Elliptic Integrals - 19.8 Quadratic Transformations

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
19.8#Ex1 a n + 1 = a n + g n 2 subscript π‘Ž 𝑛 1 subscript π‘Ž 𝑛 subscript 𝑔 𝑛 2 {\displaystyle{\displaystyle a_{n+1}=\frac{a_{n}+g_{n}}{2}}}
a_{n+1} = \frac{a_{n}+g_{n}}{2}		 

a[n + 1] = (a[n]+ g[n])/(2)
 
Subscript[a, n + 1] == Divide[Subscript[a, n]+ Subscript[g, n],2]
 Skipped - no semantic math Skipped - no semantic math - -
19.8#Ex2 g n + 1 = a n ⁒ g n subscript 𝑔 𝑛 1 subscript π‘Ž 𝑛 subscript 𝑔 𝑛 {\displaystyle{\displaystyle g_{n+1}=\sqrt{a_{n}g_{n}}}}
g_{n+1} = \sqrt{a_{n}g_{n}}		 

g[n + 1] = sqrt(a[n]*g[n])
 
Subscript[g, n + 1] == Sqrt[Subscript[a, n]*Subscript[g, n]]
 Skipped - no semantic math Skipped - no semantic math - -
19.8.E2 c n = a n 2 - g n 2 subscript 𝑐 𝑛 superscript subscript π‘Ž 𝑛 2 superscript subscript 𝑔 𝑛 2 {\displaystyle{\displaystyle c_{n}=\sqrt{a_{n}^{2}-g_{n}^{2}}}}
c_{n} = \sqrt{a_{n}^{2}-g_{n}^{2}}		 

c[n] = sqrt((a[n])^(2)- (g[n])^(2))
 
Subscript[c, n] == Sqrt[(Subscript[a, n])^(2)- (Subscript[g, n])^(2)]
 Skipped - no semantic math Skipped - no semantic math - -
19.8.E3 c n + 1 = a n - g n 2 subscript 𝑐 𝑛 1 subscript π‘Ž 𝑛 subscript 𝑔 𝑛 2 {\displaystyle{\displaystyle c_{n+1}=\frac{a_{n}-g_{n}}{2}}}
c_{n+1} = \frac{a_{n}-g_{n}}{2}		 

c[n + 1] = (a[n]- g[n])/(2)
 
Subscript[c, n + 1] == Divide[Subscript[a, n]- Subscript[g, n],2]
 Skipped - no semantic math Skipped - no semantic math - -
19.8.E4 1 M ⁑ ( a 0 , g 0 ) = 2 Ο€ ⁒ ∫ 0 Ο€ / 2 d ΞΈ a 0 2 ⁒ cos 2 ⁑ ΞΈ + g 0 2 ⁒ sin 2 ⁑ ΞΈ 1 arithmetic-geometric-mean subscript π‘Ž 0 subscript 𝑔 0 2 πœ‹ superscript subscript 0 πœ‹ 2 πœƒ superscript subscript π‘Ž 0 2 2 πœƒ superscript subscript 𝑔 0 2 2 πœƒ {\displaystyle{\displaystyle\frac{1}{M\left(a_{0},g_{0}\right)}=\frac{2}{\pi}% \int_{0}^{\pi/2}\frac{\mathrm{d}\theta}{\sqrt{a_{0}^{2}{\cos^{2}}\theta+g_{0}^% {2}{\sin^{2}}\theta}}}}
\frac{1}{\AGM@{a_{0}}{g_{0}}} = \frac{2}{\pi}\int_{0}^{\pi/2}\frac{\diff{\theta}}{\sqrt{a_{0}^{2}\cos^{2}@@{\theta}+g_{0}^{2}\sin^{2}@@{\theta}}}		 

(1)/(GaussAGM(a[0], g[0])) = (2)/(Pi)*int((1)/(sqrt((a[0])^(2)*(cos(theta))^(2)+ (g[0])^(2)*(sin(theta))^(2))), theta = 0..Pi/2)

Error
Failure Missing Macro Error Error Skip - symbolical successful subtest
19.8.E4 2 Ο€ ⁒ ∫ 0 Ο€ / 2 d ΞΈ a 0 2 ⁒ cos 2 ⁑ ΞΈ + g 0 2 ⁒ sin 2 ⁑ ΞΈ = 1 Ο€ ⁒ ∫ 0 ∞ d t t ⁒ ( t + a 0 2 ) ⁒ ( t + g 0 2 ) 2 πœ‹ superscript subscript 0 πœ‹ 2 πœƒ superscript subscript π‘Ž 0 2 2 πœƒ superscript subscript 𝑔 0 2 2 πœƒ 1 πœ‹ superscript subscript 0 𝑑 𝑑 𝑑 superscript subscript π‘Ž 0 2 𝑑 superscript subscript 𝑔 0 2 {\displaystyle{\displaystyle\frac{2}{\pi}\int_{0}^{\pi/2}\frac{\mathrm{d}% \theta}{\sqrt{a_{0}^{2}{\cos^{2}}\theta+g_{0}^{2}{\sin^{2}}\theta}}=\frac{1}{% \pi}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t(t+a_{0}^{2})(t+g_{0}^{2})}}}}
\frac{2}{\pi}\int_{0}^{\pi/2}\frac{\diff{\theta}}{\sqrt{a_{0}^{2}\cos^{2}@@{\theta}+g_{0}^{2}\sin^{2}@@{\theta}}} = \frac{1}{\pi}\int_{0}^{\infty}\frac{\diff{t}}{\sqrt{t(t+a_{0}^{2})(t+g_{0}^{2})}}		 

