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(mPi + phi), k)= 2m*EllipticK(k)+ EllipticF(sin(phi), k)`	`EllipticF[mPi + \[Phi], (k)^2]= 2m*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(mPi - phi), k)= 2m*EllipticK(k)- EllipticF(sin(phi), k)`	`EllipticF[mPi - \[Phi], (k)^2]= 2m*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(mPi + phi), k)= 2m*EllipticE(k)+ EllipticE(sin(phi), k)`	`EllipticE[mPi + \[Phi], (k)^2]= 2m*EllipticE[(k)^2]+ EllipticE[\[Phi], (k)^2]`	Failure	Failure	Fail `-6.303071081-.6035229402I <- {phi = 2^(1/2)+I2^(1/2), k = 1, m = 1}` `-3.999999999-.16e-8I <- {phi = 2^(1/2)+I2^(1/2), k = 1, m = 2}` `-10.30307108-.6035229462I <- {phi = 2^(1/2)+I2^(1/2), k = 1, m = 3}` `-8.695154357-1.322364992I <- {phi = 2^(1/2)+I2^(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(mPi - phi), k)= 2m*EllipticE(k)- EllipticE(sin(phi), k)`	`EllipticE[mPi - \[Phi], (k)^2]= 2m*EllipticE[(k)^2]- EllipticE[\[Phi], (k)^2]`	Failure	Failure	Fail `2.303071080+.6035229417I <- {phi = 2^(1/2)+I2^(1/2), k = 1, m = 1}` `-4.000000000-.15e-8I <- {phi = 2^(1/2)+I2^(1/2), k = 1, m = 2}` `-1.696928920+.6035229433I <- {phi = 2^(1/2)+I2^(1/2), k = 1, m = 3}` `7.069958809-4.053051928I <- {phi = 2^(1/2)+I2^(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(mPi + phi), k) - EllipticE(sin(mPi + phi), k))/(k)^2 = 2m(EllipticK(k) - EllipticE(k))/(k)^2 + (EllipticF(sin(phi), k) - EllipticE(sin(phi), k))/(k)^2`	`Divide[EllipticF[mPi + \[Phi], (k)^2] - EllipticE[mPi + \[Phi], (k)^2], (k)^4]= 2mDivide[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(mPi - phi), k) - EllipticE(sin(mPi - phi), k))/(k)^2 = 2m(EllipticK(k) - EllipticE(k))/(k)^2 - (EllipticF(sin(phi), k) - EllipticE(sin(phi), k))/(k)^2`	`Divide[EllipticF[mPi - \[Phi], (k)^2] - EllipticE[mPi - \[Phi], (k)^2], (k)^4]= 2mDivide[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 `.9591286913-1.118374137I <- {phi = 2^(1/2)+I2^(1/2), x = 1}` `1.959128691-1.118374137I <- {phi = 2^(1/2)+I2^(1/2), x = 2}` `2.959128691-1.118374137I <- {phi = 2^(1/2)+I2^(1/2), x = 3}` `.9591286913+1.118374137I <- {phi = 2^(1/2)-I2^(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))/(k1 - (k)^(2))-(ksin(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],k1 - (k)^(2)]-Divide[kSin[\[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)+I2^(1/2), k = 1}` `.3848531746-2.832892174I <- {phi = 2^(1/2)+I2^(1/2), k = 2}` `.2974950167-1.686437583I <- {phi = 2^(1/2)+I2^(1/2), k = 3}` `Float(infinity)+Float(infinity)I <- {phi = 2^(1/2)-I2^(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)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), k = 1}` `.3340407480e-1-.3675640793I <- {alpha = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), k = 2}` `-.3093953717e-1-.2652186316I <- {alpha = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), k = 3}` `Float(undefined)+Float(undefined)I <- {alpha = 2^(1/2)+I2^(1/2), phi = 2^(1/2)-I2^(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	${\displaystyle{\displaystyle(k{k^{\prime}}^{2}D_{k}^{2}+(1-3k^{2})D_{k}-k)F% \left(\phi,k\right)=\frac{-k\sin\phi\cos\phi}{(1-k^{2}{\sin^{2}}\phi)^{3/2}}}}$	`(k1 - (k)^(2) D(D[k])^(2)+(1 - 3(k)^(2))D[k]- k)* EllipticF(sin(phi), k)=(- ksin(phi)cos(phi))/((1 - (k)^(2)* (sin(phi))^(2))^(3/ 2))`	`(k1 - (k)^(2) D(Subscript[D, k])^(2)+(1 - 3(k)^(2))Subscript[D, k]- k)* EllipticF[\[Phi], (k)^2]=Divide[- kSin[\[Phi]]Cos[\[Phi]],(1 - (k)^(2)* (Sin[\[Phi]])^(2))^(3/ 2)]`	Error	Failure	-	Successful
