Results of Elliptic Integrals

DLMF	Formula	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
19.2.E4	${\displaystyle{\displaystyle F\left(\phi,k\right)=\int_{0}^{\phi}\frac{\mathrm{d}\theta}{\sqrt{1-k^{2}{\sin^{2}}\theta}}}}$	EllipticF(sin(phi), k)= int((1)/(sqrt(1 - (k)^(2)* (sin(theta))^(2))), theta = 0..phi)	EllipticF[\[Phi], (k)^2]= Integrate[Divide[1,Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)]], {\[Theta], 0, \[Phi]}]	Failure	Failure	Skip	Error
19.2.E4	${\displaystyle{\displaystyle\int_{0}^{\phi}\frac{\mathrm{d}\theta}{\sqrt{1-k^{2}{\sin^{2}}\theta}}=\int_{0}^{\sin\phi}\frac{\mathrm{d}t}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}}}}$	int((1)/(sqrt(1 - (k)^(2)* (sin(theta))^(2))), theta = 0..phi)= int((1)/(sqrt(1 - (t)^(2))*sqrt(1 - (k)^(2)* (t)^(2))), t = 0..sin(phi))	Integrate[Divide[1,Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)]], {\[Theta], 0, \[Phi]}]= Integrate[Divide[1,Sqrt[1 - (t)^(2)]*Sqrt[1 - (k)^(2)* (t)^(2)]], {t, 0, Sin[\[Phi]]}]	Failure	Failure	Skip	Error
19.2.E5	${\displaystyle{\displaystyle\int_{0}^{\phi}\sqrt{1-k^{2}{\sin^{2}}\theta}\mathrm{d}\theta\\ =\int_{0}^{\sin\phi}\frac{\sqrt{1-k^{2}t^{2}}}{\sqrt{1-t^{2}}}\mathrm{d}t}}$	int(sqrt(1 - (k)^(2)* (sin(theta))^(2)), theta = 0..phi)= int((sqrt(1 - (k)^(2)* (t)^(2)))/(sqrt(1 - (t)^(2))), t = 0..sin(phi))	Integrate[Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)], {\[Theta], 0, \[Phi]}]= Integrate[Divide[Sqrt[1 - (k)^(2)* (t)^(2)],Sqrt[1 - (t)^(2)]], {t, 0, Sin[\[Phi]]}]	Error	Error	-	-
19.2.E6	${\displaystyle{\displaystyle D\left(\phi,k\right)=\int_{0}^{\phi}\frac{{\sin^{2}}\theta\mathrm{d}\theta}{\sqrt{1-k^{2}{\sin^{2}}\theta}}}}$	(EllipticF(sin(phi), k) - EllipticE(sin(phi), k))/(k)^2 = int(((sin(theta))^(2))/(sqrt(1 - (k)^(2)* (sin(theta))^(2))), theta = 0..phi)	Divide[EllipticF[\[Phi], (k)^2] - EllipticE[\[Phi], (k)^2], (k)^4]= Integrate[Divide[(Sin[\[Theta]])^(2),Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)]], {\[Theta], 0, \[Phi]}]	Failure	Failure	Skip	Error
19.2.E6	${\displaystyle{\displaystyle\int_{0}^{\phi}\frac{{\sin^{2}}\theta\mathrm{d}\theta}{\sqrt{1-k^{2}{\sin^{2}}\theta}}=\int_{0}^{\sin\phi}\frac{t^{2}\mathrm{d}t}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}}}}$	int(((sin(theta))^(2))/(sqrt(1 - (k)^(2)* (sin(theta))^(2))), theta = 0..phi)= int(((t)^(2))/(sqrt(1 - (t)^(2))*sqrt(1 - (k)^(2)* (t)^(2))), t = 0..sin(phi))	Integrate[Divide[(Sin[\[Theta]])^(2),Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)]], {\[Theta], 0, \[Phi]}]= Integrate[Divide[(t)^(2),Sqrt[1 - (t)^(2)]*Sqrt[1 - (k)^(2)* (t)^(2)]], {t, 0, Sin[\[Phi]]}]	Failure	Failure	Skip	Error
19.2.E6	${\displaystyle{\displaystyle\int_{0}^{\sin\phi}\frac{t^{2}\mathrm{d}t}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}}=(F\left(\phi,k\right)-E\left(\phi,k\right))/k^{2}}}$	int(((t)^(2))/(sqrt(1 - (t)^(2))*sqrt(1 - (k)^(2)* (t)^(2))), t = 0..sin(phi))=(EllipticF(sin(phi), k)- EllipticE(sin(phi), k))/ (k)^(2)	Integrate[Divide[(t)^(2),Sqrt[1 - (t)^(2)]*Sqrt[1 - (k)^(2)* (t)^(2)]], {t, 0, Sin[\[Phi]]}]=(EllipticF[\[Phi], (k)^2]- EllipticE[\[Phi], (k)^2])/ (k)^(2)	Successful	Failure	-	Error
19.2#Ex1	${\displaystyle{\displaystyle K\left(k\right)=F\left(\pi/2,k\right)}}$	EllipticK(k)= EllipticF(sin(Pi/ 2), k)	EllipticK[(k)^2]= EllipticF[Pi/ 2, (k)^2]	Successful	Successful	-	-
19.2#Ex2	${\displaystyle{\displaystyle E\left(k\right)=E\left(\pi/2,k\right)}}$	EllipticE(k)= EllipticE(sin(Pi/ 2), k)	EllipticE[(k)^2]= EllipticE[Pi/ 2, (k)^2]	Successful	Successful	-	-
19.2#Ex3	${\displaystyle{\displaystyle D\left(k\right)=D\left(\pi/2,k\right)}}$	(EllipticK(k) - EllipticE(k))/(k)^2 = (EllipticF(sin(Pi/ 2), k) - EllipticE(sin(Pi/ 2), k))/(k)^2	Divide[EllipticK[(k)^2] - EllipticE[(k)^2], (k)^4]= Divide[EllipticF[Pi/ 2, (k)^2] - EllipticE[Pi/ 2, (k)^2], (k)^4]	Successful	Successful	-	-
19.2#Ex3	${\displaystyle{\displaystyle D\left(\pi/2,k\right)=(K\left(k\right)-E\left(k\right))/k^{2}}}$	(EllipticF(sin(Pi/ 2), k) - EllipticE(sin(Pi/ 2), k))/(k)^2 =(EllipticK(k)- EllipticE(k))/ (k)^(2)	Divide[EllipticF[Pi/ 2, (k)^2] - EllipticE[Pi/ 2, (k)^2], (k)^4]=(EllipticK[(k)^2]- EllipticE[(k)^2])/ (k)^(2)	Successful	Failure	-	Fail Complex[-0.08185805455243832, 0.4541460103381725] <- {Rule[k, 2]} Complex[-0.027015555974087394, 0.3299973634705001] <- {Rule[k, 3]}
19.2#Ex4	${\displaystyle{\displaystyle\Pi\left(\alpha^{2},k\right)=\Pi\left(\pi/2,\alpha^{2},k\right)}}$	EllipticPi((alpha)^(2), k)= EllipticPi(sin(Pi/ 2), (alpha)^(2), k)	EllipticPi[(\[Alpha])^(2), (k)^2]= EllipticPi[(\[Alpha])^(2), Pi/ 2,(k)^2]	Successful	Successful	-	-
19.2#Ex8	${\displaystyle{\displaystyle F\left(m\pi+\phi,k\right)=2mK\left(k\right)+F\left(\phi,k\right)}}$	EllipticF(sin(m*Pi + phi), k)= 2*m*EllipticK(k)+ EllipticF(sin(phi), k)	EllipticF[m*Pi + \[Phi], (k)^2]= 2*m*EllipticK[(k)^2]+ EllipticF[\[Phi], (k)^2]	Failure	Failure	Error	Successful
19.2#Ex8	${\displaystyle{\displaystyle F\left(m\pi-\phi,k\right)=2mK\left(k\right)-F\left(\phi,k\right)}}$	EllipticF(sin(m*Pi - phi), k)= 2*m*EllipticK(k)- EllipticF(sin(phi), k)	EllipticF[m*Pi - \[Phi], (k)^2]= 2*m*EllipticK[(k)^2]- EllipticF[\[Phi], (k)^2]	Failure	Failure	Error	Successful
19.2#Ex9	${\displaystyle{\displaystyle E\left(m\pi+\phi,k\right)=2mE\left(k\right)+E\left(\phi,k\right)}}$	EllipticE(sin(m*Pi + phi), k)= 2*m*EllipticE(k)+ EllipticE(sin(phi), k)	EllipticE[m*Pi + \[Phi], (k)^2]= 2*m*EllipticE[(k)^2]+ EllipticE[\[Phi], (k)^2]	Failure	Failure	Fail -6.303071081-.6035229402*I <- {phi = 2^(1/2)+I*2^(1/2), k = 1, m = 1} -3.999999999-.16e-8*I <- {phi = 2^(1/2)+I*2^(1/2), k = 1, m = 2} -10.30307108-.6035229462*I <- {phi = 2^(1/2)+I*2^(1/2), k = 1, m = 3} -8.695154357-1.322364992*I <- {phi = 2^(1/2)+I*2^(1/2), k = 2, m = 1} ... skip entries to safe data	Successful
19.2#Ex9	${\displaystyle{\displaystyle E\left(m\pi-\phi,k\right)=2mE\left(k\right)-E\left(\phi,k\right)}}$	EllipticE(sin(m*Pi - phi), k)= 2*m*EllipticE(k)- EllipticE(sin(phi), k)	EllipticE[m*Pi - \[Phi], (k)^2]= 2*m*EllipticE[(k)^2]- EllipticE[\[Phi], (k)^2]	Failure	Failure	Fail 2.303071080+.6035229417*I <- {phi = 2^(1/2)+I*2^(1/2), k = 1, m = 1} -4.000000000-.15e-8*I <- {phi = 2^(1/2)+I*2^(1/2), k = 1, m = 2} -1.696928920+.6035229433*I <- {phi = 2^(1/2)+I*2^(1/2), k = 1, m = 3} 7.069958809-4.053051928*I <- {phi = 2^(1/2)+I*2^(1/2), k = 2, m = 1} ... skip entries to safe data	Successful
19.2#Ex10	${\displaystyle{\displaystyle D\left(m\pi+\phi,k\right)=2mD\left(k\right)+D\left(\phi,k\right)}}$	(EllipticF(sin(m*Pi + phi), k) - EllipticE(sin(m*Pi + phi), k))/(k)^2 = 2*m*(EllipticK(k) - EllipticE(k))/(k)^2 + (EllipticF(sin(phi), k) - EllipticE(sin(phi), k))/(k)^2	Divide[EllipticF[m*Pi + \[Phi], (k)^2] - EllipticE[m*Pi + \[Phi], (k)^2], (k)^4]= 2*m*Divide[EllipticK[(k)^2] - EllipticE[(k)^2], (k)^4]+ Divide[EllipticF[\[Phi], (k)^2] - EllipticE[\[Phi], (k)^2], (k)^4]	Failure	Failure	Error	Successful
19.2#Ex10	${\displaystyle{\displaystyle D\left(m\pi-\phi,k\right)=2mD\left(k\right)-D\left(\phi,k\right)}}$	(EllipticF(sin(m*Pi - phi), k) - EllipticE(sin(m*Pi - phi), k))/(k)^2 = 2*m*(EllipticK(k) - EllipticE(k))/(k)^2 - (EllipticF(sin(phi), k) - EllipticE(sin(phi), k))/(k)^2	Divide[EllipticF[m*Pi - \[Phi], (k)^2] - EllipticE[m*Pi - \[Phi], (k)^2], (k)^4]= 2*m*Divide[EllipticK[(k)^2] - EllipticE[(k)^2], (k)^4]- Divide[EllipticF[\[Phi], (k)^2] - EllipticE[\[Phi], (k)^2], (k)^4]	Failure	Failure	Error	Successful
19.2#Ex13	${\displaystyle{\displaystyle x=\tan\phi}}$	x = tan(phi)	x = Tan[\[Phi]]	Failure	Failure	Fail .9591286913-1.118374137*I <- {phi = 2^(1/2)+I*2^(1/2), x = 1} 1.959128691-1.118374137*I <- {phi = 2^(1/2)+I*2^(1/2), x = 2} 2.959128691-1.118374137*I <- {phi = 2^(1/2)+I*2^(1/2), x = 3} .9591286913+1.118374137*I <- {phi = 2^(1/2)-I*2^(1/2), x = 1} ... skip entries to safe data	Fail Complex[0.9591286914366802, -1.1183741374008342] <- {Rule[x, 1], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.9591286914366801, -1.1183741374008342] <- {Rule[x, 2], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[2.95912869143668, -1.1183741374008342] <- {Rule[x, 3], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.9591286914366802, 1.1183741374008342] <- {Rule[x, 1], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.4#Ex1	${\displaystyle{\displaystyle\frac{\mathrm{d}K\left(k\right)}{\mathrm{d}k}=\frac{E\left(k\right)-{k^{\prime}}^{2}K\left(k\right)}{k{k^{\prime}}^{2}}}}$	diff(EllipticK(k), k)=(EllipticE(k)- 1 - (k)^(2)* EllipticK(k))/(k*1 - (k)^(2))	D[EllipticK[(k)^2], k]=Divide[EllipticE[(k)^2]- 1 - (k)^(2)* EllipticK[(k)^2],k*1 - (k)^(2)]	Failure	Failure	Error	Fail Complex[-2.4717549813624253, 3.1435959698369205] <- {Rule[k, 2]} Complex[-1.12187012081601, 1.8575646745447774] <- {Rule[k, 3]}
19.4#Ex2	${\displaystyle{\displaystyle\frac{\mathrm{d}(E\left(k\right)-{k^{\prime}}^{2}K\left(k\right))}{\mathrm{d}k}=kK\left(k\right)}}$	diff(EllipticE(k)- 1 - (k)^(2)* EllipticK(k), k)= k*EllipticK(k)	D[EllipticE[(k)^2]- 1 - (k)^(2)* EllipticK[(k)^2], k]= k*EllipticK[(k)^2]	Failure	Failure	Error	Fail Complex[-3.3189229307917216, 6.419990143492479] <- {Rule[k, 2]} Complex[-3.226352319798031, 7.107872714050419] <- {Rule[k, 3]}
19.4#Ex3	${\displaystyle{\displaystyle\frac{\mathrm{d}E\left(k\right)}{\mathrm{d}k}=\frac{E\left(k\right)-K\left(k\right)}{k}}}$	diff(EllipticE(k), k)=(EllipticE(k)- EllipticK(k))/(k)	D[EllipticE[(k)^2], k]=Divide[EllipticE[(k)^2]- EllipticK[(k)^2],k]	Successful	Successful	-	-
19.4#Ex4	${\displaystyle{\displaystyle\frac{\mathrm{d}(E\left(k\right)-K\left(k\right))}{\mathrm{d}k}=-\frac{kE\left(k\right)}{{k^{\prime}}^{2}}}}$	diff(EllipticE(k)- EllipticK(k), k)= -(k*EllipticE(k))/(1 - (k)^(2))	D[EllipticE[(k)^2]- EllipticK[(k)^2], k]= -Divide[k*EllipticE[(k)^2],1 - (k)^(2)]	Successful	Successful	-	-
19.4.E3	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}E\left(k\right)}{{\mathrm{d}k}^{2}}=-\frac{1}{k}\frac{\mathrm{d}K\left(k\right)}{\mathrm{d}k}}}$	diff(EllipticE(k), [k$(2)])= -(1)/(k)*diff(EllipticK(k), k)	D[EllipticE[(k)^2], {k, 2}]= -Divide[1,k]*D[EllipticK[(k)^2], k]	Successful	Successful	-	-
19.4.E3	${\displaystyle{\displaystyle-\frac{1}{k}\frac{\mathrm{d}K\left(k\right)}{\mathrm{d}k}=\frac{{k^{\prime}}^{2}K\left(k\right)-E\left(k\right)}{k^{2}{k^{\prime}}^{2}}}}$	-(1)/(k)*diff(EllipticK(k), k)=(1 - (k)^(2)* EllipticK(k)- EllipticE(k))/((k)^(2)* 1 - (k)^(2))	-Divide[1,k]*D[EllipticK[(k)^2], k]=Divide[1 - (k)^(2)* EllipticK[(k)^2]- EllipticE[(k)^2],(k)^(2)* 1 - (k)^(2)]	Error	Failure	-	Fail
19.4.E4	${\displaystyle{\displaystyle\frac{\partial\Pi\left(\alpha^{2},k\right)}{\partial k}=\frac{k}{{k^{\prime}}^{2}(k^{2}-\alpha^{2})}(E\left(k\right)-{k^{\prime}}^{2}\Pi\left(\alpha^{2},k\right))}}$	diff(EllipticPi((alpha)^(2), k), k)=(k)/(1 - (k)^(2)*((k)^(2)- (alpha)^(2)))*(EllipticE(k)- 1 - (k)^(2)* EllipticPi((alpha)^(2), k))	D[EllipticPi[(\[Alpha])^(2), (k)^2], k]=Divide[k,1 - (k)^(2)*((k)^(2)- (\[Alpha])^(2))]*(EllipticE[(k)^2]- 1 - (k)^(2)* EllipticPi[(\[Alpha])^(2), (k)^2])	Failure	Failure	Error	Fail Complex[-0.5687202790618282, -0.6586049032127743] <- {Rule[k, 2], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.5507473882918545, -0.09813531953107986] <- {Rule[k, 3], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.06853665940289096, 0.43401271964497673] <- {Rule[k, 2], Rule[α, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-0.022201879042164176, 0.3037111242639709] <- {Rule[k, 3], Rule[α, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.4.E5	${\displaystyle{\displaystyle\frac{\partial F\left(\phi,k\right)}{\partial k}={\frac{E\left(\phi,k\right)-{k^{\prime}}^{2}F\left(\phi,k\right)}{k{k^{\prime}}^{2}}-\frac{k\sin\phi\cos\phi}{{k^{\prime}}^{2}\sqrt{1-k^{2}{\sin^{2}}\phi}}}}}$	diff(EllipticF(sin(phi), k), k)=(EllipticE(sin(phi), k)- 1 - (k)^(2)* EllipticF(sin(phi), k))/(k*1 - (k)^(2))-(k*sin(phi)*cos(phi))/(1 - (k)^(2)*sqrt(1 - (k)^(2)* (sin(phi))^(2)))	D[EllipticF[\[Phi], (k)^2], k]=Divide[EllipticE[\[Phi], (k)^2]- 1 - (k)^(2)* EllipticF[\[Phi], (k)^2],k*1 - (k)^(2)]-Divide[k*Sin[\[Phi]]*Cos[\[Phi]],1 - (k)^(2)*Sqrt[1 - (k)^(2)* (Sin[\[Phi]])^(2)]]	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {phi = 2^(1/2)+I*2^(1/2), k = 1} .3848531746-2.832892174*I <- {phi = 2^(1/2)+I*2^(1/2), k = 2} .2974950167-1.686437583*I <- {phi = 2^(1/2)+I*2^(1/2), k = 3} Float(infinity)+Float(infinity)*I <- {phi = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[0.3848531762144859, -2.8328921744194435] <- {Rule[k, 2], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.297495017249507, -1.6864375827943054] <- {Rule[k, 3], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.38485317621448545, 2.8328921744194426] <- {Rule[k, 2], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[0.2974950172495072, 1.686437582794305] <- {Rule[k, 3], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.4.E6	${\displaystyle{\displaystyle\frac{\partial E\left(\phi,k\right)}{\partial k}=\frac{E\left(\phi,k\right)-F\left(\phi,k\right)}{k}}}$	diff(EllipticE(sin(phi), k), k)=(EllipticE(sin(phi), k)- EllipticF(sin(phi), k))/(k)	D[EllipticE[\[Phi], (k)^2], k]=Divide[EllipticE[\[Phi], (k)^2]- EllipticF[\[Phi], (k)^2],k]	Successful	Successful	-	-
19.4.E7	${\displaystyle{\displaystyle\frac{\partial\Pi\left(\phi,\alpha^{2},k\right)}{\partial k}=\frac{k}{{k^{\prime}}^{2}(k^{2}-\alpha^{2})}\left({E\left(\phi,k\right)-{k^{\prime}}^{2}\Pi\left(\phi,\alpha^{2},k\right)}-\frac{k^{2}\sin\phi\cos\phi}{\sqrt{1-k^{2}{\sin^{2}}\phi}}\right)}}$	diff(EllipticPi(sin(phi), (alpha)^(2), k), k)=(k)/(1 - (k)^(2)*((k)^(2)- (alpha)^(2)))*(EllipticE(sin(phi), k)- 1 - (k)^(2)* EllipticPi(sin(phi), (alpha)^(2), k)-((k)^(2)* sin(phi)*cos(phi))/(sqrt(1 - (k)^(2)* (sin(phi))^(2))))	D[EllipticPi[(\[Alpha])^(2), \[Phi],(k)^2], k]=Divide[k,1 - (k)^(2)*((k)^(2)- (\[Alpha])^(2))]*(EllipticE[\[Phi], (k)^2]- 1 - (k)^(2)* EllipticPi[(\[Alpha])^(2), \[Phi],(k)^2]-Divide[(k)^(2)* Sin[\[Phi]]*Cos[\[Phi]],Sqrt[1 - (k)^(2)* (Sin[\[Phi]])^(2)]])	Failure	Failure	Fail Float(undefined)+Float(undefined)*I <- {alpha = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), k = 1} .3340407480e-1-.3675640793*I <- {alpha = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), k = 2} -.3093953717e-1-.2652186316*I <- {alpha = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), k = 3} Float(undefined)+Float(undefined)*I <- {alpha = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[0.0334040747326544, -0.36756407921347006] <- {Rule[k, 2], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.030939537020799865, -0.265218631561315] <- {Rule[k, 3], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.15465722767175843, 0.23001025865583416] <- {Rule[k, 2], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-0.030260984981937494, 0.2341460018375755] <- {Rule[k, 3], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.4.E8	${\displaystyle{\displaystyle(k{k^{\prime}}^{2}D_{k}^{2}+(1-3k^{2})D_{k}-k)F\left(\phi,k\right)=\frac{-k\sin\phi\cos\phi}{(1-k^{2}{\sin^{2}}\phi)^{3/2}}}}$	(k*1 - (k)^(2)* D(D[k])^(2)+(1 - 3*(k)^(2))*D[k]- k)* EllipticF(sin(phi), k)=(- k*sin(phi)*cos(phi))/((1 - (k)^(2)* (sin(phi))^(2))^(3/ 2))	(k*1 - (k)^(2)* D(Subscript[D, k])^(2)+(1 - 3*(k)^(2))*Subscript[D, k]- k)* EllipticF[\[Phi], (k)^2]=Divide[- k*Sin[\[Phi]]*Cos[\[Phi]],(1 - (k)^(2)* (Sin[\[Phi]])^(2))^(3/ 2)]	Error	Failure	-	Successful