(2)/(Pi)*int((1)/(sqrt((a[0])^(2)*(cos(theta))^(2)+ (g[0])^(2)*(sin(theta))^(2))), theta = 0..Pi/2) = (1)/(Pi)*int((1)/(sqrt(t*(t + (a[0])^(2))*(t + (g[0])^(2)))), t = 0..infinity)

Divide[2,Pi]*Integrate[Divide[1,Sqrt[(Subscript[a, 0])^(2)*(Cos[\[Theta]])^(2)+ (Subscript[g, 0])^(2)*(Sin[\[Theta]])^(2)]], {\[Theta], 0, Pi/2}, GenerateConditions->None] == Divide[1,Pi]*Integrate[Divide[1,Sqrt[t*(t + (Subscript[a, 0])^(2))*(t + (Subscript[g, 0])^(2))]], {t, 0, Infinity}, GenerateConditions->None]
Aborted Aborted Skipped - Because timed out Skipped - Because timed out
19.8.E5 K ⁑ ( k ) = Ο€ 2 ⁒ M ⁑ ( 1 , k β€² ) complete-elliptic-integral-first-kind-K π‘˜ πœ‹ 2 arithmetic-geometric-mean 1 superscript π‘˜ β€² {\displaystyle{\displaystyle K\left(k\right)=\frac{\pi}{2M\left(1,k^{\prime}% \right)}}}
\compellintKk@{k} = \frac{\pi}{2\AGM@{1}{k^{\prime}}}		 - ∞ < k 2 , k 2 < 1 formulae-sequence superscript π‘˜ 2 superscript π‘˜ 2 1 {\displaystyle{\displaystyle-\infty<k^{2},k^{2}<1}} 
EllipticK(k) = (Pi)/(2*GaussAGM(1, sqrt(1 - (k)^(2))))

Error
Failure Missing Macro Error Error -
19.8.E6 E ⁑ ( k ) = Ο€ 2 ⁒ M ⁑ ( 1 , k β€² ) ⁒ ( a 0 2 - βˆ‘ n = 0 ∞ 2 n - 1 ⁒ c n 2 ) complete-elliptic-integral-second-kind-E π‘˜ πœ‹ 2 arithmetic-geometric-mean 1 superscript π‘˜ β€² superscript subscript π‘Ž 0 2 superscript subscript 𝑛 0 superscript 2 𝑛 1 superscript subscript 𝑐 𝑛 2 {\displaystyle{\displaystyle E\left(k\right)=\frac{\pi}{2M\left(1,k^{\prime}% \right)}\left(a_{0}^{2}-\sum_{n=0}^{\infty}2^{n-1}c_{n}^{2}\right)}}
\compellintEk@{k} = \frac{\pi}{2\AGM@{1}{k^{\prime}}}\left(a_{0}^{2}-\sum_{n=0}^{\infty}2^{n-1}c_{n}^{2}\right)		 - ∞ < k 2 , k 2 < 1 , a 0 = 1 , g 0 = k β€² formulae-sequence superscript π‘˜ 2 formulae-sequence superscript π‘˜ 2 1 formulae-sequence subscript π‘Ž 0 1 subscript 𝑔 0 superscript π‘˜ β€² {\displaystyle{\displaystyle-\infty<k^{2},k^{2}<1,a_{0}=1,g_{0}=k^{\prime}}} 
EllipticE(k) = (Pi)/(2*GaussAGM(1, sqrt(1 - (k)^(2))))*((a[0])^(2)- sum((2)^(n - 1)* (c[n])^(2), n = 0..infinity))

