Elliptic Integrals - 19.7 Connection Formulas

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
19.7.E1 E ⁡ ( k ) ⁢ K ′ ⁡ ( k ) + E ′ ⁡ ( k ) ⁢ K ⁡ ( k ) - K ⁡ ( k ) ⁢ K ′ ⁡ ( k ) = 1 2 ⁢ π complete-elliptic-integral-second-kind-E 𝑘 complementary-complete-elliptic-integral-first-kind-K 𝑘 complementary-complete-elliptic-integral-second-kind-E 𝑘 complete-elliptic-integral-first-kind-K 𝑘 complete-elliptic-integral-first-kind-K 𝑘 complementary-complete-elliptic-integral-first-kind-K 𝑘 1 2 𝜋 {\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}}
\compellintEk@{k}\ccompellintKk@{k}+\ccompellintEk@{k}\compellintKk@{k}-\compellintKk@{k}\ccompellintKk@{k} = \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
Failed [1 / 3]
Result: Indeterminate
Test Values: {Rule[k, 1]}

19.7#Ex1 K ⁡ ( i ⁢ k / k ′ ) = k ′ ⁢ K ⁡ ( k ) complete-elliptic-integral-first-kind-K 𝑖 𝑘 superscript 𝑘 ′ superscript 𝑘 ′ complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle K\left(ik/k^{\prime}\right)=k^{\prime}K\left(k% \right)}}
\compellintKk@{ik/k^{\prime}} = k^{\prime}\compellintKk@{k}		 

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
Failed [3 / 3]
Result: Indeterminate
Test Values: {Rule[k, 1]}


Result: Complex[-2.220446049250313*^-16, -2.9198052634126777]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.7#Ex2 K ⁡ ( - i ⁢ k ′ / k ) = k ⁢ K ⁡ ( k ′ ) complete-elliptic-integral-first-kind-K 𝑖 superscript 𝑘 ′ 𝑘 𝑘 complete-elliptic-integral-first-kind-K superscript 𝑘 ′ {\displaystyle{\displaystyle K\left(-ik^{\prime}/k\right)=kK\left(k^{\prime}% \right)}}
\compellintKk@{-ik^{\prime}/k} = k\compellintKk@{k^{\prime}}		 

EllipticK(- I*sqrt(1 - (k)^(2))/k) = k*EllipticK(sqrt(1 - (k)^(2)))

EllipticK[(- I*Sqrt[1 - (k)^(2)]/k)^2] == k*EllipticK[(Sqrt[1 - (k)^(2)])^2]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
19.7#Ex3 E ⁡ ( i ⁢ k / k ′ ) = ( 1 / k ′ ) ⁢ E ⁡ ( k ) complete-elliptic-integral-second-kind-E 𝑖 𝑘 superscript 𝑘 ′ 1 superscript 𝑘 ′ complete-elliptic-integral-second-kind-E 𝑘 {\displaystyle{\displaystyle E\left(ik/k^{\prime}\right)=(1/k^{\prime})E\left(% k\right)}}
\compellintEk@{ik/k^{\prime}} = (1/k^{\prime})\compellintEk@{k}		 

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
Failed [3 / 3]
Result: Float(infinity)+Float(infinity)*I
Test Values: {k = 1}


Result: .6e-9+.4691535424*I
Test Values: {k = 2}

... skip entries to safe data
Failed [3 / 3]
Result: Indeterminate
Test Values: {Rule[k, 1]}


Result: Complex[-5.551115123125783*^-16, 0.46915354293820644]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.7#Ex4 E ⁡ ( - i ⁢ k ′ / k ) = ( 1 / k ) ⁢ E ⁡ ( k ′ ) complete-elliptic-integral-second-kind-E 𝑖 superscript 𝑘 ′ 𝑘 1 𝑘 complete-elliptic-integral-second-kind-E superscript 𝑘 ′ {\displaystyle{\displaystyle E\left(-ik^{\prime}/k\right)=(1/k)E\left(k^{% \prime}\right)}}
\compellintEk@{-ik^{\prime}/k} = (1/k)\compellintEk@{k^{\prime}}		 

EllipticE(- I*sqrt(1 - (k)^(2))/k) = (1/k)*EllipticE(sqrt(1 - (k)^(2)))