19.4.E9	${\displaystyle{\displaystyle(k{k^{\prime}}^{2}D_{k}^{2}+{k^{\prime}}^{2}D_{k}+k)E\left(\phi,k\right)=\frac{k\sin\phi\cos\phi}{\sqrt{1-k^{2}{\sin^{2}}\phi}}}}$	(k*1 - (k)^(2)* D(D[k])^(2)+ 1 - (k)^(2)* D[k]+ k)* EllipticE(sin(phi), k)=(k*sin(phi)*cos(phi))/(sqrt(1 - (k)^(2)* (sin(phi))^(2)))	(k*1 - (k)^(2)* D(Subscript[D, k])^(2)+ 1 - (k)^(2)* Subscript[D, k]+ k)* EllipticE[\[Phi], (k)^2]=Divide[k*Sin[\[Phi]]*Cos[\[Phi]],Sqrt[1 - (k)^(2)* (Sin[\[Phi]])^(2)]]	Error	Failure	-	Successful
19.5.E1	${\displaystyle{\displaystyle K\left(k\right)=\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k^{2m}}}$	EllipticK(k)=(Pi)/(2)*sum((pochhammer((1)/(2), m)*pochhammer((1)/(2), m))/(factorial(m)*factorial(m))*(k)^(2*m), m = 0..infinity)	EllipticK[(k)^2]=Divide[Pi,2]*Sum[Divide[Pochhammer[Divide[1,2], m]*Pochhammer[Divide[1,2], m],(m)!*(m)!]*(k)^(2*m), {m, 0, Infinity}]	Failure	Successful	Skip	-
19.5.E1	${\displaystyle{\displaystyle\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k^{2m}=\frac{\pi}{2}{{}_{2}F_{1}}\left(\tfrac{1}{2},\tfrac{1}{2};1;k^{2}\right)}}$	(Pi)/(2)*sum((pochhammer((1)/(2), m)*pochhammer((1)/(2), m))/(factorial(m)*factorial(m))*(k)^(2*m), m = 0..infinity)=(Pi)/(2)*hypergeom([(1)/(2),(1)/(2)], [1], (k)^(2))	Divide[Pi,2]*Sum[Divide[Pochhammer[Divide[1,2], m]*Pochhammer[Divide[1,2], m],(m)!*(m)!]*(k)^(2*m), {m, 0, Infinity}]=Divide[Pi,2]*HypergeometricPFQ[{Divide[1,2],Divide[1,2]}, {1}, (k)^(2)]	Failure	Successful	Skip	-
19.5.E2	${\displaystyle{\displaystyle E\left(k\right)=\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{{\left(-\tfrac{1}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k^{2m}}}$	EllipticE(k)=(Pi)/(2)*sum((pochhammer(-(1)/(2), m)*pochhammer((1)/(2), m))/(factorial(m)*factorial(m))*(k)^(2*m), m = 0..infinity)	EllipticE[(k)^2]=Divide[Pi,2]*Sum[Divide[Pochhammer[-Divide[1,2], m]*Pochhammer[Divide[1,2], m],(m)!*(m)!]*(k)^(2*m), {m, 0, Infinity}]	Failure	Successful	Skip	-
19.5.E2	${\displaystyle{\displaystyle\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{{\left(-\tfrac{1}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k^{2m}=\frac{\pi}{2}{{}_{2}F_{1}}\left(-\tfrac{1}{2},\tfrac{1}{2};1;k^{2}\right)}}$	(Pi)/(2)*sum((pochhammer(-(1)/(2), m)*pochhammer((1)/(2), m))/(factorial(m)*factorial(m))*(k)^(2*m), m = 0..infinity)=(Pi)/(2)*hypergeom([-(1)/(2),(1)/(2)], [1], (k)^(2))	Divide[Pi,2]*Sum[Divide[Pochhammer[-Divide[1,2], m]*Pochhammer[Divide[1,2], m],(m)!*(m)!]*(k)^(2*m), {m, 0, Infinity}]=Divide[Pi,2]*HypergeometricPFQ[{-Divide[1,2],Divide[1,2]}, {1}, (k)^(2)]	Failure	Successful	Skip	-
19.5.E3	${\displaystyle{\displaystyle D\left(k\right)=\frac{\pi}{4}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{3}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{(m+1)!\;m!}k^{2m}}}$	(EllipticK(k) - EllipticE(k))/(k)^2 =(Pi)/(4)*sum((pochhammer((3)/(2), m)*pochhammer((1)/(2), m))/(factorial(m + 1)*factorial(m))*(k)^(2*m), m = 0..infinity)	Divide[EllipticK[(k)^2] - EllipticE[(k)^2], (k)^4]=Divide[Pi,4]*Sum[Divide[Pochhammer[Divide[3,2], m]*Pochhammer[Divide[1,2], m],(m + 1)!*(m)!]*(k)^(2*m), {m, 0, Infinity}]	Failure	Failure	Skip	Fail Complex[-0.08185805455243848, 0.4541460103381727] <- {Rule[k, 2]} Complex[-0.02701555597408747, 0.3299973634705002] <- {Rule[k, 3]}
19.5.E3	${\displaystyle{\displaystyle\frac{\pi}{4}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{3}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{(m+1)!\;m!}k^{2m}=\frac{\pi}{4}{{}_{2}F_{1}}\left(\tfrac{3}{2},\tfrac{1}{2};2;k^{2}\right)}}$	(Pi)/(4)*sum((pochhammer((3)/(2), m)*pochhammer((1)/(2), m))/(factorial(m + 1)*factorial(m))*(k)^(2*m), m = 0..infinity)=(Pi)/(4)*hypergeom([(3)/(2),(1)/(2)], [2], (k)^(2))	Divide[Pi,4]*Sum[Divide[Pochhammer[Divide[3,2], m]*Pochhammer[Divide[1,2], m],(m + 1)!*(m)!]*(k)^(2*m), {m, 0, Infinity}]=Divide[Pi,4]*HypergeometricPFQ[{Divide[3,2],Divide[1,2]}, {2}, (k)^(2)]	Failure	Successful	Skip	-
19.5.E5	${\displaystyle{\displaystyle q=\exp\left(-\pi{K^{\prime}}\left(k\right)/K\left(k\right)\right)}}$	q = exp(- Pi*EllipticCK(k)/ EllipticK(k))	q = Exp[- Pi*EllipticK[1-(k)^2]/ EllipticK[(k)^2]]	Failure	Failure	Skip	Fail Complex[0.41421356237309515, 1.4142135623730951] <- {Rule[k, 1], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.41421356237309515, -1.4142135623730951] <- {Rule[k, 1], Rule[q, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-2.414213562373095, -1.4142135623730951] <- {Rule[k, 1], Rule[q, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-2.414213562373095, 1.4142135623730951] <- {Rule[k, 1], Rule[q, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
19.5.E8	${\displaystyle{\displaystyle K\left(k\right)=\frac{\pi}{2}\left(1+2\sum_{n=1}^{\infty}q^{n^{2}}\right)^{2}}}$	EllipticK(k)=(Pi)/(2)*(1 + 2*sum((q)^((n)^(2)), n = 1..infinity))^(2)	EllipticK[(k)^2]=Divide[Pi,2]*(1 + 2*Sum[(q)^((n)^(2)), {n, 1, Infinity}])^(2)	Failure	Failure	Skip	Fail DirectedInfinity[] <- {Rule[k, 1], Rule[q, Rational[1, 2]]} Complex[-6.276558134259685, -1.0782578237498217] <- {Rule[k, 2], Rule[q, Rational[1, 2]]} Complex[-6.580304399791072, -0.842875177406298] <- {Rule[k, 3], Rule[q, Rational[1, 2]]}
19.5.E9	${\displaystyle{\displaystyle E\left(k\right)=K\left(k\right)+\frac{2\pi^{2}}{K\left(k\right)}\,\frac{\sum_{n=1}^{\infty}(-1)^{n}n^{2}q^{n^{2}}}{1+2\sum_{n=1}^{\infty}(-1)^{n}q^{n^{2}}}}}$	EllipticE(k)= EllipticK(k)+(2*(Pi)^(2))/(EllipticK(k))*(sum((- 1)^(n)* (n)^(2)* (q)^((n)^(2)), n = 1..infinity))/(1 + 2*sum((- 1)^(n)* (q)^((n)^(2)), n = 1..infinity))	EllipticE[(k)^2]= EllipticK[(k)^2]+Divide[2*(Pi)^(2),EllipticK[(k)^2]]*Divide[Sum[(- 1)^(n)* (n)^(2)* (q)^((n)^(2)), {n, 1, Infinity}],1 + 2*Sum[(- 1)^(n)* (q)^((n)^(2)), {n, 1, Infinity}]]	Failure	Failure	Skip	Error
19.5.E10	${\displaystyle{\displaystyle K\left(k\right)=\frac{\pi}{2}\prod_{m=1}^{\infty}(1+k_{m})}}$	EllipticK(k)=(Pi)/(2)*product(1 + k[m], m = 1..infinity)	EllipticK[(k)^2]=Divide[Pi,2]*Product[1 + Subscript[k, m], {m, 1, Infinity}]	Failure	Failure	Skip	Skip
19.6#Ex1	${\displaystyle{\displaystyle K\left(0\right)=E\left(0\right)}}$	EllipticK(0)= EllipticE(0)	EllipticK[(0)^2]= EllipticE[(0)^2]	Successful	Successful	-	-
19.6#Ex1	${\displaystyle{\displaystyle E\left(0\right)={K^{\prime}}\left(1\right)}}$	EllipticE(0)= EllipticCK(1)	EllipticE[(0)^2]= EllipticK[1-(1)^2]	Successful	Successful	-	-
19.6#Ex1	${\displaystyle{\displaystyle{K^{\prime}}\left(1\right)={E^{\prime}}\left(1\right)}}$	EllipticCK(1)= EllipticCE(1)	EllipticK[1-(1)^2]= EllipticE[1-(1)^2]	Successful	Successful	-	-
19.6#Ex1	${\displaystyle{\displaystyle{E^{\prime}}\left(1\right)=\tfrac{1}{2}\pi}}$	EllipticCE(1)=(1)/(2)*Pi	EllipticE[1-(1)^2]=Divide[1,2]*Pi	Successful	Successful	-	-
19.6#Ex2	${\displaystyle{\displaystyle K\left(1\right)={K^{\prime}}\left(0\right)}}$	EllipticK(1)= EllipticCK(0)	EllipticK[(1)^2]= EllipticK[1-(0)^2]	Error	Failure	-	Fail Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[Indeterminate, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[Indeterminate, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[Indeterminate, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[Indeterminate, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
19.6#Ex2	${\displaystyle{\displaystyle{K^{\prime}}\left(0\right)=\infty}}$	EllipticCK(0)= infinity	EllipticK[1-(0)^2]= Infinity	Error	Failure	-	Fail Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[Indeterminate, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[Indeterminate, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[Indeterminate, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[Indeterminate, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
19.6#Ex3	${\displaystyle{\displaystyle E\left(1\right)={E^{\prime}}\left(0\right)}}$	EllipticE(1)= EllipticCE(0)	EllipticE[(1)^2]= EllipticE[1-(0)^2]	Successful	Successful	-	-
19.6#Ex3	${\displaystyle{\displaystyle{E^{\prime}}\left(0\right)=1}}$	EllipticCE(0)= 1	EllipticE[1-(0)^2]= 1	Successful	Successful	-	-
19.6#Ex4	${\displaystyle{\displaystyle\Pi\left(k^{2},k\right)=E\left(k\right)/{k^{\prime}}^{2}}}$	EllipticPi((k)^(2), k)= EllipticE(k)/ 1 - (k)^(2)	EllipticPi[(k)^(2), (k)^2]= EllipticE[(k)^2]/ 1 - (k)^(2)	Failure	Failure	Fail Float(infinity) <- {k = 1} 3.458268152-1.791805641*I <- {k = 2} 8.701204041-2.810641644*I <- {k = 3}	Fail DirectedInfinity[] <- {Rule[k, 1]} Complex[3.4582681513867195, -1.791805641849464] <- {Rule[k, 2]} Complex[8.701204041408065, -2.8106416436990806] <- {Rule[k, 3]}
19.6#Ex5	${\displaystyle{\displaystyle\Pi\left(-k,k\right)=\tfrac{1}{4}\pi(1+k)^{-1}+\tfrac{1}{2}K\left(k\right)}}$	EllipticPi(- k, k)=(1)/(4)*Pi*(1 + k)^(- 1)+(1)/(2)*EllipticK(k)	EllipticPi[- k, (k)^2]=Divide[1,4]*Pi*(1 + k)^(- 1)+Divide[1,2]*EllipticK[(k)^2]	Failure	Failure	Error	Successful
19.6.E3	${\displaystyle{\displaystyle\Pi\left(\alpha^{2},0\right)=\pi/(2\sqrt{1-\alpha^{2}}),\quad\Pi\left(0,k\right)}}$	EllipticPi((alpha)^(2), 0)= Pi/(2*sqrt(1 - (alpha)^(2))), EllipticPi(0, k)	EllipticPi[(\[Alpha])^(2), (0)^2]= Pi/(2*Sqrt[1 - (\[Alpha])^(2)]), EllipticPi[0, (k)^2]	Error	Failure	-	Error
19.6.E3	${\displaystyle{\displaystyle\pi/(2\sqrt{1-\alpha^{2}}),\quad\Pi\left(0,k\right)=K\left(k\right)}}$	Pi/(2*sqrt(1 - (alpha)^(2))), EllipticPi(0, k)= EllipticK(k)	Pi/(2*Sqrt[1 - (\[Alpha])^(2)]), EllipticPi[0, (k)^2]= EllipticK[(k)^2]	Error	Failure	-	Error
19.6.E5	${\displaystyle{\displaystyle\Pi\left(\alpha^{2},k\right)=K\left(k\right)-\Pi\left(k^{2}/\alpha^{2},k\right)}}$	EllipticPi((alpha)^(2), k)= EllipticK(k)- EllipticPi((k)^(2)/ (alpha)^(2), k)	EllipticPi[(\[Alpha])^(2), (k)^2]= EllipticK[(k)^2]- EllipticPi[(k)^(2)/ (\[Alpha])^(2), (k)^2]	Failure	Failure	Error	Fail Complex[0.6269006702249605, 0.17364143326773873] <- {Rule[k, 2], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.4911490398753759, 0.04265145338289439] <- {Rule[k, 3], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.6269006702249589, -0.17364143326773473] <- {Rule[k, 2], Rule[α, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[0.49114903987537695, -0.042651453382894666] <- {Rule[k, 3], Rule[α, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.6#Ex8	${\displaystyle{\displaystyle\Pi\left(\alpha^{2},0\right)=0}}$	EllipticPi((alpha)^(2), 0)= 0	EllipticPi[(\[Alpha])^(2), (0)^2]= 0	Failure	Failure	Fail .6097433517+.4760732230*I <- {alpha = 2^(1/2)+I*2^(1/2)} .6097433517-.4760732230*I <- {alpha = 2^(1/2)-I*2^(1/2)} .6097433517+.4760732230*I <- {alpha = -2^(1/2)-I*2^(1/2)} .6097433517-.4760732230*I <- {alpha = -2^(1/2)+I*2^(1/2)}	Fail Complex[0.6097433514448427, 0.4760732227700887] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.6097433514448427, -0.4760732227700887] <- {Rule[α, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[0.6097433514448427, 0.4760732227700887] <- {Rule[α, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[0.6097433514448427, -0.4760732227700887] <- {Rule[α, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
19.6#Ex11	${\displaystyle{\displaystyle F\left(0,k\right)=0}}$	EllipticF(sin(0), k)= 0	EllipticF[0, (k)^2]= 0	Successful	Successful	-	-
19.6#Ex12	${\displaystyle{\displaystyle F\left(\phi,0\right)=\phi}}$	EllipticF(sin(phi), 0)= phi	EllipticF[\[Phi], (0)^2]= \[Phi]	Failure	Successful	Successful	-
19.6#Ex13	${\displaystyle{\displaystyle F\left(\tfrac{1}{2}\pi,1\right)=\infty}}$	EllipticF(sin((1)/(2)*Pi), 1)= infinity	EllipticF[Divide[1,2]*Pi, (1)^2]= Infinity	Error	Failure	-	Fail Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[Indeterminate, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[Indeterminate, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[Indeterminate, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[Indeterminate, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
19.6#Ex14	${\displaystyle{\displaystyle F\left(\tfrac{1}{2}\pi,k\right)=K\left(k\right)}}$	EllipticF(sin((1)/(2)*Pi), k)= EllipticK(k)	EllipticF[Divide[1,2]*Pi, (k)^2]= EllipticK[(k)^2]	Successful	Successful	-	-
19.6#Ex15	${\displaystyle{\displaystyle\lim_{\phi\to 0}\ifrac{F\left(\phi,k\right)}{\phi}=1}}$	limit((EllipticF(sin(phi), k))/(phi), phi = 0)= 1	Limit[Divide[EllipticF[\[Phi], (k)^2],\[Phi]], \[Phi] -> 0]= 1	Successful	Successful	-	-
19.6#Ex16	${\displaystyle{\displaystyle E\left(0,k\right)=0}}$	EllipticE(sin(0), k)= 0	EllipticE[0, (k)^2]= 0	Successful	Successful	-	-
19.6#Ex17	${\displaystyle{\displaystyle E\left(\phi,0\right)=\phi}}$	EllipticE(sin(phi), 0)= phi	EllipticE[\[Phi], (0)^2]= \[Phi]	Failure	Successful	Successful	-
19.6#Ex18	${\displaystyle{\displaystyle E\left(\tfrac{1}{2}\pi,1\right)=1}}$	EllipticE(sin((1)/(2)*Pi), 1)= 1	EllipticE[Divide[1,2]*Pi, (1)^2]= 1	Successful	Successful	-	-
19.6#Ex19	${\displaystyle{\displaystyle E\left(\phi,1\right)=\sin\phi}}$	EllipticE(sin(phi), 1)= sin(phi)	EllipticE[\[Phi], (1)^2]= Sin[\[Phi]]	Successful	Failure	-	Successful
19.6#Ex20	${\displaystyle{\displaystyle E\left(\tfrac{1}{2}\pi,k\right)=E\left(k\right)}}$	EllipticE(sin((1)/(2)*Pi), k)= EllipticE(k)	EllipticE[Divide[1,2]*Pi, (k)^2]= EllipticE[(k)^2]	Successful	Successful	-	-
19.6.E10	${\displaystyle{\displaystyle\lim_{\phi\to 0}\ifrac{E\left(\phi,k\right)}{\phi}=1}}$	limit((EllipticE(sin(phi), k))/(phi), phi = 0)= 1	Limit[Divide[EllipticE[\[Phi], (k)^2],\[Phi]], \[Phi] -> 0]= 1	Successful	Successful	-	-
19.6#Ex21	${\displaystyle{\displaystyle\Pi\left(0,\alpha^{2},k\right)=0}}$	EllipticPi(sin(0), (alpha)^(2), k)= 0	EllipticPi[(\[Alpha])^(2), 0,(k)^2]= 0	Successful	Successful	-	-
19.6#Ex22	${\displaystyle{\displaystyle\Pi\left(\phi,0,0\right)=\phi}}$	EllipticPi(sin(phi), 0, 0)= phi	EllipticPi[0, \[Phi],(0)^2]= \[Phi]	Failure	Successful	Successful	-
19.6#Ex23	${\displaystyle{\displaystyle\Pi\left(\phi,1,0\right)=\tan\phi}}$	EllipticPi(sin(phi), 1, 0)= tan(phi)	EllipticPi[1, \[Phi],(0)^2]= Tan[\[Phi]]	Failure	Successful	Successful	-
19.6#Ex27	${\displaystyle{\displaystyle\Pi\left(\phi,0,k\right)=F\left(\phi,k\right)}}$	EllipticPi(sin(phi), 0, k)= EllipticF(sin(phi), k)	EllipticPi[0, \[Phi],(k)^2]= EllipticF[\[Phi], (k)^2]	Successful	Successful	-	-