Error
Failure Missing Macro Error Error -
19.8.E6 Ο€ 2 ⁒ M ⁑ ( 1 , k β€² ) ⁒ ( a 0 2 - βˆ‘ n = 0 ∞ 2 n - 1 ⁒ c n 2 ) = K ⁑ ( k ) ⁒ ( a 1 2 - βˆ‘ n = 2 ∞ 2 n - 1 ⁒ c n 2 ) πœ‹ 2 arithmetic-geometric-mean 1 superscript π‘˜ β€² superscript subscript π‘Ž 0 2 superscript subscript 𝑛 0 superscript 2 𝑛 1 superscript subscript 𝑐 𝑛 2 complete-elliptic-integral-first-kind-K π‘˜ superscript subscript π‘Ž 1 2 superscript subscript 𝑛 2 superscript 2 𝑛 1 superscript subscript 𝑐 𝑛 2 {\displaystyle{\displaystyle\frac{\pi}{2M\left(1,k^{\prime}\right)}\left(a_{0}% ^{2}-\sum_{n=0}^{\infty}2^{n-1}c_{n}^{2}\right)=K\left(k\right)\left(a_{1}^{2}% -\sum_{n=2}^{\infty}2^{n-1}c_{n}^{2}\right)}}
\frac{\pi}{2\AGM@{1}{k^{\prime}}}\left(a_{0}^{2}-\sum_{n=0}^{\infty}2^{n-1}c_{n}^{2}\right) = \compellintKk@{k}\left(a_{1}^{2}-\sum_{n=2}^{\infty}2^{n-1}c_{n}^{2}\right)		 - ∞ < k 2 , k 2 < 1 , a 0 = 1 , g 0 = k β€² formulae-sequence superscript π‘˜ 2 formulae-sequence superscript π‘˜ 2 1 formulae-sequence subscript π‘Ž 0 1 subscript 𝑔 0 superscript π‘˜ β€² {\displaystyle{\displaystyle-\infty<k^{2},k^{2}<1,a_{0}=1,g_{0}=k^{\prime}}} 
(Pi)/(2*GaussAGM(1, sqrt(1 - (k)^(2))))*((a[0])^(2)- sum((2)^(n - 1)* (c[n])^(2), n = 0..infinity)) = EllipticK(k)*((a[1])^(2)- sum((2)^(n - 1)* (c[n])^(2), n = 2..infinity))

Error
Failure Missing Macro Error Error -
19.8.E7 Ξ  ⁑ ( Ξ± 2 , k ) = Ο€ 4 ⁒ M ⁑ ( 1 , k β€² ) ⁒ ( 2 + Ξ± 2 1 - Ξ± 2 ⁒ βˆ‘ n = 0 ∞ Q n ) complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 π‘˜ πœ‹ 4 arithmetic-geometric-mean 1 superscript π‘˜ β€² 2 superscript 𝛼 2 1 superscript 𝛼 2 superscript subscript 𝑛 0 subscript 𝑄 𝑛 {\displaystyle{\displaystyle\Pi\left(\alpha^{2},k\right)=\frac{\pi}{4M\left(1,% k^{\prime}\right)}\left(2+\frac{\alpha^{2}}{1-\alpha^{2}}\sum_{n=0}^{\infty}Q_% {n}\right)}}
\compellintPik@{\alpha^{2}}{k} = \frac{\pi}{4\AGM@{1}{k^{\prime}}}\left(2+\frac{\alpha^{2}}{1-\alpha^{2}}\sum_{n=0}^{\infty}Q_{n}\right)		 - ∞ < k 2 , k 2 < 1 , - ∞ < Ξ± 2 , Ξ± 2 < 1 formulae-sequence superscript π‘˜ 2 formulae-sequence superscript π‘˜ 2 1 formulae-sequence superscript 𝛼 2 superscript 𝛼 2 1 {\displaystyle{\displaystyle-\infty<k^{2},k^{2}<1,-\infty<\alpha^{2},\alpha^{2% }<1}} 
EllipticPi((alpha)^(2), k) = (Pi)/(4*GaussAGM(1, sqrt(1 - (k)^(2))))*(2 +((alpha)^(2))/(1 - (alpha)^(2))*sum(Q[n], n = 0..infinity))

Error
Failure Missing Macro Error Error -
19.8#Ex3 p n + 1 = p n 2 + a n ⁒ g n 2 ⁒ p n subscript 𝑝 𝑛 1 superscript subscript 𝑝 𝑛 2 subscript π‘Ž 𝑛 subscript 𝑔 𝑛 2 subscript 𝑝 𝑛 {\displaystyle{\displaystyle p_{n+1}=\frac{p_{n}^{2}+a_{n}g_{n}}{2p_{n}}}}
p_{n+1} = \frac{p_{n}^{2}+a_{n}g_{n}}{2p_{n}}		 

p[n + 1] = ((p[n])^(2)+ a[n]*g[n])/(2*p[n])
 