EllipticE[(- I*Sqrt[1 - (k)^(2)]/k)^2] == (1/k)*EllipticE[(Sqrt[1 - (k)^(2)])^2]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
19.7#Ex5 K ⁡ ( 1 / k ) = k ⁢ ( K ⁡ ( k ) - i ⁢ K ⁡ ( k ′ ) ) complete-elliptic-integral-first-kind-K 1 𝑘 𝑘 complete-elliptic-integral-first-kind-K 𝑘 imaginary-unit complete-elliptic-integral-first-kind-K superscript 𝑘 ′ {\displaystyle{\displaystyle K\left(1/k\right)=k(K\left(k\right)-\mathrm{i}K% \left(k^{\prime}\right))}}
\compellintKk@{1/k} = k(\compellintKk@{k}-\iunit\compellintKk@{k^{\prime}})		 

EllipticK(1/k) = k*(EllipticK(k)- I*EllipticK(sqrt(1 - (k)^(2))))

EllipticK[(1/k)^2] == k*(EllipticK[(k)^2]- I*EllipticK[(Sqrt[1 - (k)^(2)])^2])
Failure Failure Error
Failed [3 / 3]
Result: Indeterminate
Test Values: {Rule[k, 1]}


Result: Complex[-2.220446049250313*^-16, 4.313031294999287]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.7#Ex5 K ⁡ ( 1 / k ) = k ⁢ ( K ⁡ ( k ) + i ⁢ K ⁡ ( k ′ ) ) complete-elliptic-integral-first-kind-K 1 𝑘 𝑘 complete-elliptic-integral-first-kind-K 𝑘 imaginary-unit complete-elliptic-integral-first-kind-K superscript 𝑘 ′ {\displaystyle{\displaystyle K\left(1/k\right)=k(K\left(k\right)+\mathrm{i}K% \left(k^{\prime}\right))}}
\compellintKk@{1/k} = k(\compellintKk@{k}+\iunit\compellintKk@{k^{\prime}})		 

EllipticK(1/k) = k*(EllipticK(k)+ I*EllipticK(sqrt(1 - (k)^(2))))

EllipticK[(1/k)^2] == k*(EllipticK[(k)^2]+ I*EllipticK[(Sqrt[1 - (k)^(2)])^2])
Failure Failure Error
Failed [1 / 3]
Result: Indeterminate
Test Values: {Rule[k, 1]}

19.7#Ex6 K ⁡ ( 1 / k ′ ) = k ′ ⁢ ( K ⁡ ( k ′ ) + i ⁢ K ⁡ ( k ) ) complete-elliptic-integral-first-kind-K 1 superscript 𝑘 ′ superscript 𝑘 ′ complete-elliptic-integral-first-kind-K superscript 𝑘 ′ imaginary-unit complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle K\left(1/k^{\prime}\right)=k^{\prime}(K\left(k^{% \prime}\right)+\mathrm{i}K\left(k\right))}}
\compellintKk@{1/k^{\prime}} = k^{\prime}(\compellintKk@{k^{\prime}}+\iunit\compellintKk@{k})		 

EllipticK(1/(sqrt(1 - (k)^(2)))) = sqrt(1 - (k)^(2))*(EllipticK(sqrt(1 - (k)^(2)))+ I*EllipticK(k))

EllipticK[(1/(Sqrt[1 - (k)^(2)]))^2] == Sqrt[1 - (k)^(2)]*(EllipticK[(Sqrt[1 - (k)^(2)])^2]+ I*EllipticK[(k)^2])
Failure Failure Error
Failed [3 / 3]
Result: Indeterminate
Test Values: {Rule[k, 1]}


Result: Complex[2.9198052634126785, -3.7351946687866775]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.7#Ex6 K ⁡ ( 1 / k ′ ) = k ′ ⁢ ( K ⁡ ( k ′ ) - i ⁢ K ⁡ ( k ) ) complete-elliptic-integral-first-kind-K 1 superscript 𝑘 ′ superscript 𝑘 ′ complete-elliptic-integral-first-kind-K superscript 𝑘 ′ imaginary-unit complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle K\left(1/k^{\prime}\right)=k^{\prime}(K\left(k^{% \prime}\right)-\mathrm{i}K\left(k\right))}}
\compellintKk@{1/k^{\prime}} = k^{\prime}(\compellintKk@{k^{\prime}}-\iunit\compellintKk@{k})		 