19.6#Ex28	${\displaystyle{\displaystyle\Pi\left(\phi,k^{2},k\right)=\frac{1}{{k^{\prime}}^{2}}\left(E\left(\phi,k\right)-\frac{k^{2}}{\Delta}\sin\phi\cos\phi\right)}}$	EllipticPi(sin(phi), (k)^(2), k)=(1)/(1 - (k)^(2))*(EllipticE(sin(phi), k)-((k)^(2))/(Delta)*sin(phi)*cos(phi))	EllipticPi[(k)^(2), \[Phi],(k)^2]=Divide[1,1 - (k)^(2)]*(EllipticE[\[Phi], (k)^2]-Divide[(k)^(2),\[CapitalDelta]]*Sin[\[Phi]]*Cos[\[Phi]])	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), k = 1} 2.583555547+2.729095606*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), k = 2} 1.800281293+2.241476784*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), k = 3} Float(infinity)+Float(infinity)*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[2.5835555495392164, 2.729095607128086] <- {Rule[k, 2], Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.8002812942897588, 2.24147678315869] <- {Rule[k, 3], Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-1.1970424154535648, -1.4962733566009967] <- {Rule[k, 2], Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-1.3895982386729004, -1.2012830092764584] <- {Rule[k, 3], Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.6#Ex29	${\displaystyle{\displaystyle\Pi\left(\phi,1,k\right)=F\left(\phi,k\right)-\frac{1}{{k^{\prime}}^{2}}(E\left(\phi,k\right)-\Delta\tan\phi)}}$	EllipticPi(sin(phi), 1, k)= EllipticF(sin(phi), k)-(1)/(1 - (k)^(2))*(EllipticE(sin(phi), k)- Delta*tan(phi))	EllipticPi[1, \[Phi],(k)^2]= EllipticF[\[Phi], (k)^2]-Divide[1,1 - (k)^(2)]*(EllipticE[\[Phi], (k)^2]- \[CapitalDelta]*Tan[\[Phi]])	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), k = 1} -2.077524150+.3723387150*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), k = 2} -1.086812375+.1094418732*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), k = 3} Float(infinity)+Float(infinity)*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[-2.0775241504865902, 0.37233871531005636] <- {Rule[k, 2], Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-1.0868123753385022, 0.1094418733619304] <- {Rule[k, 3], Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-1.0231109018735427, -0.3338048760552679] <- {Rule[k, 2], Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-0.6914074071086094, -0.09499168364138477] <- {Rule[k, 3], Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.6#Ex30	${\displaystyle{\displaystyle\Pi\left(\tfrac{1}{2}\pi,\alpha^{2},k\right)=\Pi\left(\alpha^{2},k\right)}}$	EllipticPi(sin((1)/(2)*Pi), (alpha)^(2), k)= EllipticPi((alpha)^(2), k)	EllipticPi[(\[Alpha])^(2), Divide[1,2]*Pi,(k)^2]= EllipticPi[(\[Alpha])^(2), (k)^2]	Successful	Successful	-	-
19.6#Ex31	${\displaystyle{\displaystyle\lim_{\phi\to 0}\ifrac{\Pi\left(\phi,\alpha^{2},k\right)}{\phi}=1}}$	limit((EllipticPi(sin(phi), (alpha)^(2), k))/(phi), phi = 0)= 1	Limit[Divide[EllipticPi[(\[Alpha])^(2), \[Phi],(k)^2],\[Phi]], \[Phi] -> 0]= 1	Successful	Successful	-	-
19.7.E1	${\displaystyle{\displaystyle E\left(k\right){K^{\prime}}\left(k\right)+{E^{\prime}}\left(k\right)K\left(k\right)-K\left(k\right){K^{\prime}}\left(k\right)=\tfrac{1}{2}\pi}}$	EllipticE(k)*EllipticCK(k)+ EllipticCE(k)*EllipticK(k)- EllipticK(k)*EllipticCK(k)=(1)/(2)*Pi	EllipticE[(k)^2]*EllipticK[1-(k)^2]+ EllipticE[1-(k)^2]*EllipticK[(k)^2]- EllipticK[(k)^2]*EllipticK[1-(k)^2]=Divide[1,2]*Pi	Failure	Failure	Error	Successful
19.7#Ex1	${\displaystyle{\displaystyle K\left(ik/k^{\prime}\right)=k^{\prime}K\left(k\right)}}$	EllipticK(I*k/sqrt(1 - (k)^(2)))=sqrt(1 - (k)^(2))*EllipticK(k)	EllipticK[(I*k/Sqrt[1 - (k)^(2)])^2]=Sqrt[1 - (k)^(2)]*EllipticK[(k)^2]	Failure	Failure	Error	Fail Complex[-2.220446049250313*^-16, -2.9198052634126777] <- {Rule[k, 2]} Complex[0.0, -3.0497736761637926] <- {Rule[k, 3]}
19.7#Ex3	${\displaystyle{\displaystyle E\left(ik/k^{\prime}\right)=(1/k^{\prime})E\left(k\right)}}$	EllipticE(I*k/sqrt(1 - (k)^(2)))=(1/sqrt(1 - (k)^(2)))* EllipticE(k)	EllipticE[(I*k/Sqrt[1 - (k)^(2)])^2]=(1/Sqrt[1 - (k)^(2)])* EllipticE[(k)^2]	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {k = 1} .6e-9+.4691535424*I <- {k = 2} -.5e-9+.1878050212*I <- {k = 3}	Fail Complex[-5.551115123125783*^-16, 0.46915354293820644] <- {Rule[k, 2]} Complex[2.220446049250313*^-16, 0.18780502089910417] <- {Rule[k, 3]}
19.7#Ex9	${\displaystyle{\displaystyle F\left(\phi,k_{1}\right)=kF\left(\beta,k\right)}}$	EllipticF(sin(phi), k[1])= k*EllipticF(sin(beta), k)	EllipticF[\[Phi], (Subscript[k, 1])^2]= k*EllipticF[\[Beta], (k)^2]	Failure	Failure	Fail .1478755578-.6820014149*I <- {beta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), k = 1} .1609634146-1.274502936*I <- {beta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), k = 2} .1632109318-1.647695481*I <- {beta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), k = 3} .8504514854-1.466163968*I <- {beta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Successful
19.7#Ex10	${\displaystyle{\displaystyle E\left(\phi,k_{1}\right)=(E\left(\beta,k\right)-{k^{\prime}}^{2}F\left(\beta,k\right))/k}}$	EllipticE(sin(phi), k[1])=(EllipticE(sin(beta), k)- 1 - (k)^(2)* EllipticF(sin(beta), k))/ k	EllipticE[\[Phi], (Subscript[k, 1])^2]=(EllipticE[\[Beta], (k)^2]- 1 - (k)^(2)* EllipticF[\[Beta], (k)^2])/ k	Failure	Failure	Fail 3.373306475+3.452491889*I <- {beta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), k = 1} 3.041115012+4.688090748*I <- {beta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), k = 2} 2.905177117+5.217063396*I <- {beta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), k = 3} -1.662478744+4.734343989*I <- {beta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Successful
19.7#Ex11	${\displaystyle{\displaystyle\Pi\left(\phi,\alpha^{2},k_{1}\right)=k\Pi\left(\beta,k^{2}\alpha^{2},k\right)}}$	EllipticPi(sin(phi), (alpha)^(2), k[1])= k*EllipticPi(sin(beta), (k)^(2)* (alpha)^(2), k)	EllipticPi[(\[Alpha])^(2), \[Phi],(Subscript[k, 1])^2]= k*EllipticPi[(k)^(2)* (\[Alpha])^(2), \[Beta],(k)^2]	Failure	Failure	Skip	Successful
19.7#Ex17	${\displaystyle{\displaystyle\sin\theta=\frac{\sqrt{1+k^{2}}\sin\phi}{\sqrt{1+k^{2}{\sin^{2}}\phi}}}}$	sin(theta)=(sqrt(1 + (k)^(2))*sin(phi))/(sqrt(1 + (k)^(2)* (sin(phi))^(2)))	Sin[\[Theta]]=Divide[Sqrt[1 + (k)^(2)]*Sin[\[Phi]],Sqrt[1 + (k)^(2)* (Sin[\[Phi]])^(2)]]	Failure	Failure	Fail .863648898+.2706006291*I <- {phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 1} 1.061002746+.2942020964*I <- {phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 2} 1.109187756+.2984612377*I <- {phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 3} .863648898-.3329223119*I <- {phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[0.86364889928252, 0.27060062852924793] <- {Rule[k, 1], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.0610027472814447, 0.29420209574283873] <- {Rule[k, 2], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.1091877569768196, 0.2984612371222133] <- {Rule[k, 3], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.86364889928252, 0.3329223112062738] <- {Rule[k, 1], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.7#Ex19	${\displaystyle{\displaystyle F\left(i\phi,k\right)=iF\left(\psi,k^{\prime}\right)}}$	EllipticF(sin(I*phi), k)= I*EllipticF(sin(psi), sqrt(1 - (k)^(2)))	EllipticF[I*\[Phi], (k)^2]= I*EllipticF[\[Psi], (Sqrt[1 - (k)^(2)])^2]	Failure	Failure	Fail .9251391454+.76168273e-1*I <- {phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), k = 1} .3353318864+.3237108e-2*I <- {phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), k = 2} .2150344377+.8523035e-3*I <- {phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), k = 3} -1.903287979+.76168273e-1*I <- {phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[0.9251391460853863, 0.0761682733812794] <- {Rule[k, 1], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.33533188682392046, 0.0032371074313650716] <- {Rule[k, 2], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.21503443789021323, 8.523034067818847*^-4] <- {Rule[k, 3], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-1.9032879786608041, 0.0761682733812794] <- {Rule[k, 1], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.7#Ex20	${\displaystyle{\displaystyle E\left(i\phi,k\right)=i\left(F\left(\psi,k^{\prime}\right)-E\left(\psi,k^{\prime}\right)+(\tan\psi)\sqrt{1-{k^{\prime}}^{2}{\sin^{2}}\psi}\right)}}$	EllipticE(sin(I*phi), k)= I*(EllipticF(sin(psi), sqrt(1 - (k)^(2)))- EllipticE(sin(psi), sqrt(1 - (k)^(2)))+(tan(psi))*sqrt(1 - 1 - (k)^(2)* (sin(psi))^(2)))	EllipticE[I*\[Phi], (k)^2]= I*(EllipticF[\[Psi], (Sqrt[1 - (k)^(2)])^2]- EllipticE[\[Psi], (Sqrt[1 - (k)^(2)])^2]+(Tan[\[Psi]])*Sqrt[1 - 1 - (k)^(2)* (Sin[\[Psi]])^(2)])	Failure	Failure	Fail -1.901989389-2.116793618*I <- {phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), k = 1} -6.480561985-4.744566218*I <- {phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), k = 2} -10.28790921-7.196969925*I <- {phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), k = 3} -2.401081691-2.116793618*I <- {phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[-1.9019893907701486, -2.1167936214122864] <- {Rule[k, 1], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-6.480561989871777, -4.744566223003809] <- {Rule[k, 2], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-10.28790921849954, -7.196969931680352] <- {Rule[k, 3], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-2.401081691908424, -2.1167936214122864] <- {Rule[k, 1], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.7#Ex21	${\displaystyle{\displaystyle\Pi\left(i\phi,\alpha^{2},k\right)=i\left(F\left(\psi,k^{\prime}\right)-\alpha^{2}\Pi\left(\psi,1-\alpha^{2},k^{\prime}\right)\right)/{(1-\alpha^{2})}}}$	EllipticPi(sin(I*phi), (alpha)^(2), k)= I*(EllipticF(sin(psi), sqrt(1 - (k)^(2)))- (alpha)^(2)* EllipticPi(sin(psi), 1 - (alpha)^(2), sqrt(1 - (k)^(2))))/(1 - (alpha)^(2))	EllipticPi[(\[Alpha])^(2), I*\[Phi],(k)^2]= I*(EllipticF[\[Psi], (Sqrt[1 - (k)^(2)])^2]- (\[Alpha])^(2)* EllipticPi[1 - (\[Alpha])^(2), \[Psi],(Sqrt[1 - (k)^(2)])^2])/(1 - (\[Alpha])^(2))	Failure	Failure	Skip	Skip
19.8.E4	${\displaystyle{\displaystyle\frac{1}{M\left(a_{0},g_{0}\right)}=\frac{2}{\pi}\int_{0}^{\pi/2}\frac{\mathrm{d}\theta}{\sqrt{a_{0}^{2}{\cos^{2}}\theta+g_{0}^{2}{\sin^{2}}\theta}}}}$	(1)/(GaussAGM(a[0], g[0]))int((1)/(sqrt(a(a[0])^(2)*(cos(theta))^(2)+ g(g[0])^(2)*(sin(theta))^(2))), theta = 0..Pi/ 2)	Error	Failure	Error	Skip	-
19.8.E4	${\displaystyle{\displaystyle\frac{2}{\pi}\int_{0}^{\pi/2}\frac{\mathrm{d}\theta}{\sqrt{a_{0}^{2}{\cos^{2}}\theta+g_{0}^{2}{\sin^{2}}\theta}}=\frac{1}{\pi}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t(t+a_{0}^{2})(t+g_{0}^{2})}}}}$	int((1)/(sqrt(a(a[0])^(2)*(cos(theta))^(2)+ g(g[0])^(2)*(sin(theta))^(2))), theta = 0..Pi/ 2)int((1)/(sqrt(t*(t + a(a[0])^(2))*(t + g(g[0])^(2)))), t = 0..infinity)	Integrate[Divide[1,Sqrt[a(Subscript[a, 0])^(2)*(Cos[\[Theta]])^(2)+ g(Subscript[g, 0])^(2)*(Sin[\[Theta]])^(2)]], {\[Theta], 0, Pi/ 2}]Integrate[Divide[1,Sqrt[t*(t + a(Subscript[a, 0])^(2))*(t + g(Subscript[g, 0])^(2))]], {t, 0, Infinity}]	Failure	Failure	Skip	Error
19.8.E5	${\displaystyle{\displaystyle K\left(k\right)=\frac{\pi}{2\!M\left(1,k^{\prime}\right)}}}$	EllipticK(k)=(Pi)/(2*GaussAGM(1, sqrt(1 - (k)^(2))))	Error	Failure	Error	Skip	-
19.8.E6	${\displaystyle{\displaystyle E\left(k\right)=\frac{\pi}{2\!M\left(1,k^{\prime}\right)}\left(a_{0}^{2}-\sum_{n=0}^{\infty}2^{n-1}c_{n}^{2}\right)}}$	EllipticE(k)(a(a[0])^(2)- sum((2)^(n - 1)* c(c[n])^(2), n = 0..infinity))	Error	Failure	Error	Skip	-
19.8.E6	${\displaystyle{\displaystyle\frac{\pi}{2\!M\left(1,k^{\prime}\right)}\left(a_{0}^{2}-\sum_{n=0}^{\infty}2^{n-1}c_{n}^{2}\right)=K\left(k\right)\left(a_{1}^{2}-\sum_{n=2}^{\infty}2^{n-1}c_{n}^{2}\right)}}$	(a(a[0])^(2)- sum((2)^(n - 1)* c(c[n])^(2), n = 0..infinity))(a(a[1])^(2)- sum((2)^(n - 1)* c(c[n])^(2), n = 2..infinity))	Error	Failure	Error	Skip	-
19.8.E7	${\displaystyle{\displaystyle\Pi\left(\alpha^{2},k\right)=\frac{\pi}{4\!M\left(1,k^{\prime}\right)}\left(2+\frac{\alpha^{2}}{1-\alpha^{2}}\sum_{n=0}^{\infty}Q_{n}\right)}}$	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	Error	Skip	-
19.8.E9	${\displaystyle{\displaystyle\Pi\left(\alpha^{2},k\right)=\frac{\pi}{4\!M\left(1,k^{\prime}\right)}\frac{k^{2}}{k^{2}-\alpha^{2}}\sum_{n=0}^{\infty}Q_{n}}}$	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	Error	Skip	-
19.8#Ex7	${\displaystyle{\displaystyle\phi_{1}=\phi+\operatorname{arctan}\left(k^{\prime}\tan\phi\right)}}$	phi[1]= phi + arctan(sqrt(1 - (k)^(2))*tan(phi))	Subscript[\[Phi], 1]= \[Phi]+ ArcTan[Sqrt[1 - (k)^(2)]*Tan[\[Phi]]]	Failure	Failure	Fail 1.094670290-.1488495390e-1*I <- {phi = 2^(1/2)+I*2^(1/2), phi[1] = 2^(1/2)+I*2^(1/2), k = 2} 1.264955024-.1049221987e-1*I <- {phi = 2^(1/2)+I*2^(1/2), phi[1] = 2^(1/2)+I*2^(1/2), k = 3} -2.828427124*I <- {phi = 2^(1/2)+I*2^(1/2), phi[1] = 2^(1/2)-I*2^(1/2), k = 1} 1.094670290-2.843312078*I <- {phi = 2^(1/2)+I*2^(1/2), phi[1] = 2^(1/2)-I*2^(1/2), k = 2} ... skip entries to safe data	Successful
19.8#Ex7	${\displaystyle{\displaystyle\phi+\operatorname{arctan}\left(k^{\prime}\tan\phi\right)=\operatorname{arcsin}\left((1+k^{\prime})\frac{\sin\phi\cos\phi}{\sqrt{1-k^{2}{\sin^{2}}\phi}}\right)}}$	phi + arctan(sqrt(1 - (k)^(2))*tan(phi))= arcsin((1 +sqrt(1 - (k)^(2)))*(sin(phi)*cos(phi))/(sqrt(1 - (k)^(2)* (sin(phi))^(2))))	\[Phi]+ ArcTan[Sqrt[1 - (k)^(2)]*Tan[\[Phi]]]= ArcSin[(1 +Sqrt[1 - (k)^(2)])*Divide[Sin[\[Phi]]*Cos[\[Phi]],Sqrt[1 - (k)^(2)* (Sin[\[Phi]])^(2)]]]	Failure	Failure	Fail 1.876175051-2.798657216*I <- {phi = 2^(1/2)-I*2^(1/2), k = 2} 2.216744518-2.807442684*I <- {phi = 2^(1/2)-I*2^(1/2), k = 3} -1.876175051+2.798657216*I <- {phi = -2^(1/2)+I*2^(1/2), k = 2} -2.216744518+2.807442684*I <- {phi = -2^(1/2)+I*2^(1/2), k = 3}	Fail Complex[1.8761750519919396, -2.798657217034252] <- {Rule[k, 2], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[2.216744518353377, -2.8074426850863325] <- {Rule[k, 3], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-1.8761750519919396, 2.798657217034252] <- {Rule[k, 2], Rule[ϕ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} Complex[-2.216744518353377, 2.8074426850863325] <- {Rule[k, 3], Rule[ϕ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
19.8#Ex8	${\displaystyle{\displaystyle K\left(k\right)=(1+k_{1})K\left(k_{1}\right)}}$	EllipticK(k)=(1 + k[1])* EllipticK(k[1])	EllipticK[(k)^2]=(1 + Subscript[k, 1])* EllipticK[(Subscript[k, 1])^2]	Failure	Failure	Error	Fail Complex[-1.685750354812596, -1.0782578237498217] <- {Rule[k, 2]} Complex[-1.6173867356247322, -0.842875177406298] <- {Rule[k, 3]}