Subscript[p, n + 1] == Divide[(Subscript[p, n])^(2)+ Subscript[a, n]*Subscript[g, n],2*Subscript[p, n]]
 Skipped - no semantic math Skipped - no semantic math - -
19.8#Ex4 Ξ΅ n = p n 2 - a n ⁒ g n p n 2 + a n ⁒ g n subscript πœ€ 𝑛 superscript subscript 𝑝 𝑛 2 subscript π‘Ž 𝑛 subscript 𝑔 𝑛 superscript subscript 𝑝 𝑛 2 subscript π‘Ž 𝑛 subscript 𝑔 𝑛 {\displaystyle{\displaystyle\varepsilon_{n}=\frac{p_{n}^{2}-a_{n}g_{n}}{p_{n}^% {2}+a_{n}g_{n}}}}
\varepsilon_{n} = \frac{p_{n}^{2}-a_{n}g_{n}}{p_{n}^{2}+a_{n}g_{n}}		 

varepsilon[n] = ((p[n])^(2)- a[n]*g[n])/((p[n])^(2)+ a[n]*g[n])
 
Subscript[\[CurlyEpsilon], n] == Divide[(Subscript[p, n])^(2)- Subscript[a, n]*Subscript[g, n],(Subscript[p, n])^(2)+ Subscript[a, n]*Subscript[g, n]]
 Skipped - no semantic math Skipped - no semantic math - -
19.8#Ex5 Q n + 1 = 1 2 ⁒ Q n ⁒ Ξ΅ n subscript 𝑄 𝑛 1 1 2 subscript 𝑄 𝑛 subscript πœ€ 𝑛 {\displaystyle{\displaystyle Q_{n+1}=\tfrac{1}{2}Q_{n}\varepsilon_{n}}}
Q_{n+1} = \tfrac{1}{2}Q_{n}\varepsilon_{n}		 

Q[n + 1] = (1)/(2)*Q[n]*varepsilon[n]
 
Subscript[Q, n + 1] == Divide[1,2]*Subscript[Q, n]*Subscript[\[CurlyEpsilon], n]
 Skipped - no semantic math Skipped - no semantic math - -
19.8.E9 Ξ  ⁑ ( Ξ± 2 , k ) = Ο€ 4 ⁒ M ⁑ ( 1 , k β€² ) ⁒ k 2 k 2 - Ξ± 2 ⁒ βˆ‘ n = 0 ∞ Q n complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 π‘˜ πœ‹ 4 arithmetic-geometric-mean 1 superscript π‘˜ β€² superscript π‘˜ 2 superscript π‘˜ 2 superscript 𝛼 2 superscript subscript 𝑛 0 subscript 𝑄 𝑛 {\displaystyle{\displaystyle\Pi\left(\alpha^{2},k\right)=\frac{\pi}{4M\left(1,% k^{\prime}\right)}\frac{k^{2}}{k^{2}-\alpha^{2}}\sum_{n=0}^{\infty}Q_{n}}}
\compellintPik@{\alpha^{2}}{k} = \frac{\pi}{4\AGM@{1}{k^{\prime}}}\frac{k^{2}}{k^{2}-\alpha^{2}}\sum_{n=0}^{\infty}Q_{n}		 - ∞ < k 2 , k 2 < 1 , 1 < Ξ± 2 , Ξ± 2 < ∞ formulae-sequence superscript π‘˜ 2 formulae-sequence superscript π‘˜ 2 1 formulae-sequence 1 superscript 𝛼 2 superscript 𝛼 2 {\displaystyle{\displaystyle-\infty<k^{2},k^{2}<1,1<\alpha^{2},\alpha^{2}<% \infty}} 
EllipticPi((alpha)^(2), k) = (Pi)/(4*GaussAGM(1, sqrt(1 - (k)^(2))))*((k)^(2))/((k)^(2)- (alpha)^(2))*sum(Q[n], n = 0..infinity)

Error
Failure Missing Macro Error Error -
19.8.E10 p 0 2 = 1 - ( k 2 / Ξ± 2 ) superscript subscript 𝑝 0 2 1 superscript π‘˜ 2 superscript 𝛼 2 {\displaystyle{\displaystyle p_{0}^{2}=1-(k^{2}/\alpha^{2})}}
p_{0}^{2} = 1-(k^{2}/\alpha^{2})		 

(p[0])^(2) = 1 -((k)^(2)/(alpha)^(2))
 
(Subscript[p, 0])^(2) == 1 -((k)^(2)/\[Alpha]^(2))
 Skipped - no semantic math Skipped - no semantic math - -
19.8#Ex8 K ⁑ ( k ) = ( 1 + k 1 ) ⁒ K ⁑ ( k 1 ) complete-elliptic-integral-first-kind-K π‘˜ 1 subscript π‘˜ 1 complete-elliptic-integral-first-kind-K subscript π‘˜ 1 {\displaystyle{\displaystyle K\left(k\right)=(1+k_{1})K\left(k_{1}\right)}}
\compellintKk@{k} = (1+k_{1})\compellintKk@{k_{1}}		 

EllipticK(k) = (1 + k[1])*EllipticK(k[1])