EllipticK(1/(sqrt(1 - (k)^(2)))) = sqrt(1 - (k)^(2))*(EllipticK(sqrt(1 - (k)^(2)))- I*EllipticK(k))

EllipticK[(1/(Sqrt[1 - (k)^(2)]))^2] == Sqrt[1 - (k)^(2)]*(EllipticK[(Sqrt[1 - (k)^(2)])^2]- I*EllipticK[(k)^2])
Failure Failure Error
Failed [1 / 3]
Result: Indeterminate
Test Values: {Rule[k, 1]}

19.7#Ex7 E ⁡ ( 1 / k ) = ( 1 / k ) ⁢ ( E ⁡ ( k ) + i ⁢ E ⁡ ( k ′ ) - k ′ 2 ⁢ K ⁡ ( k ) - i ⁢ k 2 ⁢ K ⁡ ( k ′ ) ) complete-elliptic-integral-second-kind-E 1 𝑘 1 𝑘 complete-elliptic-integral-second-kind-E 𝑘 imaginary-unit complete-elliptic-integral-second-kind-E superscript 𝑘 ′ superscript superscript 𝑘 ′ 2 complete-elliptic-integral-first-kind-K 𝑘 imaginary-unit superscript 𝑘 2 complete-elliptic-integral-first-kind-K superscript 𝑘 ′ {\displaystyle{\displaystyle E\left(1/k\right)=(1/k)\left(E\left(k\right)+% \mathrm{i}E\left(k^{\prime}\right)-{k^{\prime}}^{2}K\left(k\right)-\mathrm{i}k% ^{2}K\left(k^{\prime}\right)\right)}}
\compellintEk@{1/k} = (1/k)\left(\compellintEk@{k}+\iunit\compellintEk@{k^{\prime}}-{k^{\prime}}^{2}\compellintKk@{k}-\iunit k^{2}\compellintKk@{k^{\prime}}\right)		 

EllipticE(1/k) = (1/k)*(EllipticE(k)+ I*EllipticE(sqrt(1 - (k)^(2)))-1 - (k)^(2)*EllipticK(k)- I*(k)^(2)* EllipticK(sqrt(1 - (k)^(2))))

EllipticE[(1/k)^2] == (1/k)*(EllipticE[(k)^2]+ I*EllipticE[(Sqrt[1 - (k)^(2)])^2]-1 - (k)^(2)*EllipticK[(k)^2]- I*(k)^(2)* EllipticK[(Sqrt[1 - (k)^(2)])^2])
Failure Failure Error
Failed [3 / 3]
Result: DirectedInfinity[]
Test Values: {Rule[k, 1]}


Result: Complex[3.4500631209220436, -1.8829831432620088]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.7#Ex7 E ⁡ ( 1 / k ) = ( 1 / k ) ⁢ ( E ⁡ ( k ) - i ⁢ E ⁡ ( k ′ ) - k ′ 2 ⁢ K ⁡ ( k ) + i ⁢ k 2 ⁢ K ⁡ ( k ′ ) ) complete-elliptic-integral-second-kind-E 1 𝑘 1 𝑘 complete-elliptic-integral-second-kind-E 𝑘 imaginary-unit complete-elliptic-integral-second-kind-E superscript 𝑘 ′ superscript superscript 𝑘 ′ 2 complete-elliptic-integral-first-kind-K 𝑘 imaginary-unit superscript 𝑘 2 complete-elliptic-integral-first-kind-K superscript 𝑘 ′ {\displaystyle{\displaystyle E\left(1/k\right)=(1/k)\left(E\left(k\right)-% \mathrm{i}E\left(k^{\prime}\right)-{k^{\prime}}^{2}K\left(k\right)+\mathrm{i}k% ^{2}K\left(k^{\prime}\right)\right)}}
\compellintEk@{1/k} = (1/k)\left(\compellintEk@{k}-\iunit\compellintEk@{k^{\prime}}-{k^{\prime}}^{2}\compellintKk@{k}+\iunit k^{2}\compellintKk@{k^{\prime}}\right)		 

EllipticE(1/k) = (1/k)*(EllipticE(k)- I*EllipticE(sqrt(1 - (k)^(2)))-1 - (k)^(2)*EllipticK(k)+ I*(k)^(2)* EllipticK(sqrt(1 - (k)^(2))))