19.8#Ex9	${\displaystyle{\displaystyle E\left(k\right)=(1+k^{\prime})E\left(k_{1}\right)-k^{\prime}K\left(k\right)}}$	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	Fail Complex[0.8064340115138717, 0.2620377583302733] <- {Rule[k, 2]} Complex[1.1233981879763189, -0.2935366302694502] <- {Rule[k, 3]}
19.8#Ex10	${\displaystyle{\displaystyle F\left(\phi,k\right)=\tfrac{1}{2}(1+k_{1})F\left(\phi_{1},k_{1}\right)}}$	EllipticF(sin(phi), k)=(1)/(2)*(1 + k[1])* EllipticF(sin(phi[1]), k[1])	EllipticF[\[Phi], (k)^2]=Divide[1,2]*(1 + Subscript[k, 1])* EllipticF[Subscript[\[Phi], 1], (Subscript[k, 1])^2]	Failure	Failure	Fail .2918190600+.64188702e-1*I <- {phi = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), phi[1] = 2^(1/2)+I*2^(1/2), k = 1} .407379233e-1-.384751455*I <- {phi = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), phi[1] = 2^(1/2)+I*2^(1/2), k = 2} -.393423424e-1-.6075011659*I <- {phi = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), phi[1] = 2^(1/2)+I*2^(1/2), k = 3} -1.145001001+.5724275385*I <- {phi = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), phi[1] = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Successful
19.8#Ex11	${\displaystyle{\displaystyle E\left(\phi,k\right)=\tfrac{1}{2}(1+k^{\prime})E\left(\phi_{1},k_{1}\right)-k^{\prime}F\left(\phi,k\right)+\tfrac{1}{2}(1-k^{\prime})\sin\phi_{1}}}$	EllipticE(sin(phi), k)=(1)/(2)*(1 +sqrt(1 - (k)^(2)))* EllipticE(sin(phi[1]), k[1])-sqrt(1 - (k)^(2))*EllipticF(sin(phi), k)+(1)/(2)*(1 -sqrt(1 - (k)^(2)))* sin(phi[1])	EllipticE[\[Phi], (k)^2]=Divide[1,2]*(1 +Sqrt[1 - (k)^(2)])* EllipticE[Subscript[\[Phi], 1], (Subscript[k, 1])^2]-Sqrt[1 - (k)^(2)]*EllipticF[\[Phi], (k)^2]+Divide[1,2]*(1 -Sqrt[1 - (k)^(2)])* Sin[Subscript[\[Phi], 1]]	Failure	Failure	Fail -.942116029-.9810550268*I <- {phi = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), phi[1] = 2^(1/2)+I*2^(1/2), k = 1} .743033975-3.185064608*I <- {phi = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), phi[1] = 2^(1/2)+I*2^(1/2), k = 2} 3.178569085-4.992225484*I <- {phi = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), phi[1] = 2^(1/2)+I*2^(1/2), k = 3} 1.575776580+2.225504017*I <- {phi = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), phi[1] = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Successful
19.8#Ex14	${\displaystyle{\displaystyle c_{1}={\csc^{2}}\phi_{1}}}$	c[1]= (csc(phi[1]))^(2)	Subscript[c, 1]= (Csc[Subscript[\[Phi], 1]])^(2)	Failure	Failure	Fail 1.210530730+1.472494686*I <- {c[1] = 2^(1/2)+I*2^(1/2), phi[1] = 2^(1/2)+I*2^(1/2)} 1.210530730+1.355932438*I <- {c[1] = 2^(1/2)+I*2^(1/2), phi[1] = 2^(1/2)-I*2^(1/2)} 1.210530730+1.472494686*I <- {c[1] = 2^(1/2)+I*2^(1/2), phi[1] = -2^(1/2)-I*2^(1/2)} 1.210530730+1.355932438*I <- {c[1] = 2^(1/2)+I*2^(1/2), phi[1] = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[1.210530730056432, 1.472494685832072] <- {Rule[Subscript[c, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ϕ, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.210530730056432, 1.3559324389141183] <- {Rule[Subscript[c, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ϕ, 1], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[1.210530730056432, 1.472494685832072] <- {Rule[Subscript[c, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ϕ, 1], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[1.210530730056432, 1.3559324389141183] <- {Rule[Subscript[c, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ϕ, 1], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.8#Ex16	${\displaystyle{\displaystyle 2\phi_{2}=\phi+\operatorname{arcsin}\left(k\sin\phi\right)}}$	2*phi[2]= phi + arcsin(k*sin(phi))	2*Subscript[\[Phi], 2]= \[Phi]+ ArcSin[k*Sin[\[Phi]]]	Failure	Failure	Fail -.13450037e-1-.735044026*I <- {phi = 2^(1/2)+I*2^(1/2), phi[2] = 2^(1/2)+I*2^(1/2), k = 2} -.15590800e-1-1.147765401*I <- {phi = 2^(1/2)+I*2^(1/2), phi[2] = 2^(1/2)+I*2^(1/2), k = 3} -5.656854248*I <- {phi = 2^(1/2)+I*2^(1/2), phi[2] = 2^(1/2)-I*2^(1/2), k = 1} -.13450037e-1-6.391898274*I <- {phi = 2^(1/2)+I*2^(1/2), phi[2] = 2^(1/2)-I*2^(1/2), k = 2} ... skip entries to safe data	Successful
19.8#Ex17	${\displaystyle{\displaystyle F\left(\phi,k\right)=\frac{2}{1+k}F\left(\phi_{2},k_{2}\right)}}$	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	Fail -.1478755578+.6820014149*I <- {phi = 2^(1/2)+I*2^(1/2), k[2] = 2^(1/2)+I*2^(1/2), phi[2] = 2^(1/2)+I*2^(1/2), k = 1} -.1866400363+.5025213980*I <- {phi = 2^(1/2)+I*2^(1/2), k[2] = 2^(1/2)+I*2^(1/2), phi[2] = 2^(1/2)+I*2^(1/2), k = 2} -.1605619730+.4145017571*I <- {phi = 2^(1/2)+I*2^(1/2), k[2] = 2^(1/2)+I*2^(1/2), phi[2] = 2^(1/2)+I*2^(1/2), k = 3} -.8504514854+1.514599702*I <- {phi = 2^(1/2)+I*2^(1/2), k[2] = 2^(1/2)+I*2^(1/2), phi[2] = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Successful
19.8#Ex18	${\displaystyle{\displaystyle E\left(\phi,k\right)=(1+k)E\left(\phi_{2},k_{2}\right)+(1-k)F\left(\phi_{2},k_{2}\right)-k\sin\phi}}$	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	Fail -3.768464116-3.924220109*I <- {phi = 2^(1/2)+I*2^(1/2), k[2] = 2^(1/2)+I*2^(1/2), phi[2] = 2^(1/2)+I*2^(1/2), k = 1} -3.226003445-6.062382948*I <- {phi = 2^(1/2)+I*2^(1/2), k[2] = 2^(1/2)+I*2^(1/2), phi[2] = 2^(1/2)+I*2^(1/2), k = 2} -2.601575251-8.024788758*I <- {phi = 2^(1/2)+I*2^(1/2), k[2] = 2^(1/2)+I*2^(1/2), phi[2] = 2^(1/2)+I*2^(1/2), k = 3} 6.303106322+7.694970191*I <- {phi = 2^(1/2)+I*2^(1/2), k[2] = 2^(1/2)+I*2^(1/2), phi[2] = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Successful
19.8#Ex20	${\displaystyle{\displaystyle\sin\psi_{1}=\frac{(1+k^{\prime})\sin\phi}{1+\Delta}}}$	sin(psi[1])=((1 +sqrt(1 - (k)^(2)))* sin(phi))/(1 + Delta)	Sin[Subscript[\[Psi], 1]]=Divide[(1 +Sqrt[1 - (k)^(2)])* Sin[\[Phi]],1 + \[CapitalDelta]]	Failure	Failure	Fail 1.433508616+.5973782105*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), psi[1] = 2^(1/2)+I*2^(1/2), k = 1} .921485403-.6462809029*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), psi[1] = 2^(1/2)+I*2^(1/2), k = 2} .597378210-1.433508616*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), psi[1] = 2^(1/2)+I*2^(1/2), k = 3} 1.433508616-.61447305e-2*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), psi[1] = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[1.4335086174698362, 0.5973782103005845] <- {Rule[k, 1], Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ψ, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.921485403472285, -0.6462809030436932] <- {Rule[k, 2], Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ψ, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.5973782103005842, -1.433508617469836] <- {Rule[k, 3], Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ψ, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.4335086174698362, -0.006144729434937324] <- {Rule[k, 1], Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ψ, 1], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.8#Ex21	${\displaystyle{\displaystyle\Delta=\sqrt{1-k^{2}{\sin^{2}}\phi}}}$	Delta =sqrt(1 - (k)^(2)* (sin(phi))^(2))	\[CapitalDelta]=Sqrt[1 - (k)^(2)* (Sin[\[Phi]])^(2)]	Failure	Failure	Fail 1.074539570+3.325606671*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), k = 1} .7940624558+5.601906026*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), k = 2} .4980873551+7.792433846*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), k = 3} 1.074539570-.497179547*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[1.0745395706783705, 3.3256066725373055] <- {Rule[k, 1], Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.7940624576072436, 5.601906029178632] <- {Rule[k, 2], Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.4980873572993363, 7.792433849582317] <- {Rule[k, 3], Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.0745395706783705, -0.49717954779111495] <- {Rule[k, 1], Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.8#Ex22	${\displaystyle{\displaystyle F\left(\phi,k\right)=(1+k_{1})F\left(\psi_{1},k_{1}\right)}}$	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	Fail .945637034e-1-1.362004431*I <- {phi = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), psi[1] = 2^(1/2)+I*2^(1/2), k = 1} -.1565174333-1.810944588*I <- {phi = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), psi[1] = 2^(1/2)+I*2^(1/2), k = 2} -.2365976990-2.033694299*I <- {phi = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), psi[1] = 2^(1/2)+I*2^(1/2), k = 3} -2.779076419-.345526758*I <- {phi = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), psi[1] = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[-0.48907441628770865, -1.4903818357543746] <- {Rule[k, 1], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ψ, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-1.4370593625301775, -1.7478657864319769] <- {Rule[k, 2], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ψ, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-1.7006101295186067, -1.3166147323299215] <- {Rule[k, 3], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ψ, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.4890744162877091, 4.471145507263124] <- {Rule[k, 1], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[ψ, 1], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.8#Ex23	${\displaystyle{\displaystyle E\left(\phi,k\right)=(1+k^{\prime})E\left(\psi_{1},k_{1}\right)-k^{\prime}F\left(\phi,k\right)+(1-\Delta)\cot\phi}}$	EllipticE(sin(phi), k)=(1 +sqrt(1 - (k)^(2)))* EllipticE(sin(psi[1]), k[1])-sqrt(1 - (k)^(2))*EllipticF(sin(phi), k)+(1 - Delta)* 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 - \[CapitalDelta])* Cot[\[Phi]]	Failure	Failure	Fail -.607875038-2.285836406*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), psi[1] = 2^(1/2)+I*2^(1/2), k = 1} 3.299178317-9.848207688*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), psi[1] = 2^(1/2)+I*2^(1/2), k = 2} 7.141163056-15.04717488*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), psi[1] = 2^(1/2)+I*2^(1/2), k = 3} 4.427910181+3.523758744*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), k[1] = 2^(1/2)+I*2^(1/2), psi[1] = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Skip
19.8#Ex26	${\displaystyle{\displaystyle c={\csc^{2}}\phi}}$	c = (csc(phi))^(2)	c = (Csc[\[Phi]])^(2)	Failure	Failure	Fail 1.210530730+1.472494686*I <- {c = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2)} 1.210530730+1.355932438*I <- {c = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)-I*2^(1/2)} 1.210530730+1.472494686*I <- {c = 2^(1/2)+I*2^(1/2), phi = -2^(1/2)-I*2^(1/2)} 1.210530730+1.355932438*I <- {c = 2^(1/2)+I*2^(1/2), phi = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[1.210530730056432, 1.472494685832072] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.210530730056432, 1.3559324389141183] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[1.210530730056432, 1.472494685832072] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[1.210530730056432, 1.3559324389141183] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.9#Ex2	${\displaystyle{\displaystyle 1<=E\left(k\right)}}$	1 < = EllipticE(k)	1 < = EllipticE[(k)^2]	Failure	Failure	Successful	Successful
19.9#Ex2	${\displaystyle{\displaystyle E\left(k\right)<=\pi/2}}$	EllipticE(k)< = Pi/ 2	EllipticE[(k)^2]< = Pi/ 2	Failure	Failure	Successful	Successful
19.9#Ex3	${\displaystyle{\displaystyle 1<=(2/\pi)\sqrt{1-\alpha^{2}}\Pi\left(\alpha^{2},k\right)\leq 1/k^{\prime}}}$	1 < =(2/ Pi)*sqrt(1 - (alpha)^(2))*EllipticPi((alpha)^(2), k)<= 1/sqrt(1 - (k)^(2))	1 < =(2/ Pi)*Sqrt[1 - (\[Alpha])^(2)]*EllipticPi[(\[Alpha])^(2), (k)^2]<= 1/Sqrt[1 - (k)^(2)]	Failure	Failure	Error	Successful
19.9.E2	${\displaystyle{\displaystyle 1+\frac{{k^{\prime}}^{2}}{8}<\frac{K\left(k\right)}{\ln\left(4/k^{\prime}\right)}}}$	1 +(1 - (k)^(2))/(8)<(EllipticK(k))/(ln(4/sqrt(1 - (k)^(2))))	1 +Divide[1 - (k)^(2),8]<Divide[EllipticK[(k)^2],Log[4/Sqrt[1 - (k)^(2)]]]	Failure	Failure	Error	Successful
19.9.E2	${\displaystyle{\displaystyle\frac{K\left(k\right)}{\ln\left(4/k^{\prime}\right)}<1+\frac{{k^{\prime}}^{2}}{4}}}$	(EllipticK(k))/(ln(4/sqrt(1 - (k)^(2))))< 1 +(1 - (k)^(2))/(4)	Divide[EllipticK[(k)^2],Log[4/Sqrt[1 - (k)^(2)]]]< 1 +Divide[1 - (k)^(2),4]	Failure	Failure	Error	Successful
19.9.E3	${\displaystyle{\displaystyle 9+\frac{k^{2}{k^{\prime}}^{2}}{8}<\frac{(8+k^{2})K\left(k\right)}{\ln\left(4/k^{\prime}\right)}}}$	9 +((k)^(2)* 1 - (k)^(2))/(8)<((8 + (k)^(2))* EllipticK(k))/(ln(4/sqrt(1 - (k)^(2))))	9 +Divide[(k)^(2)* 1 - (k)^(2),8]<Divide[(8 + (k)^(2))* EllipticK[(k)^2],Log[4/Sqrt[1 - (k)^(2)]]]	Failure	Failure	Error	Successful
19.9.E3	${\displaystyle{\displaystyle\frac{(8+k^{2})K\left(k\right)}{\ln\left(4/k^{\prime}\right)}<9.096}}$	((8 + (k)^(2))* EllipticK(k))/(ln(4/sqrt(1 - (k)^(2))))< 9.096	Divide[(8 + (k)^(2))* EllipticK[(k)^2],Log[4/Sqrt[1 - (k)^(2)]]]< 9.096	Failure	Failure	Error	Successful
19.9.E4	${\displaystyle{\displaystyle\left(\frac{1+{k^{\prime}}^{3/2}}{2}\right)^{2/3}<=\frac{2}{\pi}E\left(k\right)}}$	((1 +(sqrt(1 - (k)^(2)))^(3/ 2))/(2))^(2/ 3)< =(2)/(Pi)*EllipticE(k)	(Divide[1 +(Sqrt[1 - (k)^(2)])^(3/ 2),2])^(2/ 3)< =Divide[2,Pi]*EllipticE[(k)^2]	Failure	Failure	Successful	Successful
19.9.E4	${\displaystyle{\displaystyle\frac{2}{\pi}E\left(k\right)<=\left(\frac{1+{k^{\prime}}^{2}}{2}\right)^{1/2}}}$	(2)/(Pi)*EllipticE(k)< =((1 + 1 - (k)^(2))/(2))^(1/ 2)	Divide[2,Pi]*EllipticE[(k)^2]< =(Divide[1 + 1 - (k)^(2),2])^(1/ 2)	Failure	Failure	Successful	Successful
19.9.E5	${\displaystyle{\displaystyle\ln\frac{(1+\sqrt{k^{\prime}})^{2}}{k}<\frac{\pi{K^{\prime}}\left(k\right)}{2\!K\left(k\right)}}}$	ln(((1 +sqrt(sqrt(1 - (k)^(2))))^(2))/(k))<(Pi*EllipticCK(k))/(2*EllipticK(k))	Log[Divide[(1 +Sqrt[Sqrt[1 - (k)^(2)]])^(2),k]]<Divide[Pi*EllipticK[1-(k)^2],2*EllipticK[(k)^2]]	Failure	Failure	Error	Successful
19.9.E5	${\displaystyle{\displaystyle\frac{\pi{K^{\prime}}\left(k\right)}{2\!K\left(k\right)}<\ln\frac{2(1+k^{\prime})}{k}}}$	(Pi*EllipticCK(k))/(2*EllipticK(k))< ln((2*(1 +sqrt(1 - (k)^(2))))/(k))	Divide[Pi*EllipticK[1-(k)^2],2*EllipticK[(k)^2]]< Log[Divide[2*(1 +Sqrt[1 - (k)^(2)]),k]]	Failure	Failure	Error	Successful
19.9.E6	${\displaystyle{\displaystyle(1-\tfrac{3}{4}k^{2})^{-1/2}<\frac{4}{\pi k^{2}}(K\left(k\right)-E\left(k\right))}}$	(1 -(3)/(4)*(k)^(2))^(- 1/ 2)<(4)/(Pi*(k)^(2))*(EllipticK(k)- EllipticE(k))	(1 -Divide[3,4]*(k)^(2))^(- 1/ 2)<Divide[4,Pi*(k)^(2)]*(EllipticK[(k)^2]- EllipticE[(k)^2])	Failure	Failure	Error	Successful
19.9.E6	${\displaystyle{\displaystyle\frac{4}{\pi k^{2}}(K\left(k\right)-E\left(k\right))<(k^{\prime})^{-3/4}}}$	(4)/(Pi*(k)^(2))*(EllipticK(k)- EllipticE(k))<(sqrt(1 - (k)^(2)))^(- 3/ 4)	Divide[4,Pi*(k)^(2)]*(EllipticK[(k)^2]- EllipticE[(k)^2])<(Sqrt[1 - (k)^(2)])^(- 3/ 4)	Failure	Failure	Error	Successful
