellintPik@{\psi_{1}}{\alpha_{1}^{2}}{k_{1}}+(\rho-1)\incellintFk@{\phi}{k}-\CarlsonellintRC@{c-1}{c-\alpha^{2}}		Error	\[Rho]*EllipticPi[\[Alpha]^(2), \[Phi],(k)^2] == Divide[4,1 +Sqrt[1 - (k)^(2)]]*EllipticPi[(Subscript[\[Alpha], 1])^(2), Subscript[\[Psi], 1],(Subscript[k, 1])^2]+(\[Rho]- 1)*EllipticF[\[Phi], (k)^2]- 1/Sqrt[((Csc[\[Phi]])^(2))- \[Alpha]^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-(((Csc[\[Phi]])^(2))- 1)/(((Csc[\[Phi]])^(2))- \[Alpha]^(2))]	Missing Macro Error	Failure	-	Skipped - Because timed out