EllipticE[(1/k)^2] == (1/k)*(EllipticE[(k)^2]- I*EllipticE[(Sqrt[1 - (k)^(2)])^2]-1 - (k)^(2)*EllipticK[(k)^2]+ I*(k)^(2)* EllipticK[(Sqrt[1 - (k)^(2)])^2])
Failure Failure Error
Failed [3 / 3]
Result: DirectedInfinity[]
Test Values: {Rule[k, 1]}


Result: Complex[3.4500631209220436, -3.773902383124376]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.7#Ex8 E ⁡ ( 1 / k ′ ) = ( 1 / k ′ ) ⁢ ( E ⁡ ( k ′ ) - i ⁢ E ⁡ ( k ) - k 2 ⁢ K ⁡ ( k ′ ) + i ⁢ k ′ 2 ⁢ K ⁡ ( k ) ) complete-elliptic-integral-second-kind-E 1 superscript 𝑘 ′ 1 superscript 𝑘 ′ complete-elliptic-integral-second-kind-E superscript 𝑘 ′ imaginary-unit complete-elliptic-integral-second-kind-E 𝑘 superscript 𝑘 2 complete-elliptic-integral-first-kind-K superscript 𝑘 ′ imaginary-unit superscript superscript 𝑘 ′ 2 complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle E\left(1/k^{\prime}\right)=(1/k^{\prime})\left(E% \left(k^{\prime}\right)-\mathrm{i}E\left(k\right)-k^{2}K\left(k^{\prime}\right% )+\mathrm{i}{k^{\prime}}^{2}K\left(k\right)\right)}}
\compellintEk@{1/k^{\prime}} = (1/k^{\prime})\left(\compellintEk@{k^{\prime}}-\iunit\compellintEk@{k}-k^{2}\compellintKk@{k^{\prime}}+\iunit{k^{\prime}}^{2}\compellintKk@{k}\right)		 

EllipticE(1/(sqrt(1 - (k)^(2)))) = (1/(sqrt(1 - (k)^(2))))*(EllipticE(sqrt(1 - (k)^(2)))- I*EllipticE(k)- (k)^(2)* EllipticK(sqrt(1 - (k)^(2)))+ I*1 - (k)^(2)*EllipticK(k))

EllipticE[(1/(Sqrt[1 - (k)^(2)]))^2] == (1/(Sqrt[1 - (k)^(2)]))*(EllipticE[(Sqrt[1 - (k)^(2)])^2]- I*EllipticE[(k)^2]- (k)^(2)* EllipticK[(Sqrt[1 - (k)^(2)])^2]+ I*1 - (k)^(2)*EllipticK[(k)^2])
Failure Failure Error
Failed [3 / 3]
Result: Indeterminate
Test Values: {Rule[k, 1]}


Result: Complex[-1.1384238737361991, -2.262384972182541]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.7#Ex8 E ⁡ ( 1 / k ′ ) = ( 1 / k ′ ) ⁢ ( E ⁡ ( k ′ ) + i ⁢ E ⁡ ( k ) - k 2 ⁢ K ⁡ ( k ′ ) - i ⁢ k ′ 2 ⁢ K ⁡ ( k ) ) complete-elliptic-integral-second-kind-E 1 superscript 𝑘 ′ 1 superscript 𝑘 ′ complete-elliptic-integral-second-kind-E superscript 𝑘 ′ imaginary-unit complete-elliptic-integral-second-kind-E 𝑘 superscript 𝑘 2 complete-elliptic-integral-first-kind-K superscript 𝑘 ′ imaginary-unit superscript superscript 𝑘 ′ 2 complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle E\left(1/k^{\prime}\right)=(1/k^{\prime})\left(E% \left(k^{\prime}\right)+\mathrm{i}E\left(k\right)-k^{2}K\left(k^{\prime}\right% )-\mathrm{i}{k^{\prime}}^{2}K\left(k\right)\right)}}
\compellintEk@{1/k^{\prime}} = (1/k^{\prime})\left(\compellintEk@{k^{\prime}}+\iunit\compellintEk@{k}-k^{2}\compellintKk@{k^{\prime}}-\iunit{k^{\prime}}^{2}\compellintKk@{k}\right)		 

