Elliptic Integrals - 19.6 Special Cases

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
19.6#Ex1 K ⁡ ( 0 ) = E ⁡ ( 0 ) complete-elliptic-integral-first-kind-K 0 complete-elliptic-integral-second-kind-E 0 {\displaystyle{\displaystyle K\left(0\right)=E\left(0\right)}}
\compellintKk@{0} = \compellintEk@{0}		 

EllipticK(0) = EllipticE(0)

EllipticK[(0)^2] == EllipticE[(0)^2]
Successful Successful - Successful [Tested: 1]
19.6#Ex1 E ⁡ ( 0 ) = K ′ ⁡ ( 1 ) complete-elliptic-integral-second-kind-E 0 complementary-complete-elliptic-integral-first-kind-K 1 {\displaystyle{\displaystyle E\left(0\right)={K^{\prime}}\left(1\right)}}
\compellintEk@{0} = \ccompellintKk@{1}		 

EllipticE(0) = EllipticCK(1)

EllipticE[(0)^2] == EllipticK[1-(1)^2]
Successful Successful - Successful [Tested: 1]
19.6#Ex1 K ′ ⁡ ( 1 ) = E ′ ⁡ ( 1 ) complementary-complete-elliptic-integral-first-kind-K 1 complementary-complete-elliptic-integral-second-kind-E 1 {\displaystyle{\displaystyle{K^{\prime}}\left(1\right)={E^{\prime}}\left(1% \right)}}
\ccompellintKk@{1} = \ccompellintEk@{1}		 

EllipticCK(1) = EllipticCE(1)

EllipticK[1-(1)^2] == EllipticE[1-(1)^2]
Successful Successful - Successful [Tested: 1]
19.6#Ex1 E ′ ⁡ ( 1 ) = 1 2 ⁢ π complementary-complete-elliptic-integral-second-kind-E 1 1 2 𝜋 {\displaystyle{\displaystyle{E^{\prime}}\left(1\right)=\tfrac{1}{2}\pi}}
\ccompellintEk@{1} = \tfrac{1}{2}\pi		 

EllipticCE(1) = (1)/(2)*Pi

EllipticE[1-(1)^2] == Divide[1,2]*Pi
Successful Successful - Successful [Tested: 1]
19.6#Ex2 K ⁡ ( 1 ) = K ′ ⁡ ( 0 ) complete-elliptic-integral-first-kind-K 1 complementary-complete-elliptic-integral-first-kind-K 0 {\displaystyle{\displaystyle K\left(1\right)={K^{\prime}}\left(0\right)}}
\compellintKk@{1} = \ccompellintKk@{0}		 

EllipticK(1) = EllipticCK(0)

EllipticK[(1)^2] == EllipticK[1-(0)^2]
Error Failure -
Failed [1 / 1]
Result: Indeterminate
Test Values: {}

19.6#Ex2 K ′ ⁡ ( 0 ) = ∞ complementary-complete-elliptic-integral-first-kind-K 0 {\displaystyle{\displaystyle{K^{\prime}}\left(0\right)=\infty}}
\ccompellintKk@{0} = \infty		 

EllipticCK(0) = infinity

EllipticK[1-(0)^2] == Infinity
Error Failure -
Failed [1 / 1]
Result: Indeterminate
Test Values: {}

19.6#Ex3 E ⁡ ( 1 ) = E ′ ⁡ ( 0 ) complete-elliptic-integral-second-kind-E 1 complementary-complete-elliptic-integral-second-kind-E 0 {\displaystyle{\displaystyle E\left(1\right)={E^{\prime}}\left(0\right)}}
\compellintEk@{1} = \ccompellintEk@{0}		 

EllipticE(1) = EllipticCE(0)

EllipticE[(1)^2] == EllipticE[1-(0)^2]
Successful Successful - Successful [Tested: 1]
19.6#Ex3 E ′ ⁡ ( 0 ) = 1 complementary-complete-elliptic-integral-second-kind-E 0 1 {\displaystyle{\displaystyle{E^{\prime}}\left(0\right)=1}}
\ccompellintEk@{0} = 1		 