EllipticK[(k)^2] == (1 + Subscript[k, 1])*EllipticK[(Subscript[k, 1])^2]
Failure Failure Error
Failed [30 / 30]
Result: DirectedInfinity[]
Test Values: {Rule[k, 1], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-1.44075376931664, -1.6191557371087932]
Test Values: {Rule[k, 2], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.8#Ex9 E ⁑ ( k ) = ( 1 + k β€² ) ⁒ E ⁑ ( k 1 ) - k β€² ⁒ K ⁑ ( k ) complete-elliptic-integral-second-kind-E π‘˜ 1 superscript π‘˜ β€² complete-elliptic-integral-second-kind-E subscript π‘˜ 1 superscript π‘˜ β€² complete-elliptic-integral-first-kind-K π‘˜ {\displaystyle{\displaystyle E\left(k\right)=(1+k^{\prime})E\left(k_{1}\right)% -k^{\prime}K\left(k\right)}}
\compellintEk@{k} = (1+k^{\prime})\compellintEk@{k_{1}}-k^{\prime}\compellintKk@{k}		 

EllipticE(k) = (1 +sqrt(1 - (k)^(2)))*EllipticE(k[1])-sqrt(1 - (k)^(2))*EllipticK(k)

EllipticE[(k)^2] == (1 +Sqrt[1 - (k)^(2)])*EllipticE[(Subscript[k, 1])^2]-Sqrt[1 - (k)^(2)]*EllipticK[(k)^2]
Failure Failure Error
Failed [30 / 30]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[0.595329372049606, 0.2521613076710463]
Test Values: {Rule[k, 2], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.8#Ex10 F ⁑ ( Ο• , k ) = 1 2 ⁒ ( 1 + k 1 ) ⁒ F ⁑ ( Ο• 1 , k 1 ) elliptic-integral-first-kind-F italic-Ο• π‘˜ 1 2 1 subscript π‘˜ 1 elliptic-integral-first-kind-F subscript italic-Ο• 1 subscript π‘˜ 1 {\displaystyle{\displaystyle F\left(\phi,k\right)=\tfrac{1}{2}(1+k_{1})F\left(% \phi_{1},k_{1}\right)}}
\incellintFk@{\phi}{k} = \tfrac{1}{2}(1+k_{1})\incellintFk@{\phi_{1}}{k_{1}}		 

EllipticF(sin(phi), k) = (1)/(2)*(1 + k[1])*EllipticF(sin(phi + arctan(sqrt(1 - (k)^(2))*tan(phi))), k[1])

EllipticF[\[Phi], (k)^2] == Divide[1,2]*(1 + Subscript[k, 1])*EllipticF[\[Phi]+ ArcTan[Sqrt[1 - (k)^(2)]*Tan[\[Phi]]], (Subscript[k, 1])^2]
Failure Failure
Failed [300 / 300]
Result: .2591790565-.226164263e-1*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, k[1] = 1/2*3^(1/2)+1/2*I, k = 1}


Result: .8581261265-.11942686e-2*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, k[1] = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.15619877563526813, 0.03685530383845256]
Test Values: {Rule[k, 1], Rule[Ο•, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[0.6672885103059906, -0.24203301849204312]
Test Values: {Rule[k, 2], Rule[Ο•, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.8#Ex11 E ⁑ ( Ο• , k ) = 1 2 ⁒ ( 1 + k β€² ) ⁒ E ⁑ ( Ο• 1 , k 1 ) - k β€² ⁒ F ⁑ ( Ο• , k ) + 1 2 ⁒ ( 1 - k β€² ) ⁒ sin ⁑ Ο• 1 elliptic-integral-second-kind-E italic-Ο• π‘˜ 1 2 1 superscript π‘˜ β€² elliptic-integral-second-kind-E subscript italic-Ο• 1 subscript π‘˜ 1 superscript π‘˜ β€² elliptic-integral-first-kind-F italic-Ο• π‘˜ 1 2 1 superscript π‘˜ β€² subscript italic-Ο• 1 {\displaystyle{\displaystyle E\left(\phi,k\right)=\tfrac{1}{2}(1+k^{\prime})E% \left(\phi_{1},k_{1}\right)-k^{\prime}F\left(\phi,k\right)+\tfrac{1}{2}(1-k^{% \prime})\sin\phi_{1}}}
\incellintEk@{\phi}{k} = \tfrac{1}{2}(1+k^{\prime})\incellintEk@{\phi_{1}}{k_{1}}-k^{\prime}\incellintFk@{\phi}{k}+\tfrac{1}{2}(1-k^{\prime})\sin@@{\phi_{1}}		 

EllipticE(sin(phi), k) = (1)/(2)*(1 +sqrt(1 - (k)^(2)))*EllipticE(sin(phi + arctan(sqrt(1 - (k)^(2))*tan(phi))), k[1])-sqrt(1 - (k)^(2))*EllipticF(sin(phi), k)+(1)/(2)*(1 -sqrt(1 - (k)^(2)))*sin(phi + arctan(sqrt(1 - (k)^(2))*tan(phi)))