EllipticE(1/(sqrt(1 - (k)^(2)))) = (1/(sqrt(1 - (k)^(2))))*(EllipticE(sqrt(1 - (k)^(2)))+ I*EllipticE(k)- (k)^(2)* EllipticK(sqrt(1 - (k)^(2)))- I*1 - (k)^(2)*EllipticK(k))

EllipticE[(1/(Sqrt[1 - (k)^(2)]))^2] == (1/(Sqrt[1 - (k)^(2)]))*(EllipticE[(Sqrt[1 - (k)^(2)])^2]+ I*EllipticE[(k)^2]- (k)^(2)* EllipticK[(Sqrt[1 - (k)^(2)])^2]- I*1 - (k)^(2)*EllipticK[(k)^2])
Failure Failure Error
Failed [3 / 3]
Result: Indeterminate
Test Values: {Rule[k, 1]}


Result: Complex[-0.45287687829515355, -3.814134176668458]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.7#Ex19 F ⁡ ( i ⁢ ϕ , k ) = i ⁢ F ⁡ ( ψ , k ′ ) elliptic-integral-first-kind-F 𝑖 italic-ϕ 𝑘 𝑖 elliptic-integral-first-kind-F 𝜓 superscript 𝑘 ′ {\displaystyle{\displaystyle F\left(i\phi,k\right)=iF\left(\psi,k^{\prime}% \right)}}
\incellintFk@{i\phi}{k} = i\incellintFk@{\psi}{k^{\prime}}		 

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
Failed [300 / 300]
Result: .1428695990-.263545696e-1*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k = 1}


Result: .749290340e-1-.334629029e-1*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.020142137049999537, -0.0010462389457662757]
Test Values: {Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[0.015860617706546204, -0.003938067237051424]
Test Values: {Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.7#Ex20 E ⁡ ( i ⁢ ϕ , k ) = i ⁢ ( F ⁡ ( ψ , k ′ ) - E ⁡ ( ψ , k ′ ) + ( tan ⁡ ψ ) ⁢ 1 - k ′ 2 ⁢ sin 2 ⁡ ψ ) elliptic-integral-second-kind-E 𝑖 italic-ϕ 𝑘 𝑖 elliptic-integral-first-kind-F 𝜓 superscript 𝑘 ′ elliptic-integral-second-kind-E 𝜓 superscript 𝑘 ′ 𝜓 1 superscript superscript 𝑘 ′ 2 2 𝜓 {\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)}}
\incellintEk@{i\phi}{k} = i\left(\incellintFk@{\psi}{k^{\prime}}-\incellintEk@{\psi}{k^{\prime}}+(\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
Failed [300 / 300]
Result: -.9970133474-.1125517221*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -2.257467281-.7782721018*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.3893501368763376, 0.20738614458301174]
Test Values: {Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-0.6710974690872284, 0.0060773305020283]
Test Values: {Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.7#Ex21 Π ⁡ ( i ⁢ ϕ , α 2 , k ) = i ⁢ ( F ⁡ ( ψ , k ′ ) - α 2 ⁢ Π ⁡ ( ψ , 1 - α 2 , k ′ ) ) / ( 1 - α 2 ) elliptic-integral-third-kind-Pi 𝑖 italic-ϕ superscript 𝛼 2 𝑘 𝑖 elliptic-integral-first-kind-F 𝜓 superscript 𝑘 ′ superscript 𝛼 2 elliptic-integral-third-kind-Pi 𝜓 1 superscript 𝛼 2 superscript 𝑘 ′ 1 superscript 𝛼 2 {\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})}}}
\incellintPik@{i\phi}{\alpha^{2}}{k} = i\left(\incellintFk@{\psi}{k^{\prime}}-\alpha^{2}\incellintPik@{\psi}{1-\alpha^{2}}{k^{\prime}}\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
Failed [292 / 300]
Result: .926834363e-2-.484444094e-1*I
Test Values: {alpha = 3/2, phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -.130749569e-2-.277524276e-1*I
Test Values: {alpha = 3/2, phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [298 / 300]
Result: Complex[0.013291772923717082, -0.006719909387202905]
Test Values: {Rule[k, 1], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[0.00953602334602252, -0.007394575555177196]
Test Values: {Rule[k, 2], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
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