19.9.E7	${\displaystyle{\displaystyle(1-\tfrac{1}{4}k^{2})^{-1/2}<\frac{4}{\pi k^{2}}(E\left(k\right)-{k^{\prime}}^{2}K\left(k\right))}}$	(1 -(1)/(4)*(k)^(2))^(- 1/ 2)<(4)/(Pi*(k)^(2))*(EllipticE(k)- 1 - (k)^(2)* EllipticK(k))	(1 -Divide[1,4]*(k)^(2))^(- 1/ 2)<Divide[4,Pi*(k)^(2)]*(EllipticE[(k)^2]- 1 - (k)^(2)* EllipticK[(k)^2])	Failure	Failure	Error	Successful
19.9.E8	${\displaystyle{\displaystyle k^{\prime}<\frac{E\left(k\right)}{K\left(k\right)}}}$	sqrt(1 - (k)^(2))<(EllipticE(k))/(EllipticK(k))	Sqrt[1 - (k)^(2)]<Divide[EllipticE[(k)^2],EllipticK[(k)^2]]	Failure	Failure	Error	Successful
19.9.E8	${\displaystyle{\displaystyle\frac{E\left(k\right)}{K\left(k\right)}<\left(\frac{1+k^{\prime}}{2}\right)^{2}}}$	(EllipticE(k))/(EllipticK(k))<((1 +sqrt(1 - (k)^(2)))/(2))^(2)	Divide[EllipticE[(k)^2],EllipticK[(k)^2]]<(Divide[1 +Sqrt[1 - (k)^(2)],2])^(2)	Failure	Failure	Error	Successful
19.9.E9	${\displaystyle{\displaystyle L(a,b)=4aE\left(k\right)}}$	L*(a , b)= 4*a*EllipticE(k)	L*(a , b)= 4*a*EllipticE[(k)^2]	Failure	Failure	Skip	Error
19.9.E11	${\displaystyle{\displaystyle\phi<=F\left(\phi,k\right)}}$	phi < = EllipticF(sin(phi), k)	\[Phi]< = EllipticF[\[Phi], (k)^2]	Failure	Failure	Successful	Successful
19.9.E12	${\displaystyle{\displaystyle E\left(\phi,k\right)<=\phi}}$	EllipticE(sin(phi), k)< = phi	EllipticE[\[Phi], (k)^2]< = \[Phi]	Failure	Failure	Successful	Successful
19.9.E13	${\displaystyle{\displaystyle\Pi\left(\phi,\alpha^{2},0\right)<=\Pi\left(\phi,\alpha^{2},k\right)}}$	EllipticPi(sin(phi), (alpha)^(2), 0)< = EllipticPi(sin(phi), (alpha)^(2), k)	EllipticPi[(\[Alpha])^(2), \[Phi],(0)^2]< = EllipticPi[(\[Alpha])^(2), \[Phi],(k)^2]	Failure	Failure	Successful	Successful
19.9.E14	${\displaystyle{\displaystyle\frac{3}{1+\Delta+\cos\phi}<\frac{F\left(\phi,k\right)}{\sin\phi}}}$	(3)/(1 + Delta + cos(phi))<(EllipticF(sin(phi), k))/(sin(phi))	Divide[3,1 + \[CapitalDelta]+ Cos[\[Phi]]]<Divide[EllipticF[\[Phi], (k)^2],Sin[\[Phi]]]	Failure	Failure	Successful	Successful
19.9.E14	${\displaystyle{\displaystyle\frac{F\left(\phi,k\right)}{\sin\phi}<\frac{1}{(\Delta\cos\phi)^{1/3}}}}$	(EllipticF(sin(phi), k))/(sin(phi))<(1)/((Delta*cos(phi))^(1/ 3))	Divide[EllipticF[\[Phi], (k)^2],Sin[\[Phi]]]<Divide[1,(\[CapitalDelta]*Cos[\[Phi]])^(1/ 3)]	Failure	Failure	Successful	Successful
19.9.E15	${\displaystyle{\displaystyle 1<F\left(\phi,k\right)\bigg{/}\left((\sin\phi)\ln\left(\frac{4}{\Delta+\cos\phi}\right)\right)}}$	1 < EllipticF(sin(phi), k)/((sin(phi))*ln((4)/(Delta + cos(phi))))	1 < EllipticF[\[Phi], (k)^2]/((Sin[\[Phi]])*Log[Divide[4,\[CapitalDelta]+ Cos[\[Phi]]]])	Failure	Failure	Successful	Successful
19.9.E15	${\displaystyle{\displaystyle F\left(\phi,k\right)\bigg{/}\left((\sin\phi)\ln\left(\frac{4}{\Delta+\cos\phi}\right)\right)<\frac{4}{2+(1+k^{2}){\sin^{2}}\phi}}}$	EllipticF(sin(phi), k)/((sin(phi))*ln((4)/(Delta + cos(phi))))<(4)/(2 +(1 + (k)^(2))* (sin(phi))^(2))	EllipticF[\[Phi], (k)^2]/((Sin[\[Phi]])*Log[Divide[4,\[CapitalDelta]+ Cos[\[Phi]]]])<Divide[4,2 +(1 + (k)^(2))* (Sin[\[Phi]])^(2)]	Failure	Failure	Successful	Successful
19.9.E17	${\displaystyle{\displaystyle L<=F\left(\phi,k\right)}}$	L < = EllipticF(sin(phi), k)	L < = EllipticF[\[Phi], (k)^2]	Failure	Failure	Successful	Successful
19.9.E17	${\displaystyle{\displaystyle F\left(\phi,k\right)<=\sqrt{UL}}}$	EllipticF(sin(phi), k)< =sqrt(U*L)	EllipticF[\[Phi], (k)^2]< =Sqrt[U*L]	Failure	Failure	Successful	Successful
19.9.E17	${\displaystyle{\displaystyle\sqrt{UL}<=\tfrac{1}{2}(U+L)}}$	sqrt(U*L)< =(1)/(2)*(U + L)	Sqrt[U*L]< =Divide[1,2]*(U + L)	Failure	Failure	Skip	Successful
19.9.E17	${\displaystyle{\displaystyle\tfrac{1}{2}(U+L)<=U}}$	(1)/(2)*(U + L)< = U	Divide[1,2]*(U + L)< = U	Failure	Failure	Skip	Successful
19.9#Ex4	${\displaystyle{\displaystyle L=(1/\sigma)\operatorname{arctanh}\left(\sigma\sin\phi\right)}}$	L =(1/ sigma)* arctanh(sigma*sin(phi))	L =(1/ \[Sigma])* ArcTanh[\[Sigma]*Sin[\[Phi]]]	Failure	Failure	Fail .8766496524+.9719354625*I <- {L = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), sigma = 2^(1/2)+I*2^(1/2)} .8447671788+1.854846236*I <- {L = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), sigma = 2^(1/2)-I*2^(1/2)} .8766496524+.9719354625*I <- {L = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), sigma = -2^(1/2)-I*2^(1/2)} .8447671788+1.854846236*I <- {L = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), sigma = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[0.8766496526774387, 0.9719354626867103] <- {Rule[L, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[σ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.8447671793191418, 0.9735808886528274] <- {Rule[L, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[σ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[1.9517774720687515, 1.85649166205948] <- {Rule[L, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[σ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[1.9836599454270485, 1.8548462360933629] <- {Rule[L, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[σ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.9#Ex5	${\displaystyle{\displaystyle U=\tfrac{1}{2}\operatorname{arctanh}\left(\sin\phi\right)+\tfrac{1}{2}k^{-1}\operatorname{arctanh}\left(k\sin\phi\right)}}$	U =(1)/(2)*arctanh(sin(phi))+(1)/(2)*(k)^(- 1)* arctanh(k*sin(phi))	U =Divide[1,2]*ArcTanh[Sin[\[Phi]]]+Divide[1,2]*(k)^(- 1)* ArcTanh[k*Sin[\[Phi]]]	Failure	Failure	Fail .9251391454-.76168273e-1*I <- {U = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), k = 1} 1.111745370+.2847489883*I <- {U = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), k = 2} 1.144167573+.4108582524*I <- {U = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), k = 3} .9251391454+2.904595397*I <- {U = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[0.9251391460853864, -0.0761682733812794] <- {Rule[k, 1], Rule[U, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.1117453704027949, 0.2847489881021125] <- {Rule[k, 2], Rule[U, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.1441675729373713, 0.41085825249377617] <- {Rule[k, 3], Rule[U, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.9251391460853866, 2.9045953981274697] <- {Rule[k, 1], Rule[U, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.11.E1	${\displaystyle{\displaystyle F\left(\theta,k\right)+F\left(\phi,k\right)=F\left(\psi,k\right)}}$	EllipticF(sin(theta), k)+ EllipticF(sin(phi), k)= EllipticF(sin(psi), k)	EllipticF[\[Theta], (k)^2]+ EllipticF[\[Phi], (k)^2]= EllipticF[\[Psi], (k)^2]	Failure	Failure	Fail .4890744166+1.490381835*I <- {phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 1} .2379932799+1.041441678*I <- {phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 2} .1579130142+.8186919671*I <- {phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 3} .4890744166-1.490381835*I <- {phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[0.48907441628770865, 1.4903818357543746] <- {Rule[k, 1], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.23799327973838402, 1.0414416780368256] <- {Rule[k, 2], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.15791301410980507, 0.8186919671649275] <- {Rule[k, 3], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.4890744162877084, 4.471145507263124] <- {Rule[k, 1], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.11.E2	${\displaystyle{\displaystyle E\left(\theta,k\right)+E\left(\phi,k\right)=E\left(\psi,k\right)+k^{2}\sin\theta\sin\phi\sin\psi}}$	EllipticE(sin(theta), k)+ EllipticE(sin(phi), k)= EllipticE(sin(psi), k)+ (k)^(2)* sin(theta)*sin(phi)*sin(psi)	EllipticE[\[Theta], (k)^2]+ EllipticE[\[Phi], (k)^2]= EllipticE[\[Psi], (k)^2]+ (k)^(2)* Sin[\[Theta]]*Sin[\[Phi]]*Sin[\[Psi]]	Failure	Failure	Fail -7.220392390-3.861416894*I <- {phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 1} -33.54643343-17.33538519*I <- {phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 2} -78.53436280-38.95995319*I <- {phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 3} -8.004067570-1.726125452*I <- {phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Skip
19.11#Ex1	${\displaystyle{\displaystyle\sin\psi=\frac{(\sin\theta\cos\phi)\Delta(\phi)+(\sin\phi\cos\theta)\Delta(\theta)}{1-k^{2}{\sin^{2}}\theta{\sin^{2}}\phi}}}$	sin(psi)=((sin(theta)*cos(phi))* Delta*(phi)+(sin(phi)*cos(theta))* Delta*(theta))/(1 - (k)^(2)* (sin(theta))^(2)* (sin(phi))^(2))	Sin[\[Psi]]=Divide[(Sin[\[Theta]]*Cos[\[Phi]])* \[CapitalDelta]*(\[Phi])+(Sin[\[Phi]]*Cos[\[Theta]])* \[CapitalDelta]*(\[Theta]),1 - (k)^(2)* (Sin[\[Theta]])^(2)* (Sin[\[Phi]])^(2)]	Failure	Failure	Fail 3.669776527-.1132600603*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 1} 2.522072859+.2078762259*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 2} 2.315484084+.2608036709*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 3} 2.972771438+1.122997368*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Skip
19.11#Ex2	${\displaystyle{\displaystyle\Delta(\theta)=\sqrt{1-k^{2}{\sin^{2}}\theta}}}$	Delta*(theta)=sqrt(1 - (k)^(2)* (sin(theta))^(2))	\[CapitalDelta]*(\[Theta])=Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)]	Failure	Failure	Fail -.3396739925+5.911393107*I <- {Delta = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 1} -.6201511062+8.187692462*I <- {Delta = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 2} -.9161262069+10.37822028*I <- {Delta = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 3} 3.660326006-1.911393109*I <- {Delta = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[-0.33967399169472473, 5.9113931101642105] <- {Rule[k, 1], Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.6201511047658516, 8.187692466805537] <- {Rule[k, 2], Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.9161262050737589, 10.378220287209222] <- {Rule[k, 3], Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[3.660326008305275, -1.91139311016421] <- {Rule[k, 1], Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.11#Ex3	${\displaystyle{\displaystyle\cos\psi=\frac{\cos\theta\cos\phi-(\sin\theta\sin\phi)\Delta(\theta)\Delta(\phi)}{1-k^{2}{\sin^{2}}\theta{\sin^{2}}\phi}}}$	cos(psi)=(cos(theta)*cos(phi)-(sin(theta)*sin(phi))* Delta*(theta)* Delta*(phi))/(1 - (k)^(2)* (sin(theta))^(2)* (sin(phi))^(2))	Cos[\[Psi]]=Divide[Cos[\[Theta]]*Cos[\[Phi]]-(Sin[\[Theta]]*Sin[\[Phi]])* \[CapitalDelta]*(\[Theta])* \[CapitalDelta]*(\[Phi]),1 - (k)^(2)* (Sin[\[Theta]])^(2)* (Sin[\[Phi]])^(2)]	Failure	Failure	Fail 3.530512453-2.923665161*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 1} 1.119055805-2.142711452*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 2} .6845705541-2.012504035*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 3} .5167796909-5.460396276*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Skip
19.11#Ex4	${\displaystyle{\displaystyle\tan\left(\tfrac{1}{2}\psi\right)=\frac{(\sin\theta)\Delta(\phi)+(\sin\phi)\Delta(\theta)}{\cos\theta+\cos\phi}}}$	tan((1)/(2)*psi)=((sin(theta))* Delta*(phi)+(sin(phi))* Delta*(theta))/(cos(theta)+ cos(phi))	Tan[Divide[1,2]*\[Psi]]=Divide[(Sin[\[Theta]])* \[CapitalDelta]*(\[Phi])+(Sin[\[Phi]])* \[CapitalDelta]*(\[Theta]),Cos[\[Theta]]+ Cos[\[Phi]]]	Failure	Failure	Fail 4.896680821+.6655470386*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2)} -14.02182515-13.61597715*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)-I*2^(1/2)} -4.050312273+.9925175081*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)-I*2^(1/2)} 14.86819369+15.27404169*I <- {Delta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[4.896680823473283, 0.6655470393878908] <- {Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[4.896680823473283, -0.992517507894449] <- {Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[4.050312275733391, -0.992517507894449] <- {Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[4.050312275733391, 0.6655470393878908] <- {Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.11#Ex5	${\displaystyle{\displaystyle\gamma=(({\csc^{2}}\theta)-\alpha^{2})(({\csc^{2}}\phi)-\alpha^{2})(({\csc^{2}}\psi)-\alpha^{2})}}$	gamma =(((csc(theta))^(2))- (alpha)^(2))*(((csc(phi))^(2))- (alpha)^(2))*(((csc(psi))^(2))- (alpha)^(2))	\[Gamma]=(((Csc[\[Theta]])^(2))- (\[Alpha])^(2))*(((Csc[\[Phi]])^(2))- (\[Alpha])^(2))*(((Csc[\[Psi]])^(2))- (\[Alpha])^(2))	Failure	Failure	Fail 10.63251777-66.33335806*I <- {alpha = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2)} 10.43981655-64.41845493*I <- {alpha = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)-I*2^(1/2)} 10.63251777-66.33335806*I <- {alpha = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)-I*2^(1/2)} 10.43981655-64.41845493*I <- {alpha = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Skip