EllipticE[\[Phi], (k)^2] == Divide[1,2]*(1 +Sqrt[1 - (k)^(2)])*EllipticE[\[Phi]+ ArcTan[Sqrt[1 - (k)^(2)]*Tan[\[Phi]]], (Subscript[k, 1])^2]-Sqrt[1 - (k)^(2)]*EllipticF[\[Phi], (k)^2]+Divide[1,2]*(1 -Sqrt[1 - (k)^(2)])*Sin[\[Phi]+ ArcTan[Sqrt[1 - (k)^(2)]*Tan[\[Phi]]]]
Failure Failure
Failed [300 / 300]
Result: -.627821156e-1-.413169945e-1*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, k[1] = 1/2*3^(1/2)+1/2*I, k = 1}


Result: .886069620e-1-.4575597e-3*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, k[1] = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.0022565574667213206, -0.009009769525654576]
Test Values: {Rule[k, 1], Rule[Ο•, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[0.11756483394447081, -0.05872123913100852]
Test Values: {Rule[k, 2], Rule[Ο•, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.8.E14 2 ⁒ ( k 2 - Ξ± 2 ) ⁒ Ξ  ⁑ ( Ο• , Ξ± 2 , k ) = Ο‰ 2 - Ξ± 2 1 + k β€² ⁒ Ξ  ⁑ ( Ο• 1 , Ξ± 1 2 , k 1 ) + k 2 ⁒ F ⁑ ( Ο• , k ) - ( 1 + k β€² ) ⁒ Ξ± 1 2 ⁒ R C ⁑ ( c 1 , c 1 - Ξ± 1 2 ) 2 superscript π‘˜ 2 superscript 𝛼 2 elliptic-integral-third-kind-Pi italic-Ο• superscript 𝛼 2 π‘˜ superscript πœ” 2 superscript 𝛼 2 1 superscript π‘˜ β€² elliptic-integral-third-kind-Pi subscript italic-Ο• 1 superscript subscript 𝛼 1 2 subscript π‘˜ 1 superscript π‘˜ 2 elliptic-integral-first-kind-F italic-Ο• π‘˜ 1 superscript π‘˜ β€² superscript subscript 𝛼 1 2 Carlson-integral-RC subscript 𝑐 1 subscript 𝑐 1 superscript subscript 𝛼 1 2 {\displaystyle{\displaystyle 2(k^{2}-\alpha^{2})\Pi\left(\phi,\alpha^{2},k% \right)=\frac{\omega^{2}-\alpha^{2}}{1+k^{\prime}}\Pi\left(\phi_{1},\alpha_{1}% ^{2},k_{1}\right)+k^{2}F\left(\phi,k\right)-{(1+k^{\prime})\alpha_{1}^{2}R_{C}% \left(c_{1},c_{1}-\alpha_{1}^{2}\right)}}}
2(k^{2}-\alpha^{2})\incellintPik@{\phi}{\alpha^{2}}{k} = \frac{\omega^{2}-\alpha^{2}}{1+k^{\prime}}\incellintPik@{\phi_{1}}{\alpha_{1}^{2}}{k_{1}}+k^{2}\incellintFk@{\phi}{k}-{(1+k^{\prime})\alpha_{1}^{2}\CarlsonellintRC@{c_{1}}{c_{1}-\alpha_{1}^{2}}}		 

Error

2*((k)^(2)- \[Alpha]^(2))*EllipticPi[\[Alpha]^(2), \[Phi],(k)^2] == Divide[\[Omega]^(2)- \[Alpha]^(2),1 +Sqrt[1 - (k)^(2)]]*EllipticPi[(Subscript[\[Alpha], 1])^(2), \[Phi]+ ArcTan[Sqrt[1 - (k)^(2)]*Tan[\[Phi]]],(Subscript[k, 1])^2]+ (k)^(2)* EllipticF[\[Phi], (k)^2]-(1 +Sqrt[1 - (k)^(2)])*(Subscript[\[Alpha], 1])^(2)*1/Sqrt[((Csc[Subscript[\[Phi], 1]])^(2))- (Subscript[\[Alpha], 1])^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-((Csc[Subscript[\[Phi], 1]])^(2))/(((Csc[Subscript[\[Phi], 1]])^(2))- (Subscript[\[Alpha], 1])^(2))]
Missing Macro Error Aborted -
Failed [300 / 300]
Result: Complex[-1.4115811709537147, -1.2227387134851169]
Test Values: {Rule[k, 1], Rule[Ξ±, 1.5], Rule[Ο•, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Ο‰, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[Ξ±, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[1.5976966939439394, -1.230515427208163]
Test Values: {Rule[k, 2], Rule[Ξ±, 1.5], Rule[Ο•, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Ο‰, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[Ξ±, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.8#Ex17 F ⁑ ( Ο• , k ) = 2 1 + k ⁒ F ⁑ ( Ο• 2 , k 2 ) elliptic-integral-first-kind-F italic-Ο• π‘˜ 2 1 π‘˜ elliptic-integral-first-kind-F subscript italic-Ο• 2 subscript π‘˜ 2 {\displaystyle{\displaystyle F\left(\phi,k\right)=\frac{2}{1+k}F\left(\phi_{2}% ,k_{2}\right)}}
\incellintFk@{\phi}{k} = \frac{2}{1+k}\incellintFk@{\phi_{2}}{k_{2}}		 

