# 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 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