19.11.E7	${\displaystyle{\displaystyle F\left(\phi,k\right)=K\left(k\right)-F\left(\theta,k\right)}}$	EllipticF(sin(phi), k)= EllipticK(k)- EllipticF(sin(theta), k)	EllipticF[\[Phi], (k)^2]= EllipticK[(k)^2]- EllipticF[\[Theta], (k)^2]	Failure	Failure	Error	Fail DirectedInfinity[] <- {Rule[k, 1], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.3668886179295301, 3.161141179823473] <- {Rule[k, 2], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.2233028836553006, 2.480259111736153] <- {Rule[k, 3], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[k, 1], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.11.E8	${\displaystyle{\displaystyle E\left(\phi,k\right)=E\left(k\right)-E\left(\theta,k\right)+k^{2}\sin\theta\sin\phi}}$	EllipticE(sin(phi), k)= EllipticE(k)- EllipticE(sin(theta), k)+ (k)^(2)* sin(theta)*sin(phi)	EllipticE[\[Phi], (k)^2]= EllipticE[(k)^2]- EllipticE[\[Theta], (k)^2]+ (k)^(2)* Sin[\[Theta]]*Sin[\[Phi]]	Failure	Failure	Fail -1.234974115-.6949781160*I <- {phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 1} -10.67592308-7.903201927*I <- {phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 2} -29.48202603-17.16755346*I <- {phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 3} -1.417094085+0.*I <- {phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[-1.234974118270812, -0.6949781151190215] <- {Rule[k, 1], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-10.675923102954668, -7.903201921490194] <- {Rule[k, 2], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-29.48202606417501, -17.167553444002383] <- {Rule[k, 3], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-1.4170940876643146, 0.0] <- {Rule[k, 1], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.11.E9	${\displaystyle{\displaystyle\tan\theta=1/(k^{\prime}\tan\phi)}}$	tan(theta)= 1/(sqrt(1 - (k)^(2))*tan(phi))	Tan[\[Theta]]= 1/(Sqrt[1 - (k)^(2)]*Tan[\[Phi]])	Failure	Failure	Fail Float(infinity)+Float(infinity)*I <- {phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 1} .5564234783+1.137215141*I <- {phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 2} .3565812465+1.129911849*I <- {phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 3} Float(infinity)+Float(infinity)*I <- {phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail DirectedInfinity[] <- {Rule[k, 1], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.5564234779243372, 1.1372151415780356] <- {Rule[k, 2], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.3565812462431767, 1.1299118490197817] <- {Rule[k, 3], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} DirectedInfinity[] <- {Rule[k, 1], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.11#Ex7	${\displaystyle{\displaystyle\gamma=(1-\alpha^{2})(({\csc^{2}}\theta)-\alpha^{2})(({\csc^{2}}\phi)-\alpha^{2})}}$	gamma =(1 - (alpha)^(2))*(((csc(theta))^(2))- (alpha)^(2))*(((csc(phi))^(2))- (alpha)^(2))	\[Gamma]=(1 - (\[Alpha])^(2))*(((Csc[\[Theta]])^(2))- (\[Alpha])^(2))*(((Csc[\[Phi]])^(2))- (\[Alpha])^(2))	Failure	Failure	Fail 23.61819217-64.05943146*I <- {alpha = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2)} 23.05018289-62.19100371*I <- {alpha = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)-I*2^(1/2)} 23.61819217-64.05943146*I <- {alpha = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)-I*2^(1/2)} 23.05018289-62.19100371*I <- {alpha = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[24.455190091722997, -62.64521797387181] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[γ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[23.887180810974133, -60.776790236964885] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[γ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[24.455190091722997, -62.64521797387181] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[γ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[23.887180810974133, -60.776790236964885] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[γ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.11.E12	${\displaystyle{\displaystyle F\left(\psi,k\right)=2\!F\left(\theta,k\right)}}$	EllipticF(sin(psi), k)= 2*EllipticF(sin(theta), k)	EllipticF[\[Psi], (k)^2]= 2*EllipticF[\[Theta], (k)^2]	Failure	Failure	Fail -.4890744166-1.490381835*I <- {psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 1} -.2379932799-1.041441678*I <- {psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 2} -.1579130142-.8186919671*I <- {psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 3} -.4890744166+4.471145505*I <- {psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[-0.48907441628770865, -1.4903818357543746] <- {Rule[k, 1], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.23799327973838402, -1.0414416780368256] <- {Rule[k, 2], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.15791301410980507, -0.8186919671649275] <- {Rule[k, 3], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.4890744162877084, -4.471145507263124] <- {Rule[k, 1], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.11.E13	${\displaystyle{\displaystyle E\left(\psi,k\right)=2\!E\left(\theta,k\right)-k^{2}{\sin^{2}}\theta\sin\psi}}$	EllipticE(sin(psi), k)= 2*EllipticE(sin(theta), k)- (k)^(2)* (sin(theta))^(2)* sin(psi)	EllipticE[\[Psi], (k)^2]= 2*EllipticE[\[Theta], (k)^2]- (k)^(2)* (Sin[\[Theta]])^(2)* Sin[\[Psi]]	Failure	Failure	Fail 7.220392390+3.861416894*I <- {psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 1} 33.54643343+17.33538519*I <- {psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 2} 78.53436280+38.95995319*I <- {psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 3} 8.004067570-.5190795705*I <- {psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[7.220392409769752, 3.8614168902831674] <- {Rule[k, 1], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[33.54643351078474, 17.335385175946172] <- {Rule[k, 2], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[78.53436298397799, 38.95995315264784] <- {Rule[k, 3], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[8.004067583645242, 0.5190795702179967] <- {Rule[k, 1], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.11.E14	${\displaystyle{\displaystyle\sin\psi=(\sin 2\theta)\Delta(\theta)/(1-k^{2}{\sin^{4}}\theta)}}$	sin(psi)=(sin(2*theta))* Delta*(theta)/(1 - (k)^(2)* (sin(theta))^(4))	Sin[\[Psi]]=(Sin[2*\[Theta]])* \[CapitalDelta]*(\[Theta])/(1 - (k)^(2)* (Sin[\[Theta]])^(4))	Failure	Failure	Fail 3.669776527-.1132600606*I <- {Delta = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 1} 2.522072859+.2078762258*I <- {Delta = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 2} 2.315484084+.2608036708*I <- {Delta = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 3} 1.736514009+1.820002458*I <- {Delta = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[3.669776527136964, -0.11326005969545538] <- {Rule[k, 1], Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[2.5220728602932043, 0.20787622554236157] <- {Rule[k, 2], Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[2.315484085450414, 0.2608036703182187] <- {Rule[k, 3], Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[3.669776527136964, -0.7167829994309771] <- {Rule[k, 1], Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.11#Ex9	${\displaystyle{\displaystyle\cos\psi=(\cos\left(2\theta\right)+k^{2}{\sin^{4}}\theta)/(1-k^{2}{\sin^{4}}\theta)}}$	cos(psi)=(cos(2*theta)+ (k)^(2)* (sin(theta))^(4))/(1 - (k)^(2)* (sin(theta))^(4))	Cos[\[Psi]]=(Cos[2*\[Theta]]+ (k)^(2)* (Sin[\[Theta]])^(4))/(1 - (k)^(2)* (Sin[\[Theta]])^(4))	Failure	Failure	Fail .9973549969-1.831129846*I <- {psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 1} 1.256189052-1.893457097*I <- {psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 2} 1.302739865-1.903587220*I <- {psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2), k = 3} .9973549969-1.991656372*I <- {psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Fail Complex[0.9973549964500539, -1.8311298475339681] <- {Rule[k, 1], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.256189051449076, -1.8934570978250853] <- {Rule[k, 2], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.3027398642662416, -1.9035872217081093] <- {Rule[k, 3], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.9973549964500539, 1.9916563727944525] <- {Rule[k, 1], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.11#Ex10	${\displaystyle{\displaystyle\tan\left(\tfrac{1}{2}\psi\right)=(\tan\theta)\Delta(\theta)}}$	tan((1)/(2)*psi)=(tan(theta))* Delta*(theta)	Tan[Divide[1,2]*\[Psi]]=(Tan[\[Theta]])* \[CapitalDelta]*(\[Theta])	Failure	Failure	Fail 4.896680820+.6655470387*I <- {Delta = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2)} .2596990392+5.302528819*I <- {Delta = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)-I*2^(1/2)} 4.896680820+.6655470387*I <- {Delta = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)-I*2^(1/2)} .2596990392+5.302528819*I <- {Delta = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Skip
19.11#Ex11	${\displaystyle{\displaystyle\sin\theta=(\sin\psi)/\sqrt{(1+\cos\psi)(1+\Delta(\psi))}}}$	sin(theta)=(sin(psi))/sqrt((1 + cos(psi))*(1 + Delta*(psi)))	Sin[\[Theta]]=(Sin[\[Psi]])/Sqrt[(1 + Cos[\[Psi]])*(1 + \[CapitalDelta]*(\[Psi]))]	Failure	Failure	Fail 1.451875812+.3324449118*I <- {Delta = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2)} 1.451875812-.2710780292*I <- {Delta = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)-I*2^(1/2)} -2.851195268-.2710780292*I <- {Delta = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)-I*2^(1/2)} -2.851195268+.3324449118*I <- {Delta = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[1.4518758131252132, 0.33244491134580767] <- {Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.6336006158758192, 0.6708026401371392] <- {Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[2.3509952614630127, 0.9730892801912212] <- {Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[2.6279656435120184, 0.97041258344749] <- {Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.11#Ex12	${\displaystyle{\displaystyle\cos\theta=\sqrt{\frac{(\cos\psi)+\Delta(\psi)}{1+\Delta(\psi)}}}}$	cos(theta)=sqrt(((cos(psi))+ Delta*(psi))/(1 + Delta*(psi)))	Cos[\[Theta]]=Sqrt[Divide[(Cos[\[Psi]])+ \[CapitalDelta]*(\[Psi]),1 + \[CapitalDelta]*(\[Psi])]]	Failure	Failure	Fail -.3760894634-1.941386214*I <- {Delta = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2)} -.3760894634+1.881400004*I <- {Delta = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)-I*2^(1/2)} -.3760894634-1.941386214*I <- {Delta = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)-I*2^(1/2)} -.3760894634+1.881400004*I <- {Delta = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[-0.376089464085568, -1.9413862149418206] <- {Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.6133021347588845, -2.111964043381314] <- {Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-0.8534281043855928, -1.7991622645406038] <- {Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-0.7997455475928336, -1.631807314610897] <- {Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.11#Ex13	${\displaystyle{\displaystyle\tan\theta=\tan\left(\tfrac{1}{2}\psi\right)\sqrt{\frac{1+\cos\psi}{(\cos\psi)+\Delta(\psi)}}}}$	tan(theta)= tan((1)/(2)*psi)*sqrt((1 + cos(psi))/((cos(psi))+ Delta*(psi)))	Tan[\[Theta]]= Tan[Divide[1,2]*\[Psi]]*Sqrt[Divide[1 + Cos[\[Psi]],(Cos[\[Psi]])+ \[CapitalDelta]*(\[Psi])]]	Failure	Failure	Fail -.9331234340+1.202056207*I <- {Delta = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2)} -.9331234340-1.034692067*I <- {Delta = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)-I*2^(1/2)} -1.014866051-1.034692067*I <- {Delta = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)-I*2^(1/2)} -1.014866051+1.202056207*I <- {Delta = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[-0.9331234343650795, 1.202056207743813] <- {Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.40152023420113836, 1.5987346019935262] <- {Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[0.15411790621097937, 1.6917010756307755] <- {Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-0.2176994205238692, 0.46809231316455746] <- {Rule[Δ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.11#Ex14	${\displaystyle{\displaystyle\gamma=(({\csc^{2}}\theta)-\alpha^{2})^{2}(({\csc^{2}}\psi)-\alpha^{2})}}$	gamma =(((csc(theta))^(2))- (alpha)^(2))^(2)*(((csc(psi))^(2))- (alpha)^(2))	\[Gamma]=(((Csc[\[Theta]])^(2))- (\[Alpha])^(2))^(2)*(((Csc[\[Psi]])^(2))- (\[Alpha])^(2))	Failure	Failure	Fail 10.63251777-66.33335806*I <- {alpha = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)+I*2^(1/2)} 10.24988272-62.55869068*I <- {alpha = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = 2^(1/2)-I*2^(1/2)} 10.63251777-66.33335806*I <- {alpha = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)-I*2^(1/2)} 10.24988272-62.55869068*I <- {alpha = 2^(1/2)+I*2^(1/2), psi = 2^(1/2)+I*2^(1/2), theta = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[11.46951567083565, -64.91914457079643] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[γ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[11.276814452857538, -63.004241447264036] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[γ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[11.46951567083565, -64.91914457079643] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[γ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[11.276814452857538, -63.004241447264036] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[γ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ψ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.12.E1	${\displaystyle{\displaystyle K\left(k\right)=\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}{k^{\prime}}^{2m}\left(\ln\left(\frac{1}{k^{\prime}}\right)+d(m)\right)}}$	EllipticK(k)= sum((pochhammer((1)/(2), m)*pochhammer((1)/(2), m))/(factorial(m)*factorial(m))*(sqrt(1 - (k)^(2)))^(2*m)*(ln((1)/(sqrt(1 - (k)^(2))))+ d*(m)), m = 0..infinity)	EllipticK[(k)^2]= Sum[Divide[Pochhammer[Divide[1,2], m]*Pochhammer[Divide[1,2], m],(m)!*(m)!]*(Sqrt[1 - (k)^(2)])^(2*m)*(Log[Divide[1,Sqrt[1 - (k)^(2)]]]+ d*(m)), {m, 0, Infinity}]	Failure	Failure	Skip	Successful
19.12#Ex1	${\displaystyle{\displaystyle d(m)=\psi\left(1+m\right)-\psi\left(\tfrac{1}{2}+m\right)}}$	d*(m)= Psi(1 + m)- Psi((1)/(2)+ m)	d*(m)= PolyGamma[1 + m]- PolyGamma[Divide[1,2]+ m]	Failure	Failure	Fail 1.027919201+1.414213562*I <- {d = 2^(1/2)+I*2^(1/2), m = 1} 2.608799430+2.828427124*I <- {d = 2^(1/2)+I*2^(1/2), m = 2} 4.089679659+4.242640686*I <- {d = 2^(1/2)+I*2^(1/2), m = 3} 1.027919201-1.414213562*I <- {d = 2^(1/2)-I*2^(1/2), m = 1} ... skip entries to safe data	Fail Complex[1.0279192012532046, 1.4142135623730951] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[m, 1]} Complex[2.6087994302929665, 2.8284271247461903] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[m, 2]} Complex[4.089679659332729, 4.242640687119286] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[m, 3]} Complex[1.0279192012532046, -1.4142135623730951] <- {Rule[d, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[m, 1]} ... skip entries to safe data
19.14.E1	${\displaystyle{\displaystyle\int_{1}^{x}\frac{\mathrm{d}t}{\sqrt{t^{3}-1}}=3^{-1/4}F\left(\phi,k\right)}}$	int((1)/(sqrt((t)^(3)- 1)), t = 1..x)= (3)^(- 1/ 4)* EllipticF(sin(phi), k)	Integrate[Divide[1,Sqrt[(t)^(3)- 1]], {t, 1, x}]= (3)^(- 1/ 4)* EllipticF[\[Phi], (k)^2]	Failure	Failure	Skip	Error
19.14.E2	${\displaystyle{\displaystyle\int_{x}^{1}\frac{\mathrm{d}t}{\sqrt{1-t^{3}}}=3^{-1/4}F\left(\phi,k\right)}}$	int((1)/(sqrt(1 - (t)^(3))), t = x..1)= (3)^(- 1/ 4)* EllipticF(sin(phi), k)	Integrate[Divide[1,Sqrt[1 - (t)^(3)]], {t, x, 1}]= (3)^(- 1/ 4)* EllipticF[\[Phi], (k)^2]	Failure	Failure	Skip	Successful
19.14.E3	${\displaystyle{\displaystyle\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{1+t^{4}}}=\frac{\operatorname{sign}\left(x\right)}{2}F\left(\phi,k\right)}}$	int((1)/(sqrt(1 + (t)^(4))), t = 0..x)=(signum(x))/(2)*EllipticF(sin(phi), k)	Integrate[Divide[1,Sqrt[1 + (t)^(4)]], {t, 0, x}]=Divide[Sign[x],2]*EllipticF[\[Phi], (k)^2]	Failure	Failure	Skip	Successful