19.4.E9	${\displaystyle{\displaystyle(k{k^{\prime}}^{2}D_{k}^{2}+{k^{\prime}}^{2}D_{k}+% k)E\left(\phi,k\right)=\frac{k\sin\phi\cos\phi}{\sqrt{1-k^{2}{\sin^{2}}\phi}}}}$	`(k1 - (k)^(2) D(D[k])^(2)+ 1 - (k)^(2)* D[k]+ k)* EllipticE(sin(phi), k)=(ksin(phi)cos(phi))/(sqrt(1 - (k)^(2)* (sin(phi))^(2)))`	`(k1 - (k)^(2) D(Subscript[D, k])^(2)+ 1 - (k)^(2)* Subscript[D, k]+ k)* EllipticE[\[Phi], (k)^2]=Divide[kSin[\[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)^(2m), 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)^(2m), {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)^(2m), 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)^(2m), {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)^(2m), 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)^(2m), {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 + 2sum((q)^((n)^(2)), n = 1..infinity))^(2)`	`EllipticK[(k)^2]=Divide[Pi,2](1 + 2Sum[(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 + 2sum((- 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 + 2Sum[(- 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}` `3.458268152-1.791805641I <- {k = 2}` `8.701204041-2.810641644I <- {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+.4760732230I <- {alpha = 2^(1/2)+I2^(1/2)}` `.6097433517-.4760732230I <- {alpha = 2^(1/2)-I2^(1/2)}` `.6097433517+.4760732230I <- {alpha = -2^(1/2)-I2^(1/2)}` `.6097433517-.4760732230I <- {alpha = -2^(1/2)+I2^(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)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), k = 1}` `2.583555547+2.729095606I <- {Delta = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), k = 2}` `1.800281293+2.241476784I <- {Delta = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), k = 3}` `Float(infinity)+Float(infinity)I <- {Delta = 2^(1/2)+I2^(1/2), phi = 2^(1/2)-I2^(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)- Deltatan(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)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), k = 1}` `-2.077524150+.3723387150I <- {Delta = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), k = 2}` `-1.086812375+.1094418732I <- {Delta = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), k = 3}` `Float(infinity)+Float(infinity)I <- {Delta = 2^(1/2)+I2^(1/2), phi = 2^(1/2)-I2^(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(Ik/sqrt(1 - (k)^(2)))=sqrt(1 - (k)^(2))EllipticK(k)`	`EllipticK[(Ik/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(Ik/sqrt(1 - (k)^(2)))=(1/sqrt(1 - (k)^(2))) EllipticE(k)`	`EllipticE[(Ik/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+.4691535424I <- {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-.6820014149I <- {beta = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), k[1] = 2^(1/2)+I2^(1/2), k = 1}` `.1609634146-1.274502936I <- {beta = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), k[1] = 2^(1/2)+I2^(1/2), k = 2}` `.1632109318-1.647695481I <- {beta = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), k[1] = 2^(1/2)+I2^(1/2), k = 3}` `.8504514854-1.466163968I <- {beta = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), k[1] = 2^(1/2)-I2^(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+3.452491889I <- {beta = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), k[1] = 2^(1/2)+I2^(1/2), k = 1}` `3.041115012+4.688090748I <- {beta = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), k[1] = 2^(1/2)+I2^(1/2), k = 2}` `2.905177117+5.217063396I <- {beta = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), k[1] = 2^(1/2)+I2^(1/2), k = 3}` `-1.662478744+4.734343989I <- {beta = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), k[1] = 2^(1/2)-I2^(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])= kEllipticPi(sin(beta), (k)^(2) (alpha)^(2), k)`	`EllipticPi[(\[Alpha])^(2), \[Phi],(Subscript[k, 1])^2]= kEllipticPi[(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+.2706006291I <- {phi = 2^(1/2)+I2^(1/2), theta = 2^(1/2)+I2^(1/2), k = 1}` `1.061002746+.2942020964I <- {phi = 2^(1/2)+I2^(1/2), theta = 2^(1/2)+I2^(1/2), k = 2}` `1.109187756+.2984612377I <- {phi = 2^(1/2)+I2^(1/2), theta = 2^(1/2)+I2^(1/2), k = 3}` `.863648898-.3329223119I <- {phi = 2^(1/2)+I2^(1/2), theta = 2^(1/2)-I2^(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(Iphi), k)= IEllipticF(sin(psi), sqrt(1 - (k)^(2)))`	`EllipticF[I\[Phi], (k)^2]= IEllipticF[\[Psi], (Sqrt[1 - (k)^(2)])^2]`	Failure	Failure	Fail `.9251391454+.76168273e-1I <- {phi = 2^(1/2)+I2^(1/2), psi = 2^(1/2)+I2^(1/2), k = 1}` `.3353318864+.3237108e-2I <- {phi = 2^(1/2)+I2^(1/2), psi = 2^(1/2)+I2^(1/2), k = 2}` `.2150344377+.8523035e-3I <- {phi = 2^(1/2)+I2^(1/2), psi = 2^(1/2)+I2^(1/2), k = 3}` `-1.903287979+.76168273e-1I <- {phi = 2^(1/2)+I2^(1/2), psi = 2^(1/2)-I2^(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(Iphi), 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 `-1.901989389-2.116793618I <- {phi = 2^(1/2)+I2^(1/2), psi = 2^(1/2)+I2^(1/2), k = 1}` `-6.480561985-4.744566218I <- {phi = 2^(1/2)+I2^(1/2), psi = 2^(1/2)+I2^(1/2), k = 2}` `-10.28790921-7.196969925I <- {phi = 2^(1/2)+I2^(1/2), psi = 2^(1/2)+I2^(1/2), k = 3}` `-2.401081691-2.116793618I <- {phi = 2^(1/2)+I2^(1/2), psi = 2^(1/2)-I2^(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(Iphi), (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)/(4GaussAGM(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)/(4GaussAGM(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-1I <- {phi = 2^(1/2)+I2^(1/2), phi[1] = 2^(1/2)+I2^(1/2), k = 2}` `1.264955024-.1049221987e-1I <- {phi = 2^(1/2)+I2^(1/2), phi[1] = 2^(1/2)+I2^(1/2), k = 3}` `-2.828427124I <- {phi = 2^(1/2)+I2^(1/2), phi[1] = 2^(1/2)-I2^(1/2), k = 1}` `1.094670290-2.843312078I <- {phi = 2^(1/2)+I2^(1/2), phi[1] = 2^(1/2)-I2^(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 `1.876175051-2.798657216I <- {phi = 2^(1/2)-I2^(1/2), k = 2}` `2.216744518-2.807442684I <- {phi = 2^(1/2)-I2^(1/2), k = 3}` `-1.876175051+2.798657216I <- {phi = -2^(1/2)+I2^(1/2), k = 2}` `-2.216744518+2.807442684I <- {phi = -2^(1/2)+I2^(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-1I <- {phi = 2^(1/2)+I2^(1/2), k[1] = 2^(1/2)+I2^(1/2), phi[1] = 2^(1/2)+I2^(1/2), k = 1}` `.407379233e-1-.384751455I <- {phi = 2^(1/2)+I2^(1/2), k[1] = 2^(1/2)+I2^(1/2), phi[1] = 2^(1/2)+I2^(1/2), k = 2}` `-.393423424e-1-.6075011659I <- {phi = 2^(1/2)+I2^(1/2), k[1] = 2^(1/2)+I2^(1/2), phi[1] = 2^(1/2)+I2^(1/2), k = 3}` `-1.145001001+.5724275385I <- {phi = 2^(1/2)+I2^(1/2), k[1] = 2^(1/2)+I2^(1/2), phi[1] = 2^(1/2)-I2^(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-.9810550268I <- {phi = 2^(1/2)+I2^(1/2), k[1] = 2^(1/2)+I2^(1/2), phi[1] = 2^(1/2)+I2^(1/2), k = 1}` `.743033975-3.185064608I <- {phi = 2^(1/2)+I2^(1/2), k[1] = 2^(1/2)+I2^(1/2), phi[1] = 2^(1/2)+I2^(1/2), k = 2}` `3.178569085-4.992225484I <- {phi = 2^(1/2)+I2^(1/2), k[1] = 2^(1/2)+I2^(1/2), phi[1] = 2^(1/2)+I2^(1/2), k = 3}` `1.575776580+2.225504017I <- {phi = 2^(1/2)+I2^(1/2), k[1] = 2^(1/2)+I2^(1/2), phi[1] = 2^(1/2)-I2^(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+1.472494686I <- {c[1] = 2^(1/2)+I2^(1/2), phi[1] = 2^(1/2)+I2^(1/2)}` `1.210530730+1.355932438I <- {c[1] = 2^(1/2)+I2^(1/2), phi[1] = 2^(1/2)-I2^(1/2)}` `1.210530730+1.472494686I <- {c[1] = 2^(1/2)+I2^(1/2), phi[1] = -2^(1/2)-I2^(1/2)}` `1.210530730+1.355932438I <- {c[1] = 2^(1/2)+I2^(1/2), phi[1] = -2^(1/2)+I2^(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	${\displaystyle{\displaystyle 2\phi_{2}=\phi+\operatorname{arcsin}\left(k\sin% \phi\right)}}$	`2phi[2]= phi + arcsin(ksin(phi))`	`2Subscript[\[Phi], 2]= \[Phi]+ ArcSin[kSin[\[Phi]]]`	Failure	Failure	Fail `-.13450037e-1-.735044026I <- {phi = 2^(1/2)+I2^(1/2), phi[2] = 2^(1/2)+I2^(1/2), k = 2}` `-.15590800e-1-1.147765401I <- {phi = 2^(1/2)+I2^(1/2), phi[2] = 2^(1/2)+I2^(1/2), k = 3}` `-5.656854248I <- {phi = 2^(1/2)+I2^(1/2), phi[2] = 2^(1/2)-I2^(1/2), k = 1}` `-.13450037e-1-6.391898274I <- {phi = 2^(1/2)+I2^(1/2), phi[2] = 2^(1/2)-I2^(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+.6820014149I <- {phi = 2^(1/2)+I2^(1/2), k[2] = 2^(1/2)+I2^(1/2), phi[2] = 2^(1/2)+I2^(1/2), k = 1}` `-.1866400363+.5025213980I <- {phi = 2^(1/2)+I2^(1/2), k[2] = 2^(1/2)+I2^(1/2), phi[2] = 2^(1/2)+I2^(1/2), k = 2}` `-.1605619730+.4145017571I <- {phi = 2^(1/2)+I2^(1/2), k[2] = 2^(1/2)+I2^(1/2), phi[2] = 2^(1/2)+I2^(1/2), k = 3}` `-.8504514854+1.514599702I <- {phi = 2^(1/2)+I2^(1/2), k[2] = 2^(1/2)+I2^(1/2), phi[2] = 2^(1/2)-I2^(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 `-3.768464116-3.924220109I <- {phi = 2^(1/2)+I2^(1/2), k[2] = 2^(1/2)+I2^(1/2), phi[2] = 2^(1/2)+I2^(1/2), k = 1}` `-3.226003445-6.062382948I <- {phi = 2^(1/2)+I2^(1/2), k[2] = 2^(1/2)+I2^(1/2), phi[2] = 2^(1/2)+I2^(1/2), k = 2}` `-2.601575251-8.024788758I <- {phi = 2^(1/2)+I2^(1/2), k[2] = 2^(1/2)+I2^(1/2), phi[2] = 2^(1/2)+I2^(1/2), k = 3}` `6.303106322+7.694970191I <- {phi = 2^(1/2)+I2^(1/2), k[2] = 2^(1/2)+I2^(1/2), phi[2] = 2^(1/2)-I2^(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+.5973782105I <- {Delta = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), psi[1] = 2^(1/2)+I2^(1/2), k = 1}` `.921485403-.6462809029I <- {Delta = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), psi[1] = 2^(1/2)+I2^(1/2), k = 2}` `.597378210-1.433508616I <- {Delta = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), psi[1] = 2^(1/2)+I2^(1/2), k = 3}` `1.433508616-.61447305e-2I <- {Delta = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), psi[1] = 2^(1/2)-I2^(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+3.325606671I <- {Delta = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), k = 1}` `.7940624558+5.601906026I <- {Delta = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), k = 2}` `.4980873551+7.792433846I <- {Delta = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), k = 3}` `1.074539570-.497179547I <- {Delta = 2^(1/2)+I2^(1/2), phi = 2^(1/2)-I2^(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-1-1.362004431I <- {phi = 2^(1/2)+I2^(1/2), k[1] = 2^(1/2)+I2^(1/2), psi[1] = 2^(1/2)+I2^(1/2), k = 1}` `-.1565174333-1.810944588I <- {phi = 2^(1/2)+I2^(1/2), k[1] = 2^(1/2)+I2^(1/2), psi[1] = 2^(1/2)+I2^(1/2), k = 2}` `-.2365976990-2.033694299I <- {phi = 2^(1/2)+I2^(1/2), k[1] = 2^(1/2)+I2^(1/2), psi[1] = 2^(1/2)+I2^(1/2), k = 3}` `-2.779076419-.345526758I <- {phi = 2^(1/2)+I2^(1/2), k[1] = 2^(1/2)+I2^(1/2), psi[1] = 2^(1/2)-I2^(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 `-.607875038-2.285836406I <- {Delta = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), k[1] = 2^(1/2)+I2^(1/2), psi[1] = 2^(1/2)+I2^(1/2), k = 1}` `3.299178317-9.848207688I <- {Delta = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), k[1] = 2^(1/2)+I2^(1/2), psi[1] = 2^(1/2)+I2^(1/2), k = 2}` `7.141163056-15.04717488I <- {Delta = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), k[1] = 2^(1/2)+I2^(1/2), psi[1] = 2^(1/2)+I2^(1/2), k = 3}` `4.427910181+3.523758744I <- {Delta = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2), k[1] = 2^(1/2)+I2^(1/2), psi[1] = 2^(1/2)-I2^(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+1.472494686I <- {c = 2^(1/2)+I2^(1/2), phi = 2^(1/2)+I2^(1/2)}` `1.210530730+1.355932438I <- {c = 2^(1/2)+I2^(1/2), phi = 2^(1/2)-I2^(1/2)}` `1.210530730+1.472494686I <- {c = 2^(1/2)+I2^(1/2), phi = -2^(1/2)-I2^(1/2)}` `1.210530730+1.355932438I <- {c = 2^(1/2)+I2^(1/2), phi = -2^(1/2)+I2^(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))<(PiEllipticCK(k))/(2EllipticK(k))`	`Log[Divide[(1 +Sqrt[Sqrt[1 - (k)^(2)]])^(2),k]]<Divide[PiEllipticK[1-(k)^2],2EllipticK[(k)^2]]`	Failure	Failure	Error	Successful
19.9.E5	${\displaystyle{\displaystyle\frac{\pi{K^{\prime}}\left(k\right)}{2\!K\left(k% \right)}<\ln\frac{2(1+k^{\prime})}{k}}}$	`(PiEllipticCK(k))/(2EllipticK(k))< ln((2*(1 +sqrt(1 - (k)^(2))))/(k))`	`Divide[PiEllipticK[1-(k)^2],2EllipticK[(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)= 4a*EllipticE(k)`	`L(a , b)= 4a*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	${\displaystyle{\displaystyle F\left(\phi,k\right)\bigg{/}\left((\sin\phi)\ln% \left(\frac{4}{\Delta+\cos\phi}\right)\right)<\frac{4}{2+(1+k^{2}){\sin^{2}}% \phi}}}$