EllipticCE(0) = 1

EllipticE[1-(0)^2] == 1
Successful Successful - Successful [Tested: 1]
19.6#Ex4 Π ⁡ ( k 2 , k ) = E ⁡ ( k ) / k ′ 2 complete-elliptic-integral-third-kind-Pi superscript 𝑘 2 𝑘 complete-elliptic-integral-second-kind-E 𝑘 superscript superscript 𝑘 ′ 2 {\displaystyle{\displaystyle\Pi\left(k^{2},k\right)=E\left(k\right)/{k^{\prime% }}^{2}}}
\compellintPik@{k^{2}}{k} = \compellintEk@{k}/{k^{\prime}}^{2}		 k 2 < 1 superscript 𝑘 2 1 {\displaystyle{\displaystyle k^{2}<1}} 
EllipticPi((k)^(2), k) = EllipticE(k)/(1 - (k)^(2))

EllipticPi[(k)^(2), (k)^2] == EllipticE[(k)^2]/(1 - (k)^(2))
Successful Successful - Successful [Tested: 0]
19.6#Ex5 Π ⁡ ( - k , k ) = 1 4 ⁢ π ⁢ ( 1 + k ) - 1 + 1 2 ⁢ K ⁡ ( k ) complete-elliptic-integral-third-kind-Pi 𝑘 𝑘 1 4 𝜋 superscript 1 𝑘 1 1 2 complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle\Pi\left(-k,k\right)=\tfrac{1}{4}\pi(1+k)^{-1}+% \tfrac{1}{2}K\left(k\right)}}
\compellintPik@{-k}{k} = \tfrac{1}{4}\pi(1+k)^{-1}+\tfrac{1}{2}\compellintKk@{k}		 0 ≤ k 2 , k 2 < 1 formulae-sequence 0 superscript 𝑘 2 superscript 𝑘 2 1 {\displaystyle{\displaystyle 0\leq k^{2},k^{2}<1}} 
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 Skip - No test values generated
19.6.E3 Π ⁡ ( α 2 , 0 ) = π / ( 2 ⁢ 1 - α 2 ) , Π ⁡ ( 0 , k ) complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 0 𝜋 2 1 superscript 𝛼 2 complete-elliptic-integral-third-kind-Pi 0 𝑘 {\displaystyle{\displaystyle\Pi\left(\alpha^{2},0\right)=\pi/(2\sqrt{1-\alpha^% {2}}),\quad\Pi\left(0,k\right)}}
\compellintPik@{\alpha^{2}}{0} = \pi/(2\sqrt{1-\alpha^{2}}),\quad\compellintPik@{0}{k}		 - ∞ < α 2 , α 2 < 1 formulae-sequence superscript 𝛼 2 superscript 𝛼 2 1 {\displaystyle{\displaystyle-\infty<\alpha^{2},\alpha^{2}<1}} 
EllipticPi((alpha)^(2), 0) = Pi/(2*sqrt(1 - (alpha)^(2)))

EllipticPi[\[Alpha]^(2), (0)^2] == Pi/(2*Sqrt[1 - \[Alpha]^(2)])
Successful Failure Skip - symbolical successful subtest Error
19.6.E3 π / ( 2 ⁢ 1 - α 2 ) , Π ⁡ ( 0 , k ) = K ⁡ ( k ) 𝜋 2 1 superscript 𝛼 2 complete-elliptic-integral-third-kind-Pi 0 𝑘 complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle\pi/(2\sqrt{1-\alpha^{2}}),\quad\Pi\left(0,k\right% )=K\left(k\right)}}
\pi/(2\sqrt{1-\alpha^{2}}),\quad\compellintPik@{0}{k} = \compellintKk@{k}		 - ∞ < α 2 , α 2 < 1 formulae-sequence superscript 𝛼 2 superscript 𝛼 2 1 {\displaystyle{\displaystyle-\infty<\alpha^{2},\alpha^{2}<1}} 
Pi/(2*sqrt(1 - (alpha)^(2)))