19.14.E5	${\displaystyle{\displaystyle{\sin^{2}}\phi=\frac{\gamma-\alpha}{U^{2}+\gamma}}}$	(sin(phi))^(2)=(gamma - alpha)/((U)^(2)+ gamma)	(Sin[\[Phi]])^(2)=Divide[\[Gamma]- \[Alpha],(U)^(2)+ \[Gamma]]	Failure	Failure	Fail 4.913966071+1.143498437*I <- {U = 2^(1/2)+I*2^(1/2), alpha = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2)} 4.913966071-1.453503677*I <- {U = 2^(1/2)+I*2^(1/2), alpha = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)-I*2^(1/2)} 4.913966071+1.143498437*I <- {U = 2^(1/2)+I*2^(1/2), alpha = 2^(1/2)+I*2^(1/2), phi = -2^(1/2)-I*2^(1/2)} 4.913966071-1.453503677*I <- {U = 2^(1/2)+I*2^(1/2), alpha = 2^(1/2)+I*2^(1/2), phi = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[4.538045200949385, 1.2985010548545433] <- {Rule[U, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[γ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[4.538045200949385, -1.2985010548545433] <- {Rule[U, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[γ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[4.538045200949385, 1.2985010548545433] <- {Rule[U, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[γ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[4.538045200949385, -1.2985010548545433] <- {Rule[U, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[γ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.14.E7	${\displaystyle{\displaystyle{\sin^{2}}\phi=\frac{(\gamma-\alpha)x^{2}}{a_{1}a_{2}+\gamma x^{2}}}}$	(sin(phi))^(2)=((gamma - alpha)* (x)^(2))/(a[1]*a[2]+ gamma*(x)^(2))	(Sin[\[Phi]])^(2)=Divide[(\[Gamma]- \[Alpha])* (x)^(2),Subscript[a, 1]*Subscript[a, 2]+ \[Gamma]*(x)^(2)]	Failure	Failure	Fail 4.913966071+1.143498437*I <- {alpha = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), a[1] = 2^(1/2)+I*2^(1/2), a[2] = 2^(1/2)+I*2^(1/2), x = 1} 5.961217471+1.282980492*I <- {alpha = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), a[1] = 2^(1/2)+I*2^(1/2), a[2] = 2^(1/2)+I*2^(1/2), x = 2} 6.632730047+2.135696835*I <- {alpha = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), a[1] = 2^(1/2)+I*2^(1/2), a[2] = 2^(1/2)+I*2^(1/2), x = 3} 4.720906995+1.607469143*I <- {alpha = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), a[1] = 2^(1/2)+I*2^(1/2), a[2] = 2^(1/2)-I*2^(1/2), x = 1} ... skip entries to safe data	Skip
19.14.E8	${\displaystyle{\displaystyle{\sin^{2}}\phi=\frac{\gamma-\alpha}{b_{1}b_{2}y^{2}+\gamma}}}$	(sin(phi))^(2)=(gamma - alpha)/(b[1]*b[2]*(y)^(2)+ gamma)	(Sin[\[Phi]])^(2)=Divide[\[Gamma]- \[Alpha],Subscript[b, 1]*Subscript[b, 2]*(y)^(2)+ \[Gamma]]	Failure	Failure	Fail 4.913966071+1.143498437*I <- {alpha = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), b[1] = 2^(1/2)+I*2^(1/2), b[2] = 2^(1/2)+I*2^(1/2), y = 1} 4.628203424+1.249441235*I <- {alpha = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), b[1] = 2^(1/2)+I*2^(1/2), b[2] = 2^(1/2)+I*2^(1/2), y = 2} 4.577691497+1.275886795*I <- {alpha = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), b[1] = 2^(1/2)+I*2^(1/2), b[2] = 2^(1/2)+I*2^(1/2), y = 3} 4.720906995+1.607469143*I <- {alpha = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), b[1] = 2^(1/2)+I*2^(1/2), b[2] = 2^(1/2)-I*2^(1/2), y = 1} ... skip entries to safe data	Skip
19.14.E9	${\displaystyle{\displaystyle{\sin^{2}}\phi=\frac{(\gamma-\alpha)(x^{2}-y^{2})}{\gamma(x^{2}-y^{2})-a_{1}(a_{2}+b_{2}x^{2})}}}$	(sin(phi))^(2)=((gamma - alpha)*((x)^(2)- (y)^(2)))/(gamma*((x)^(2)- (y)^(2))- a[1]*(a[2]+ b[2]*(x)^(2)))	(Sin[\[Phi]])^(2)=Divide[(\[Gamma]- \[Alpha])*((x)^(2)- (y)^(2)),\[Gamma]*((x)^(2)- (y)^(2))- Subscript[a, 1]*(Subscript[a, 2]+ Subscript[b, 2]*(x)^(2))]	Failure	Failure	Skip	Skip
19.14.E10	${\displaystyle{\displaystyle{\sin^{2}}\phi=\frac{(\gamma-\alpha)(y^{2}-x^{2})}{\gamma(y^{2}-x^{2})-a_{1}(a_{2}+b_{2}y^{2})}}}$	(sin(phi))^(2)=((gamma - alpha)*((y)^(2)- (x)^(2)))/(gamma*((y)^(2)- (x)^(2))- a[1]*(a[2]+ b[2]*(y)^(2)))	(Sin[\[Phi]])^(2)=Divide[(\[Gamma]- \[Alpha])*((y)^(2)- (x)^(2)),\[Gamma]*((y)^(2)- (x)^(2))- Subscript[a, 1]*(Subscript[a, 2]+ Subscript[b, 2]*(y)^(2))]	Failure	Failure	Skip	Skip
19.16.E1	${\displaystyle{\displaystyle R_{F}\left(x,y,z\right)=\frac{1}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{s(t)}}}$	0.5*int(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+z)), t = 0..infinity)=(1)/(2)*int((1)/(s*(t)), t = 0..infinity)	Error	Error	Error	-	-
19.16#Ex3	${\displaystyle{\displaystyle c={\csc^{2}}\phi}}$	c = (csc(phi))^(2)	c = (Csc[\[Phi]])^(2)	Failure	Failure	Fail 1.210530730+1.472494686*I <- {c = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2)} 1.210530730+1.355932438*I <- {c = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)-I*2^(1/2)} 1.210530730+1.472494686*I <- {c = 2^(1/2)+I*2^(1/2), phi = -2^(1/2)-I*2^(1/2)} 1.210530730+1.355932438*I <- {c = 2^(1/2)+I*2^(1/2), phi = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[1.210530730056432, 1.472494685832072] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.210530730056432, 1.3559324389141183] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[1.210530730056432, 1.472494685832072] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[1.210530730056432, 1.3559324389141183] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.18.E14	${\displaystyle{\displaystyle\frac{{\partial}^{2}w}{{\partial x}^{2}}=\frac{{\partial}^{2}w}{{\partial y}^{2}}+\frac{1}{y}\frac{\partial w}{\partial y}}}$	diff(w, [x$(2)])= diff(w, [y$(2)])+(1)/(y)*diff(w, y)	D[w, {x, 2}]= D[w, {y, 2}]+Divide[1,y]*D[w, y]	Successful	Successful	-	-
19.18.E15	${\displaystyle{\displaystyle\frac{{\partial}^{2}W}{{\partial t}^{2}}=\frac{{\partial}^{2}W}{{\partial x}^{2}}+\frac{{\partial}^{2}W}{{\partial y}^{2}}}}$	diff(W, [t$(2)])= diff(W, [x$(2)])+ diff(W, [y$(2)])	D[W, {t, 2}]= D[W, {x, 2}]+ D[W, {y, 2}]	Successful	Successful	-	-
19.18.E16	${\displaystyle{\displaystyle\frac{{\partial}^{2}u}{{\partial x}^{2}}+\frac{{\partial}^{2}u}{{\partial y}^{2}}+\frac{1}{y}\frac{\partial u}{\partial y}=0}}$	diff(u, [x$(2)])+ diff(u, [y$(2)])+(1)/(y)*diff(u, y)= 0	D[u, {x, 2}]+ D[u, {y, 2}]+Divide[1,y]*D[u, y]= 0	Successful	Successful	-	-
19.18.E17	${\displaystyle{\displaystyle\frac{{\partial}^{2}U}{{\partial x}^{2}}+\frac{{\partial}^{2}U}{{\partial y}^{2}}+\frac{{\partial}^{2}U}{{\partial z}^{2}}=0}}$	diff(U, [x$(2)])+ diff(U, [y$(2)])+ diff(U, [z$(2)])= 0	D[U, {x, 2}]+ D[U, {y, 2}]+ D[U, {z, 2}]= 0	Successful	Successful	-	-
19.20#Ex1	${\displaystyle{\displaystyle R_{F}\left(x,x,x\right)=x^{-1/2}}}$	0.5*int(1/(sqrt(t+x)*sqrt(t+x)*sqrt(t+x)), t = 0..infinity)= (x)^(- 1/ 2)	Error	Error	Error	-	-
19.20#Ex2	${\displaystyle{\displaystyle R_{F}\left(\lambda x,\lambda y,\lambda z\right)=\lambda^{-1/2}R_{F}\left(x,y,z\right)}}$	0.5*int(1/(sqrt(t+lambda*x)*sqrt(t+lambda*y)*sqrt(t+lambda*z)), t = 0..infinity)= (lambda)^(- 1/ 2)* 0.5*int(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+z)), t = 0..infinity)	Error	Error	Error	-	-
19.20#Ex4	${\displaystyle{\displaystyle R_{F}\left(0,y,y\right)=\tfrac{1}{2}\pi y^{-1/2}}}$	0.5*int(1/(sqrt(t+0)*sqrt(t+y)*sqrt(t+y)), t = 0..infinity)=(1)/(2)*Pi*(y)^(- 1/ 2)	Error	Error	Error	-	-
19.20#Ex5	${\displaystyle{\displaystyle R_{F}\left(0,0,z\right)=\infty}}$	0.5*int(1/(sqrt(t+0)*sqrt(t+0)*sqrt(t+z)), t = 0..infinity)= infinity	Error	Error	Error	-	-
19.20.E2	${\displaystyle{\displaystyle\int_{0}^{1}\frac{\mathrm{d}t}{\sqrt{1-t^{4}}}=R_{F}\left(0,1,2\right)}}$	int((1)/(sqrt(1 - (t)^(4))), t = 0..1)= 0.5*int(1/(sqrt(t+0)*sqrt(t+1)*sqrt(t+2)), t = 0..infinity)	Error	Error	Error	-	-
19.20.E2	${\displaystyle{\displaystyle R_{F}\left(0,1,2\right)=\frac{\left(\Gamma\left(\frac{1}{4}\right)\right)^{2}}{4(2\pi)^{1/2}}}}$	0.5*int(1/(sqrt(t+0)*sqrt(t+1)*sqrt(t+2)), t = 0..infinity)=((GAMMA((1)/(4)))^(2))/(4*(2*Pi)^(1/ 2))	Error	Error	Error	-	-
19.21.E4	${\displaystyle{\displaystyle R_{F}\left(0,z-1,z\right)=R_{F}\left(0,1-z,1\right)-\mathrm{i}\!R_{F}\left(0,z,1\right)}}$	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)	Error	Error	Error	-	-
19.21.E4	${\displaystyle{\displaystyle R_{F}\left(0,z-1,z\right)=R_{F}\left(0,1-z,1\right)+\mathrm{i}\!R_{F}\left(0,z,1\right)}}$	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)	Error	Error	Error	-	-
19.22.E1	${\displaystyle{\displaystyle R_{F}\left(0,x^{2},y^{2}\right)=R_{F}\left(0,xy,a^{2}\right)}}$	0.5*int(1/(sqrt(t+0)*sqrt(t+(x)^(2))*sqrt(t+(y)^(2))), t = 0..infinity)= 0.5*int(1/(sqrt(t+0)*sqrt(t+x*y)*sqrt(t+(a)^(2))), t = 0..infinity)	Error	Error	Error	-	-
19.22.E8	${\displaystyle{\displaystyle\frac{2}{\pi}R_{F}\left(0,a_{0}^{2},g_{0}^{2}\right)=\frac{1}{M\left(a_{0},g_{0}\right)}}}$	0.5*int(1/(sqrt(t+0)*sqrt(t+a(a[0])^(2))*sqrt(t+g(g[0])^(2))), t = 0..infinity)=(1)/(GaussAGM(a[0], g[0]))	Error	Error	Error	-	-
19.22.E9	${\displaystyle{\displaystyle\frac{1}{M\left(a_{0},g_{0}\right)}\left(a_{0}^{2}-\sum_{n=0}^{\infty}2^{n-1}c_{n}^{2}\right)=\frac{1}{M\left(a_{0},g_{0}\right)}\left(a_{1}^{2}-\sum_{n=2}^{\infty}2^{n-1}c_{n}^{2}\right)}}$	(a(a[0])^(2)- sum((2)^(n - 1)* c(c[n])^(2), n = 0..infinity))(a(a[1])^(2)- sum((2)^(n - 1)* c(c[n])^(2), n = 2..infinity))	Error	Failure	Error	Skip	-
19.22.E18	${\displaystyle{\displaystyle R_{F}\left(x^{2},y^{2},z^{2}\right)=R_{F}\left(a^{2},z_{-}^{2},z_{+}^{2}\right)}}$	0.5*int(1/(sqrt(t+(a)^(2))*sqrt(t+z(z[-])^(2))*sqrt(t+z(z[+])^(2))), t = 0..infinity)	Error	Error	Error	-	-
19.23.E1	${\displaystyle{\displaystyle R_{F}\left(0,y,z\right)=\int_{0}^{\pi/2}(y{\cos^{2}}\theta+z{\sin^{2}}\theta)^{-1/2}\mathrm{d}\theta}}$	0.5*int(1/(sqrt(t+0)*sqrt(t+y)*sqrt(t+z)), t = 0..infinity)= int((y*(cos(theta))^(2)+ z*(sin(theta))^(2))^(- 1/ 2), theta = 0..Pi/ 2)	Error	Error	Error	-	-
19.23.E6	${\displaystyle{\displaystyle 4\pi R_{F}\left(x,y,z\right)=\int_{0}^{2\pi}\!\!\!\!\int_{0}^{\pi}\frac{\sin\theta\mathrm{d}\theta\mathrm{d}\phi}{(x{\sin^{2}}\theta{\cos^{2}}\phi+y{\sin^{2}}\theta{\sin^{2}}\phi+z{\cos^{2}}\theta)^{1/2}}}}$	4*Pi*0.5*int(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+z)), t = 0..infinity)= int(int((sin(theta))/((x*(sin(theta))^(2)* (cos(phi))^(2)+ y*(sin(theta))^(2)* (sin(phi))^(2)+ z*(cos(theta))^(2))^(1/ 2)), theta = 0..Pi), phi = 0..2*Pi)	Error	Error	Error	-	-
19.24.E5	${\displaystyle{\displaystyle\frac{1}{a_{n}}<=\frac{2}{\pi}R_{F}\left(0,a_{0}^{2},g_{0}^{2}\right)}}$	(1)/(a[n])< 0.5*int(1/(sqrt(t+0)*sqrt(t+a(a[0])^(2))*sqrt(t+g(g[0])^(2))), t = 0..infinity)	Error	Error	Error	-	-
19.24.E5	${\displaystyle{\displaystyle\frac{2}{\pi}R_{F}\left(0,a_{0}^{2},g_{0}^{2}\right)<=\frac{1}{g_{n}}}}$	0.5*int(1/(sqrt(t+0)*sqrt(t+a(a[0])^(2))*sqrt(t+g(g[0])^(2))), t = 0..infinity)< =(1)/(g[n])	Error	Error	Error	-	-
19.24.E10	${\displaystyle{\displaystyle\frac{3}{\sqrt{x}+\sqrt{y}+\sqrt{z}}<=R_{F}\left(x,y,z\right)}}$	(3)/(sqrt(x)+sqrt(y)+sqrt(z))< = 0.5*int(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+z)), t = 0..infinity)	Error	Error	Error	-	-
19.24.E10	${\displaystyle{\displaystyle R_{F}\left(x,y,z\right)<=\frac{1}{(xyz)^{1/6}}}}$	0.5*int(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+z)), t = 0..infinity)< =(1)/((x*y*z)^(1/ 6))	Error	Error	Error	-	-
19.25#Ex1	${\displaystyle{\displaystyle K\left(k\right)=R_{F}\left(0,{k^{\prime}}^{2},1\right)}}$	EllipticK(k)= 0.5*int(1/(sqrt(t+0)*sqrt(t+1 - (k)^(2))*sqrt(t+1)), t = 0..infinity)	Error	Error	Error	-	-
19.25#Ex4	${\displaystyle{\displaystyle K\left(k\right)-E\left(k\right)=k^{2}D\left(k\right)}}$	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	-	Fail Complex[0.3274322182097533, -1.81658404135269] <- {Rule[k, 2]} Complex[0.24314000376678657, -2.9699762712345015] <- {Rule[k, 3]}
19.25.E5	${\displaystyle{\displaystyle F\left(\phi,k\right)=R_{F}\left(c-1,c-k^{2},c\right)}}$	EllipticF(sin(phi), k)= 0.5*int(1/(sqrt(t+c - 1)*sqrt(t+c - (k)^(2))*sqrt(t+c)), t = 0..infinity)	Error	Error	Error	-	-
19.25.E17	${\displaystyle{\displaystyle F\left(\phi,k\right)=R_{F}\left(x,y,z\right)}}$	EllipticF(sin(phi), k)= 0.5*int(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+z)), t = 0..infinity)	Error	Error	Error	-	-
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)* 0.5*int(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+z)), t = 0..infinity)= EllipticF(sin(phi), k)	Error	Error	Error	-	-
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 = 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)	Error	Error	Error	-	-
19.26.E1	${\displaystyle{\displaystyle R_{F}\left(x+\lambda,y+\lambda,z+\lambda\right)+R_{F}\left(x+\mu,y+\mu,z+\mu\right)=R_{F}\left(x,y,z\right)}}$	0.5*int(1/(sqrt(t+x + lambda)*sqrt(t+y + lambda)*sqrt(t+z + lambda)), t = 0..infinity)+ 0.5*int(1/(sqrt(t+x + mu)*sqrt(t+y + mu)*sqrt(t+z + mu)), t = 0..infinity)= 0.5*int(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+z)), t = 0..infinity)	Error	Error	Error	-	-
19.26.E16	${\displaystyle{\displaystyle R_{F}\left(\lambda,y+\lambda,z+\lambda\right)={R_{F}\left(0,y,z\right)-R_{F}\left(\mu,y+\mu,z+\mu\right),}}}$	0.5*int(1/(sqrt(t+lambda)*sqrt(t+y + lambda)*sqrt(t+z + lambda)), t = 0..infinity)=0.5*int(1/(sqrt(t+0)*sqrt(t+y)*sqrt(t+z)), t = 0..infinity)- 0.5*int(1/(sqrt(t+mu)*sqrt(t+y + mu)*sqrt(t+z + mu)), t = 0..infinity),	Error	Error	Error	-	-
19.26.E18	${\displaystyle{\displaystyle R_{F}\left(x,y,z\right)=2\!R_{F}\left(x+\lambda,y+\lambda,z+\lambda\right)}}$	0.5*int(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+z)), t = 0..infinity)= 2*0.5*int(1/(sqrt(t+x + lambda)*sqrt(t+y + lambda)*sqrt(t+z + lambda)), t = 0..infinity)	Error	Error	Error	-	-
19.26.E18	${\displaystyle{\displaystyle 2\!R_{F}\left(x+\lambda,y+\lambda,z+\lambda\right)=R_{F}\left(\frac{x+\lambda}{4},\frac{y+\lambda}{4},\frac{z+\lambda}{4}\right)}}$	2*0.5*int(1/(sqrt(t+x + lambda)*sqrt(t+y + lambda)*sqrt(t+z + lambda)), t = 0..infinity)= 0.5*int(1/(sqrt(t+(x + lambda)/(4))*sqrt(t+(y + lambda)/(4))*sqrt(t+(z + lambda)/(4))), t = 0..infinity)	Error	Error	Error	-	-
19.28.E1	${\displaystyle{\displaystyle\int_{0}^{1}t^{\sigma-1}R_{F}\left(0,t,1\right)\mathrm{d}t=\tfrac{1}{2}\left(\mathrm{B}\left(\sigma,\tfrac{1}{2}\right)\right)^{2}}}$	int((t)^(sigma - 1)* 0.5*int(1/(sqrt(t+0)*sqrt(t+t)*sqrt(t+1)), t = 0..infinity), t = 0..1)=(1)/(2)*(Beta(sigma, (1)/(2)))^(2)	Error	Error	Error	-	-
19.28.E9	${\displaystyle{\displaystyle\int_{0}^{\pi/2}R_{F}\left({\sin^{2}}\theta{\cos^{2}}\left(x+y\right),{\sin^{2}}\theta{\cos^{2}}\left(x-y\right),1\right)\mathrm{d}\theta=R_{F}\left(0,{\cos^{2}}x,1\right)R_{F}\left(0,{\cos^{2}}y,1\right)}}$	int(0.5*int(1/(sqrt(t+(sin(theta))^(2)* (cos(x + y))^(2))*sqrt(t+(sin(theta))^(2)* (cos(x - y))^(2))*sqrt(t+1)), t = 0..infinity), theta = 0..Pi/ 2)= 0.5*int(1/(sqrt(t+0)*sqrt(t+(cos(x))^(2))*sqrt(t+1)), t = 0..infinity)*0.5*int(1/(sqrt(t+0)*sqrt(t+(cos(y))^(2))*sqrt(t+1)), t = 0..infinity)	Error	Error	Error	-	-
19.29.E4	${\displaystyle{\displaystyle\int_{y}^{x}\frac{\mathrm{d}t}{s(t)}=2\!R_{F}\left(U_{12}^{2},U_{13}^{2},U_{23}^{2}\right)}}$	0.5*int(1/(sqrt(t+U(U[12])^(2))*sqrt(t+U(U[13])^(2))*sqrt(t+U(U[23])^(2))), t = 0..infinity)	Error	Error	Error	-	-