EllipticF(sin(phi), k) = (2)/(1 + k)*EllipticF(sin(phi[2]), k[2])

EllipticF[\[Phi], (k)^2] == Divide[2,1 + k]*EllipticF[Subscript[\[Phi], 2], (Subscript[k, 2])^2]
Failure Failure
Failed [300 / 300]
Result: .716161018e-1+.1278882161*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, k[2] = 1/2*3^(1/2)+1/2*I, phi[2] = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -.163142760e-1+.3519262665*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, k[2] = 1/2*3^(1/2)+1/2*I, phi[2] = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.0030858847214221274, 0.01883659064247678]
Test Values: {Rule[k, 1], Rule[Ο•, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[Ο•, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[0.11075679050380455, 0.16335572999260056]
Test Values: {Rule[k, 2], Rule[Ο•, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[Ο•, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.8#Ex18 E ⁑ ( Ο• , k ) = ( 1 + k ) ⁒ E ⁑ ( Ο• 2 , k 2 ) + ( 1 - k ) ⁒ F ⁑ ( Ο• 2 , k 2 ) - k ⁒ sin ⁑ Ο• elliptic-integral-second-kind-E italic-Ο• π‘˜ 1 π‘˜ elliptic-integral-second-kind-E subscript italic-Ο• 2 subscript π‘˜ 2 1 π‘˜ elliptic-integral-first-kind-F subscript italic-Ο• 2 subscript π‘˜ 2 π‘˜ italic-Ο• {\displaystyle{\displaystyle E\left(\phi,k\right)=(1+k)E\left(\phi_{2},k_{2}% \right)+(1-k)F\left(\phi_{2},k_{2}\right)-k\sin\phi}}
\incellintEk@{\phi}{k} = (1+k)\incellintEk@{\phi_{2}}{k_{2}}+(1-k)\incellintFk@{\phi_{2}}{k_{2}}-k\sin@@{\phi}		 

EllipticE(sin(phi), k) = (1 + k)*EllipticE(sin(phi[2]), k[2])+(1 - k)*EllipticF(sin(phi[2]), k[2])- k*sin(phi)

EllipticE[\[Phi], (k)^2] == (1 + k)*EllipticE[Subscript[\[Phi], 2], (Subscript[k, 2])^2]+(1 - k)*EllipticF[Subscript[\[Phi], 2], (Subscript[k, 2])^2]- k*Sin[\[Phi]]
Failure Failure
Failed [300 / 300]
Result: -.251128463-.1652679776*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, k[2] = 1/2*3^(1/2)+1/2*I, phi[2] = 1/2*3^(1/2)+1/2*I, k = 1}


Result: .549972877-.903450862e-1*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, k[2] = 1/2*3^(1/2)+1/2*I, phi[2] = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.009026229866885283, -0.03603907810261833]
Test Values: {Rule[k, 1], Rule[Ο•, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[Ο•, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[0.42447097038130677, 0.1345883883024661]
Test Values: {Rule[k, 2], Rule[Ο•, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[Ο•, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.8#Ex22 F ⁑ ( Ο• , k ) = ( 1 + k 1 ) ⁒ F ⁑ ( ψ 1 , k 1 ) elliptic-integral-first-kind-F italic-Ο• π‘˜ 1 subscript π‘˜ 1 elliptic-integral-first-kind-F subscript πœ“ 1 subscript π‘˜ 1 {\displaystyle{\displaystyle F\left(\phi,k\right)=(1+k_{1})F\left(\psi_{1},k_{% 1}\right)}}
\incellintFk@{\phi}{k} = (1+k_{1})\incellintFk@{\psi_{1}}{k_{1}}		 

EllipticF(sin(phi), k) = (1 + k[1])*EllipticF(sin(psi[1]), k[1])