Pi/(2*Sqrt[1 - \[Alpha]^(2)])
Failure Failure Error Error
19.6.E5 Π ⁡ ( α 2 , k ) = K ⁡ ( k ) - Π ⁡ ( k 2 / α 2 , k ) complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 𝑘 complete-elliptic-integral-first-kind-K 𝑘 complete-elliptic-integral-third-kind-Pi superscript 𝑘 2 superscript 𝛼 2 𝑘 {\displaystyle{\displaystyle\Pi\left(\alpha^{2},k\right)=K\left(k\right)-\Pi% \left(k^{2}/\alpha^{2},k\right)}}
\compellintPik@{\alpha^{2}}{k} = \compellintKk@{k}-\compellintPik@{k^{2}/\alpha^{2}}{k}		 

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
Failed [9 / 9]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[α, 1.5]}


Result: Complex[-1.593078238683172, 2.4424906541753444*^-15]
Test Values: {Rule[k, 2], Rule[α, 1.5]}

... skip entries to safe data
19.6#Ex8 Π ⁡ ( α 2 , 0 ) = 0 complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 0 0 {\displaystyle{\displaystyle\Pi\left(\alpha^{2},0\right)=0}}
\compellintPik@{\alpha^{2}}{0} = 0		 

EllipticPi((alpha)^(2), 0) = 0

EllipticPi[\[Alpha]^(2), (0)^2] == 0
Failure Failure
Failed [3 / 3]
Result: -1.404962946*I
Test Values: {alpha = 3/2}


Result: 1.813799364
Test Values: {alpha = 1/2}

... skip entries to safe data
Failed [3 / 3]
Result: Complex[0.0, -1.4049629462081452]
Test Values: {Rule[α, 1.5]}


Result: 1.813799364234218
Test Values: {Rule[α, 0.5]}

... skip entries to safe data
19.6#Ex11 F ⁡ ( 0 , k ) = 0 elliptic-integral-first-kind-F 0 𝑘 0 {\displaystyle{\displaystyle F\left(0,k\right)=0}}
\incellintFk@{0}{k} = 0		 

EllipticF(sin(0), k) = 0

EllipticF[0, (k)^2] == 0
Successful Successful - Successful [Tested: 3]
19.6#Ex12 F ⁡ ( ϕ , 0 ) = ϕ elliptic-integral-first-kind-F italic-ϕ 0 italic-ϕ {\displaystyle{\displaystyle F\left(\phi,0\right)=\phi}}
\incellintFk@{\phi}{0} = \phi		 

EllipticF(sin(phi), 0) = phi

EllipticF[\[Phi], (0)^2] == \[Phi]
Failure Successful
Failed [2 / 10]
Result: .858407346
Test Values: {phi = -2}


Result: -.858407346
Test Values: {phi = 2}

Successful [Tested: 10]
19.6#Ex13 F ⁡ ( 1 2 ⁢ π , 1 ) = ∞ elliptic-integral-first-kind-F 1 2 𝜋 1 {\displaystyle{\displaystyle F\left(\tfrac{1}{2}\pi,1\right)=\infty}}
\incellintFk@{\tfrac{1}{2}\pi}{1} = \infty		 

EllipticF(sin((1)/(2)*Pi), 1) = infinity

EllipticF[Divide[1,2]*Pi, (1)^2] == Infinity
Error Failure -
Failed [1 / 1]
Result: Indeterminate
Test Values: {}

19.6#Ex14 F ⁡ ( 1 2 ⁢ π , k ) = K ⁡ ( k ) elliptic-integral-first-kind-F 1 2 𝜋 𝑘 complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle F\left(\tfrac{1}{2}\pi,k\right)=K\left(k\right)}}
\incellintFk@{\tfrac{1}{2}\pi}{k} = \compellintKk@{k}		 

EllipticF(sin((1)/(2)*Pi), k) = EllipticK(k)