19.29.E11	${\displaystyle{\displaystyle I(\mathbf{m})=\int_{y}^{x}\prod_{\alpha=1}^{h}(a_{\alpha}+b_{\alpha}t)^{-1/2}\prod_{j=1}^{n}(a_{j}+b_{j}t)^{m_{j}}\mathrm{d}t}}$	I*(m)= int(product((a[alpha]+ b[alpha]*t)^(- 1/ 2)* product((a[j]+ b[j]*t)^(m[j]), j = 1..n), alpha = 1..h), t = y..x)	I*(m)= Integrate[Product[(Subscript[a, \[Alpha]]+ Subscript[b, \[Alpha]]*t)^(- 1/ 2)* Product[(Subscript[a, j]+ Subscript[b, j]*t)^(Subscript[m, j]), {j, 1, n}], {\[Alpha], 1, h}], {t, y, x}]	Error	Failure	-	Error
19.29.E18	${\displaystyle{\displaystyle b_{j}^{q}I(q\mathbf{e}_{l})=\sum_{r=0}^{q}\genfrac{(}{)}{0.0pt}{}{q}{r}b_{l}^{r}d_{lj}^{q-r}I(r\mathbf{e}_{j})}}$	(b[j])^(q)*I*sum(binomial(q,r)*b(b[l])^(r)*d(d[l*j])^(q - r)*I*(r*e[j]), r = 0..q)	(Subscript[b, j])^(q)*I*Sum[Binomial[q,r]*b(Subscript[b, l])^(r)*d(Subscript[d, l*j])^(q - r)*I*(r*Subscript[e, j]), {r, 0, q}]	Failure	Failure	Skip	Error
19.29.E19	${\displaystyle{\displaystyle\int_{y}^{x}\frac{\mathrm{d}t}{\sqrt{Q_{1}(t)Q_{2}(t)}}=R_{F}\left(U^{2}+a_{1}b_{2},U^{2}+a_{2}b_{1},U^{2}\right)}}$	int((1)/(sqrt(Q[1]*(t)* Q[2]*(t))), t = y..x)= 0.5*int(1/(sqrt(t+(U)^(2)+ a[1]*b[2])*sqrt(t+(U)^(2)+ a[2]*b[1])*sqrt(t+(U)^(2))), t = 0..infinity)	Error	Error	Error	-	-
19.29.E23	${\displaystyle{\displaystyle\int_{y}^{\infty}\frac{\mathrm{d}t}{\sqrt{(t^{2}+a^{2})(t^{2}-b^{2})}}=R_{F}\left(y^{2}+a^{2},y^{2}-b^{2},y^{2}\right)}}$	int((1)/(sqrt(((t)^(2)+ (a)^(2))*((t)^(2)- (b)^(2)))), t = y..infinity)= 0.5*int(1/(sqrt(t+(y)^(2)+ (a)^(2))*sqrt(t+(y)^(2)- (b)^(2))*sqrt(t+(y)^(2))), t = 0..infinity)	Error	Error	Error	-	-
19.29.E24	${\displaystyle{\displaystyle\int_{y}^{x}\frac{\mathrm{d}t}{\sqrt{Q_{1}(t)Q_{2}(t)}}=4\!R_{F}\left(U,U+D_{12}+V,U+D_{12}-V\right)}}$	int((1)/(sqrt(Q[1]*(t)* Q[2]*(t))), t = y..x)= 4*0.5*int(1/(sqrt(t+U)*sqrt(t+U + D[12]+ V)*sqrt(t+U + D[12]- V)), t = 0..infinity)	Error	Error	Error	-	-
19.29.E28	${\displaystyle{\displaystyle\int_{y}^{x}\frac{\mathrm{d}t}{\sqrt{t^{3}-a^{3}}}=4\!R_{F}\left(U,U-3a+2\sqrt{3}a,U-3a-2\sqrt{3}a\right)}}$	int((1)/(sqrt((t)^(3)- (a)^(3))), t = y..x)= 4*0.5*int(1/(sqrt(t+U)*sqrt(t+U - 3*a + 2*sqrt(3)*a)*sqrt(t+U - 3*a - 2*sqrt(3)*a)), t = 0..infinity)	Error	Error	Error	-	-
19.29.E30	${\displaystyle{\displaystyle\int_{y}^{x}\frac{\mathrm{d}t}{\sqrt{Q(t^{2})}}=2\!R_{F}\left(U,U-g+2\sqrt{fh},U-g-2\sqrt{fh}\right)}}$	int((1)/(sqrt(Q*((t)^(2)))), t = y..x)= 2*0.5*int(1/(sqrt(t+U)*sqrt(t+U - g + 2*sqrt(f*h))*sqrt(t+U - g - 2*sqrt(f*h))), t = 0..infinity)	Error	Error	Error	-	-
19.29.E32	${\displaystyle{\displaystyle\int_{y}^{x}\frac{\mathrm{d}t}{\sqrt{t^{4}+a^{4}}}=2\!R_{F}\left(U,U+2a^{2},U-2a^{2}\right)}}$	int((1)/(sqrt((t)^(4)+ (a)^(4))), t = y..x)= 2*0.5*int(1/(sqrt(t+U)*sqrt(t+U + 2*(a)^(2))*sqrt(t+U - 2*(a)^(2))), t = 0..infinity)	Error	Error	Error	-	-
19.30#Ex1	${\displaystyle{\displaystyle x=a\sin\phi}}$	x = a*sin(phi)	x = a*Sin[\[Phi]]	Failure	Failure	Fail -1.615975576-3.469485904*I <- {a = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), x = 1} -.615975576-3.469485904*I <- {a = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), x = 2} .384024424-3.469485904*I <- {a = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), x = 3} -2.469485904-2.615975576*I <- {a = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)-I*2^(1/2), x = 1} ... skip entries to safe data	Skip
19.30#Ex2	${\displaystyle{\displaystyle y=b\cos\phi}}$	y = b*cos(phi)	y = b*Cos[\[Phi]]	Failure	Failure	Fail -2.183489624+2.222746490*I <- {b = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), y = 1} -1.183489624+2.222746490*I <- {b = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), y = 2} -.183489624+2.222746490*I <- {b = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), y = 3} 3.222746490-3.183489624*I <- {b = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)-I*2^(1/2), y = 1} ... skip entries to safe data	Fail Complex[-2.183489625260803, 2.2227464935806323] <- {Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[y, 1], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-1.183489625260803, 2.2227464935806323] <- {Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[y, 2], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-0.18348962526080292, 2.2227464935806323] <- {Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[y, 3], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[3.2227464935806323, -3.183489625260803] <- {Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[y, 1], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.30.E2	${\displaystyle{\displaystyle s=a\int_{0}^{\phi}\sqrt{1-k^{2}{\sin^{2}}\theta}\mathrm{d}\theta}}$	s = a*int(sqrt(1 - (k)^(2)* (sin(theta))^(2)), theta = 0..phi)	s = a*Integrate[Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)], {\[Theta], 0, \[Phi]}]	Failure	Failure	Skip	Error
19.30.E3	${\displaystyle{\displaystyle s/a=E\left(\phi,k\right)}}$	s/ a = EllipticE(sin(phi), k)	s/ a = EllipticE[\[Phi], (k)^2]	Failure	Failure	Fail -1.151535540-.3017614705*I <- {a = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), s = 2^(1/2)+I*2^(1/2), k = 1} -2.941278291+.6826717339*I <- {a = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), s = 2^(1/2)+I*2^(1/2), k = 2} -4.812988572+1.491347909*I <- {a = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), s = 2^(1/2)+I*2^(1/2), k = 3} -2.151535540-1.301761470*I <- {a = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), s = 2^(1/2)-I*2^(1/2), k = 1} ... skip entries to safe data	Skip
19.30#Ex4	${\displaystyle{\displaystyle c={\csc^{2}}\phi}}$	c = (csc(phi))^(2)	c = (Csc[\[Phi]])^(2)	Failure	Failure	Fail 1.210530730+1.472494686*I <- {c = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2)} 1.210530730+1.355932438*I <- {c = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)-I*2^(1/2)} 1.210530730+1.472494686*I <- {c = 2^(1/2)+I*2^(1/2), phi = -2^(1/2)-I*2^(1/2)} 1.210530730+1.355932438*I <- {c = 2^(1/2)+I*2^(1/2), phi = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[1.210530730056432, 1.472494685832072] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.210530730056432, 1.3559324389141183] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[1.210530730056432, 1.472494685832072] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[1.210530730056432, 1.3559324389141183] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.30.E5	${\displaystyle{\displaystyle L(a,b)=4aE\left(k\right)}}$	L*(a , b)= 4*a*EllipticE(k)	L*(a , b)= 4*a*EllipticE[(k)^2]	Failure	Failure	Error	Error
19.30.E6	${\displaystyle{\displaystyle\frac{\partial s}{\partial(1/k)}=\sqrt{a^{2}-b^{2}}F\left(\phi,k\right)}}$	diff(s, 1/ k)=sqrt((a)^(2)- (b)^(2))*EllipticF(sin(phi), k)	D[s, 1/ k]=Sqrt[(a)^(2)- (b)^(2)]*EllipticF[\[Phi], (k)^2]	Error	Failure	-	Successful
19.30.E6	${\displaystyle{\displaystyle\sqrt{a^{2}-b^{2}}F\left(\phi,k\right)=\sqrt{a^{2}-b^{2}}R_{F}\left(c-1,c-k^{2},c\right)}}$	sqrt((a)^(2)- (b)^(2))*EllipticF(sin(phi), k)=sqrt((a)^(2)- (b)^(2))*0.5*int(1/(sqrt(t+c - 1)*sqrt(t+c - (k)^(2))*sqrt(t+c)), t = 0..infinity)	Error	Error	Error	-	-
19.30.E8	${\displaystyle{\displaystyle s=\frac{1}{2}\int_{0}^{y^{2}/b^{2}}\sqrt{\frac{(a^{2}+b^{2})t+b^{2}}{t(t+1)}}\mathrm{d}t}}$	s =(1)/(2)*int(sqrt((((a)^(2)+ (b)^(2))* t + (b)^(2))/(t*(t + 1))), t = 0..(y)^(2)/ (b)^(2))	s =Divide[1,2]*Integrate[Sqrt[Divide[((a)^(2)+ (b)^(2))* t + (b)^(2),t*(t + 1)]], {t, 0, (y)^(2)/ (b)^(2)}]	Failure	Failure	Skip	Error
19.30.E9	${\displaystyle{\displaystyle s=\tfrac{1}{2}I(\mathbf{e}_{1})}}$	s =(1)/(2)*I*(e[1])	s =Divide[1,2]*I*(Subscript[e, 1])	Failure	Failure	Skip	Successful
19.30.E10	${\displaystyle{\displaystyle r^{2}=2a^{2}\cos\left(2\theta\right)}}$	(r)^(2)= 2*(a)^(2)* cos(2*theta)	(r)^(2)= 2*(a)^(2)* Cos[2*\[Theta]]	Failure	Failure	Skip	Successful
19.30.E11	${\displaystyle{\displaystyle s=2a^{2}\int_{0}^{r}\frac{\mathrm{d}t}{\sqrt{4a^{4}-t^{4}}}}}$	s = 2*(a)^(2)* int((1)/(sqrt(4*(a)^(4)- (t)^(4))), t = 0..r)	s = 2*(a)^(2)* Integrate[Divide[1,Sqrt[4*(a)^(4)- (t)^(4)]], {t, 0, r}]	Error	Failure	-	Error
19.30.E11	${\displaystyle{\displaystyle 2a^{2}\int_{0}^{r}\frac{\mathrm{d}t}{\sqrt{4a^{4}-t^{4}}}=\sqrt{2a^{2}}R_{F}\left(q-1,q,q+1\right)}}$	2*(a)^(2)* int((1)/(sqrt(4*(a)^(4)- (t)^(4))), t = 0..r)=sqrt(2*(a)^(2))*0.5*int(1/(sqrt(t+q - 1)*sqrt(t+q)*sqrt(t+q + 1)), t = 0..infinity)	Error	Error	Error	-	-
19.30.E12	${\displaystyle{\displaystyle s=aF\left(\phi,1/\sqrt{2}\right)}}$	s = a*EllipticF(sin(phi), 1/sqrt(2))	s = a*EllipticF[\[Phi], (1/Sqrt[2])^2]	Failure	Failure	Fail 2.831268677-2.040721799*I <- {a = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), s = 2^(1/2)+I*2^(1/2)} 2.831268677-4.869148923*I <- {a = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), s = 2^(1/2)-I*2^(1/2)} .2841553e-2-4.869148923*I <- {a = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), s = -2^(1/2)-I*2^(1/2)} .2841553e-2-2.040721799*I <- {a = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), s = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[2.831268678966966, -2.040721798901064] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[s, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-2.040721798901064, 2.831268678966966] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[s, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-0.0028415542207758104, 4.869148923647254] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[s, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[4.869148923647254, -0.0028415542207758104] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[s, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.30.E13	${\displaystyle{\displaystyle P=4\sqrt{2a^{2}}R_{F}\left(0,1,2\right)}}$	P = 4*sqrt(2*(a)^(2))*0.5*int(1/(sqrt(t+0)*sqrt(t+1)*sqrt(t+2)), t = 0..infinity)	Error	Error	Error	-	-
19.32.E1	${\displaystyle{\displaystyle z(p)=R_{F}\left(p-x_{1},p-x_{2},p-x_{3}\right)}}$	z*(p)= 0.5*int(1/(sqrt(t+p - x[1])*sqrt(t+p - x[2])*sqrt(t+p - x[3])), t = 0..infinity)	Error	Error	Error	-	-
19.32#Ex4	${\displaystyle{\displaystyle z(x_{3})=R_{F}\left(x_{3}-x_{1},x_{3}-x_{2},0\right)}}$	z*(x[3])= 0.5*int(1/(sqrt(t+x[3]- x[1])*sqrt(t+x[3]- x[2])*sqrt(t+0)), t = 0..infinity)	Error	Error	Error	-	-
19.32#Ex4	${\displaystyle{\displaystyle R_{F}\left(x_{3}-x_{1},x_{3}-x_{2},0\right)=-iR_{F}\left(0,x_{1}-x_{3},x_{2}-x_{3}\right)}}$	0.5*int(1/(sqrt(t+x[3]- x[1])*sqrt(t+x[3]- x[2])*sqrt(t+0)), t = 0..infinity)= - I*0.5*int(1/(sqrt(t+0)*sqrt(t+x[1]- x[3])*sqrt(t+x[2]- x[3])), t = 0..infinity)	Error	Error	Error	-	-
19.33.E2	${\displaystyle{\displaystyle\frac{S}{2\pi}=c^{2}+\frac{ab}{\sin\phi}\left(E\left(\phi,k\right){\sin^{2}}\phi+F\left(\phi,k\right){\cos^{2}}\phi\right)}}$	(S)/(2*Pi)= (c)^(2)+(a*b)/(sin(phi))*(EllipticE(sin(phi), k)*(sin(phi))^(2)+ EllipticF(sin(phi), k)*(cos(phi))^(2))	Divide[S,2*Pi]= (c)^(2)+Divide[a*b,Sin[\[Phi]]]*(EllipticE[\[Phi], (k)^2]*(Sin[\[Phi]])^(2)+ EllipticF[\[Phi], (k)^2]*(Cos[\[Phi]])^(2))	Failure	Failure	Skip	Error
19.33#Ex1	${\displaystyle{\displaystyle\cos\phi=\frac{c}{a}}}$	cos(phi)=(c)/(a)	Cos[\[Phi]]=Divide[c,a]	Failure	Failure	Fail -.6603260076-1.911393109*I <- {a = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2)} -.6603260076+1.911393109*I <- {a = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)-I*2^(1/2)} -.6603260076-1.911393109*I <- {a = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), phi = -2^(1/2)-I*2^(1/2)} -.6603260076+1.911393109*I <- {a = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), phi = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data	Fail Complex[0.13599115937806153, -1.8531119867052335] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[$0, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[0.13599115937806153, -1.9696742336231872] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[$0, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[0.13599115937806153, -1.8531119867052335] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[$0, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[0.13599115937806153, -1.9696742336231872] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[$0, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data
19.33.E5	${\displaystyle{\displaystyle V(\lambda)=QR_{F}\left(a^{2}+\lambda,b^{2}+\lambda,c^{2}+\lambda\right)}}$	V*(lambda)= Q*0.5*int(1/(sqrt(t+(a)^(2)+ lambda)*sqrt(t+(b)^(2)+ lambda)*sqrt(t+(c)^(2)+ lambda)), t = 0..infinity)	Error	Error	Error	-	-
19.33.E6	${\displaystyle{\displaystyle 1/C=R_{F}\left(a^{2},b^{2},c^{2}\right)}}$	1/ C = 0.5*int(1/(sqrt(t+(a)^(2))*sqrt(t+(b)^(2))*sqrt(t+(c)^(2))), t = 0..infinity)	Error	Error	Error	-	-
19.33.E7	${\displaystyle{\displaystyle L_{c}=2\pi abc\int_{0}^{\infty}\frac{\mathrm{d}\lambda}{\sqrt{(a^{2}+\lambda)(b^{2}+\lambda)(c^{2}+\lambda)^{3}}}}}$	L[c]= 2*Pi*a*b*c*int((1)/(sqrt(((a)^(2)+ lambda)*((b)^(2)+ lambda)*((c)^(2)+ lambda)^(3))), lambda = 0..infinity)	Subscript[L, c]= 2*Pi*a*b*c*Integrate[Divide[1,Sqrt[((a)^(2)+ \[Lambda])*((b)^(2)+ \[Lambda])*((c)^(2)+ \[Lambda])^(3)]], {\[Lambda], 0, Infinity}]	Failure	Failure	Skip	Error
19.34.E1	${\displaystyle{\displaystyle ab\int_{0}^{2\pi}(h^{2}+a^{2}+b^{2}-2ab\cos\theta)^{-1/2}\cos\theta\mathrm{d}\theta=2ab\int_{-1}^{1}\frac{t\mathrm{d}t}{\sqrt{(1+t)(1-t)(a_{3}-2abt)}}}}$	a*b*int(((h)^(2)+ (a)^(2)+ (b)^(2)- 2*a*b*cos(theta))^(- 1/ 2)* cos(theta), theta = 0..2*Pi)= 2*a*b*int((t)/(sqrt((1 + t)*(1 - t)*(a[3]- 2*a*b*t))), t = - 1..1)	a*b*Integrate[((h)^(2)+ (a)^(2)+ (b)^(2)- 2*a*b*Cos[\[Theta]])^(- 1/ 2)* Cos[\[Theta]], {\[Theta], 0, 2*Pi}]= 2*a*b*Integrate[Divide[t,Sqrt[(1 + t)*(1 - t)*(Subscript[a, 3]- 2*a*b*t)]], {t, - 1, 1}]	Failure	Failure	Skip	Error
19.34.E1	${\displaystyle{\displaystyle 2ab\int_{-1}^{1}\frac{t\mathrm{d}t}{\sqrt{(1+t)(1-t)(a_{3}-2abt)}}=2abI(\mathbf{e}_{5})}}$	2*a*b*int((t)/(sqrt((1 + t)*(1 - t)*(a[3]- 2*a*b*t))), t = - 1..1)= 2*a*b*I*(e[5])	2*a*b*Integrate[Divide[t,Sqrt[(1 + t)*(1 - t)*(Subscript[a, 3]- 2*a*b*t)]], {t, - 1, 1}]= 2*a*b*I*(Subscript[e, 5])	Failure	Failure	Skip	Error
19.36.E3	${\displaystyle{\displaystyle R_{F}\left(1,2,4\right)=R_{F}\left(z_{1},z_{2},z_{3}\right)}}$	0.5*int(1/(sqrt(t+1)*sqrt(t+2)*sqrt(t+4)), t = 0..infinity)= 0.5*int(1/(sqrt(t+z[1])*sqrt(t+z[2])*sqrt(t+z[3])), t = 0..infinity)	Error	Error	Error	-	-
19.36.E9	${\displaystyle{\displaystyle R_{F}\left(t_{0}^{2},t_{0}^{2}+\theta c_{0}^{2},t_{0}^{2}+\theta a_{0}^{2}\right)=R_{F}\left(T^{2},T^{2},T^{2}+\theta M^{2}\right)}}$	0.5*int(1/(sqrt(t+t(t[0])^(2))*sqrt(t+t(t[0])^(2)+ theta*c(c[0])^(2))*sqrt(t+t(t[0])^(2)+ theta*a(a[0])^(2))), t = 0..infinity)= 0.5*int(1/(sqrt(t+(T)^(2))*sqrt(t+(T)^(2))*sqrt(t+(T)^(2)+ theta*(M)^(2))), t = 0..infinity)	Error	Error	Error	-	-