EllipticF[\[Phi], (k)^2] == (1 + Subscript[k, 1])*EllipticF[Subscript[\[Psi], 1], (Subscript[k, 1])^2]
Failure Failure
Failed [299 / 300]
Result: -.3025119160-.7226109033*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k[1] = 1/2*3^(1/2)+1/2*I, psi[1] = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -.6401936029-.6817361311*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k[1] = 1/2*3^(1/2)+1/2*I, psi[1] = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [299 / 300]
Result: Complex[-0.11940620612760577, -0.19771875715838422]
Test Values: {Rule[k, 1], Rule[Ο•, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[ψ, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-0.15464125790413003, -0.13739720920586462]
Test Values: {Rule[k, 2], Rule[Ο•, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[ψ, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.8#Ex23 E ⁑ ( Ο• , k ) = ( 1 + k β€² ) ⁒ E ⁑ ( ψ 1 , k 1 ) - k β€² ⁒ F ⁑ ( Ο• , k ) + ( 1 - Ξ” ) ⁒ cot ⁑ Ο• elliptic-integral-second-kind-E italic-Ο• π‘˜ 1 superscript π‘˜ β€² elliptic-integral-second-kind-E subscript πœ“ 1 subscript π‘˜ 1 superscript π‘˜ β€² elliptic-integral-first-kind-F italic-Ο• π‘˜ 1 Ξ” italic-Ο• {\displaystyle{\displaystyle E\left(\phi,k\right)=(1+k^{\prime})E\left(\psi_{1% },k_{1}\right)-k^{\prime}F\left(\phi,k\right)+(1-\Delta)\cot\phi}}
\incellintEk@{\phi}{k} = (1+k^{\prime})\incellintEk@{\psi_{1}}{k_{1}}-k^{\prime}\incellintFk@{\phi}{k}+(1-\Delta)\cot@@{\phi}		 

EllipticE(sin(phi), k) = (1 +sqrt(1 - (k)^(2)))*EllipticE(sin(psi[1]), k[1])-sqrt(1 - (k)^(2))*EllipticF(sin(phi), k)+(1 -(sqrt(1 - (k)^(2)* (sin(phi))^(2))))*cot(phi)

EllipticE[\[Phi], (k)^2] == (1 +Sqrt[1 - (k)^(2)])*EllipticE[Subscript[\[Psi], 1], (Subscript[k, 1])^2]-Sqrt[1 - (k)^(2)]*EllipticF[\[Phi], (k)^2]+(1 -(Sqrt[1 - (k)^(2)* (Sin[\[Phi]])^(2)]))*Cot[\[Phi]]
Failure Failure
Failed [300 / 300]
Result: -.5555013192-.1267358774*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k[1] = 1/2*3^(1/2)+1/2*I, psi[1] = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -1.589246368-2.046785663*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k[1] = 1/2*3^(1/2)+1/2*I, psi[1] = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.22091089534718378, -0.1170454776590783]
Test Values: {Rule[k, 1], Rule[Ο•, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[ψ, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-0.9299237807056446, -0.7272990802320405]
Test Values: {Rule[k, 2], Rule[Ο•, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[ψ, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.8.E20 ρ ⁒ Ξ  ⁑ ( Ο• , Ξ± 2 , k ) = 4 1 + k β€² ⁒ Ξ  ⁑ ( ψ 1 , Ξ± 1 2 , k 1 ) + ( ρ - 1 ) ⁒ F ⁑ ( Ο• , k ) - R C ⁑ ( c - 1 , c - Ξ± 2 ) 𝜌 elliptic-integral-third-kind-Pi italic-Ο• superscript 𝛼 2 π‘˜ 4 1 superscript π‘˜ β€² elliptic-integral-third-kind-Pi subscript πœ“ 1 superscript subscript 𝛼 1 2 subscript π‘˜ 1 𝜌 1 elliptic-integral-first-kind-F italic-Ο• π‘˜ Carlson-integral-RC 𝑐 1 𝑐 superscript 𝛼 2 {\displaystyle{\displaystyle\rho\Pi\left(\phi,\alpha^{2},k\right)=\frac{4}{1+k% ^{\prime}}\Pi\left(\psi_{1},\alpha_{1}^{2},k_{1}\right)+(\rho-1)F\left(\phi,k% \right)-R_{C}\left(c-1,c-\alpha^{2}\right)}}
\rho\incellintPik@{\phi}{\alpha^{2}}{k} = \frac{4}{1+k^{\prime}}\incellintPik@{\psi_{1}}{\alpha_{1}^{2}}{k_{1}}+(\rho-1)\incellintFk@{\phi}{k}-\CarlsonellintRC@{c-1}{c-\alpha^{2}}		 

Error

\[Rho]*EllipticPi[\[Alpha]^(2), \[Phi],(k)^2] == Divide[4,1 +Sqrt[1 - (k)^(2)]]*EllipticPi[(Subscript[\[Alpha], 1])^(2), Subscript[\[Psi], 1],(Subscript[k, 1])^2]+(\[Rho]- 1)*EllipticF[\[Phi], (k)^2]- 1/Sqrt[((Csc[\[Phi]])^(2))- \[Alpha]^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-(((Csc[\[Phi]])^(2))- 1)/(((Csc[\[Phi]])^(2))- \[Alpha]^(2))]
Missing Macro Error Failure - Skipped - Because timed out
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