EllipticF[Divide[1,2]*Pi, (k)^2] == EllipticK[(k)^2]
Successful Successful - Successful [Tested: 3]
19.6#Ex15 lim ϕ → 0 ⁡ F ⁡ ( ϕ , k ) / ϕ = 1 subscript → italic-ϕ 0 elliptic-integral-first-kind-F italic-ϕ 𝑘 italic-ϕ 1 {\displaystyle{\displaystyle\lim_{\phi\to 0}\ifrac{F\left(\phi,k\right)}{\phi}% =1}}
\lim_{\phi\to 0}\ifrac{\incellintFk@{\phi}{k}}{\phi} = 1		 

limit((EllipticF(sin(phi), k))/(phi), phi = 0) = 1

Limit[Divide[EllipticF[\[Phi], (k)^2],\[Phi]], \[Phi] -> 0, GenerateConditions->None] == 1
Successful Successful - Successful [Tested: 3]
19.6.E8 F ⁡ ( ϕ , 1 ) = ( sin ⁡ ϕ ) ⁢ R C ⁡ ( 1 , cos 2 ⁡ ϕ ) elliptic-integral-first-kind-F italic-ϕ 1 italic-ϕ Carlson-integral-RC 1 2 italic-ϕ {\displaystyle{\displaystyle F\left(\phi,1\right)=(\sin\phi)R_{C}\left(1,{\cos% ^{2}}\phi\right)}}
\incellintFk@{\phi}{1} = (\sin@@{\phi})\CarlsonellintRC@{1}{\cos^{2}@@{\phi}}		 

Error

EllipticF[\[Phi], (1)^2] == (Sin[\[Phi]])*1/Sqrt[(Cos[\[Phi]])^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-(1)/((Cos[\[Phi]])^(2))]
Missing Macro Error Failure -
Failed [2 / 10]
Result: DirectedInfinity[]
Test Values: {Rule[ϕ, -2]}


Result: DirectedInfinity[]
Test Values: {Rule[ϕ, 2]}

19.6.E8 ( sin ⁡ ϕ ) ⁢ R C ⁡ ( 1 , cos 2 ⁡ ϕ ) = gd - 1 ⁡ ( ϕ ) italic-ϕ Carlson-integral-RC 1 2 italic-ϕ inverse-Gudermannian italic-ϕ {\displaystyle{\displaystyle(\sin\phi)R_{C}\left(1,{\cos^{2}}\phi\right)={% \operatorname{gd}^{-1}}\left(\phi\right)}}
(\sin@@{\phi})\CarlsonellintRC@{1}{\cos^{2}@@{\phi}} = \aGudermannian@{\phi}		 - 1 2 ⁢ π < ( ϕ ) , ( ϕ ) < 1 2 ⁢ π formulae-sequence 1 2 𝜋 italic-ϕ italic-ϕ 1 2 𝜋 {\displaystyle{\displaystyle-\frac{1}{2}\pi<(\phi),(\phi)<\frac{1}{2}\pi}} 
Error

(Sin[\[Phi]])*1/Sqrt[(Cos[\[Phi]])^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-(1)/((Cos[\[Phi]])^(2))] == InverseGudermannian[\[Phi]]
Missing Macro Error Failure - Successful [Tested: 4]
19.6#Ex16 E ⁡ ( 0 , k ) = 0 elliptic-integral-second-kind-E 0 𝑘 0 {\displaystyle{\displaystyle E\left(0,k\right)=0}}
\incellintEk@{0}{k} = 0		 

EllipticE(sin(0), k) = 0

EllipticE[0, (k)^2] == 0
Successful Successful - Successful [Tested: 3]
19.6#Ex17 E ⁡ ( ϕ , 0 ) = ϕ elliptic-integral-second-kind-E italic-ϕ 0 italic-ϕ {\displaystyle{\displaystyle E\left(\phi,0\right)=\phi}}
\incellintEk@{\phi}{0} = \phi		 

EllipticE(sin(phi), 0) = phi

EllipticE[\[Phi], (0)^2] == \[Phi]
Failure Successful
Failed [2 / 10]
Result: .858407346
Test Values: {phi = -2}


Result: -.858407346
Test Values: {phi = 2}

Successful [Tested: 10]
19.6#Ex18 E ⁡ ( 1 2 ⁢ π , 1 ) = 1 elliptic-integral-second-kind-E 1 2 𝜋 1 1 {\displaystyle{\displaystyle E\left(\tfrac{1}{2}\pi,1\right)=1}}
\incellintEk@{\tfrac{1}{2}\pi}{1} = 1		 

EllipticE(sin((1)/(2)*Pi), 1) = 1

EllipticE[Divide[1,2]*Pi, (1)^2] == 1
Successful Successful - Successful [Tested: 1]
19.6#Ex19 E ⁡ ( ϕ , 1 ) = sin ⁡ ϕ elliptic-integral-second-kind-E italic-ϕ 1 italic-ϕ {\displaystyle{\displaystyle E\left(\phi,1\right)=\sin\phi}}
\incellintEk@{\phi}{1} = \sin@@{\phi}		 

EllipticE(sin(phi), 1) = sin(phi)

EllipticE[\[Phi], (1)^2] == Sin[\[Phi]]
Successful Failure -
Failed [2 / 10]
Result: -0.1814051463486368
Test Values: {Rule[ϕ, -2]}


Result: 0.1814051463486368
Test Values: {Rule[ϕ, 2]}

19.6#Ex20 E ⁡ ( 1 2 ⁢ π , k ) = E ⁡ ( k ) elliptic-integral-second-kind-E 1 2 𝜋 𝑘 complete-elliptic-integral-second-kind-E 𝑘 {\displaystyle{\displaystyle E\left(\tfrac{1}{2}\pi,k\right)=E\left(k\right)}}
\incellintEk@{\tfrac{1}{2}\pi}{k} = \compellintEk@{k}		 

EllipticE(sin((1)/(2)*Pi), k) = EllipticE(k)

EllipticE[Divide[1,2]*Pi, (k)^2] == EllipticE[(k)^2]
Successful Successful - Successful [Tested: 3]
19.6.E10 lim ϕ → 0 ⁡ E ⁡ ( ϕ , k ) / ϕ = 1 subscript → italic-ϕ 0 elliptic-integral-second-kind-E italic-ϕ 𝑘 italic-ϕ 1 {\displaystyle{\displaystyle\lim_{\phi\to 0}\ifrac{E\left(\phi,k\right)}{\phi}% =1}}
\lim_{\phi\to 0}\ifrac{\incellintEk@{\phi}{k}}{\phi} = 1		 

limit((EllipticE(sin(phi), k))/(phi), phi = 0) = 1

Limit[Divide[EllipticE[\[Phi], (k)^2],\[Phi]], \[Phi] -> 0, GenerateConditions->None] == 1
Successful Successful - Successful [Tested: 3]
19.6#Ex21 Π ⁡ ( 0 , α 2 , k ) = 0 elliptic-integral-third-kind-Pi 0 superscript 𝛼 2 𝑘 0 {\displaystyle{\displaystyle\Pi\left(0,\alpha^{2},k\right)=0}}
\incellintPik@{0}{\alpha^{2}}{k} = 0		 

EllipticPi(sin(0), (alpha)^(2), k) = 0

EllipticPi[\[Alpha]^(2), 0,(k)^2] == 0
 Successful Successful - Successful [Tested: 9]
19.6#Ex22 Π ⁡ ( ϕ , 0 , 0 ) = ϕ elliptic-integral-third-kind-Pi italic-ϕ 0 0 italic-ϕ {\displaystyle{\displaystyle\Pi\left(\phi,0,0\right)=\phi}}
\incellintPik@{\phi}{0}{0} = \phi		 

EllipticPi(sin(phi), 0, 0) = phi
 
EllipticPi[0, \[Phi],(0)^2] == \[Phi]
 Failure Successful
Failed [2 / 10]
Result: .858407346
Test Values: {phi = -2}

Result: -.858407346
Test Values: {phi = 2}

 Successful [Tested: 10]
19.6#Ex23 Π ⁡ ( ϕ , 1 , 0 ) = tan ⁡ ϕ elliptic-integral-third-kind-Pi italic-ϕ 1 0 italic-ϕ {\displaystyle{\displaystyle\Pi\left(\phi,1,0\right)=\tan\phi}}
\incellintPik@{\phi}{1}{0} = \tan@@{\phi}		 

EllipticPi(sin(phi), 1, 0) = tan(phi)
 
EllipticPi[1, \[Phi],(0)^2] == Tan[\[Phi]]
 Failure Successful
Failed [2 / 10]
Result: -4.370079726
Test Values: {phi = -2}

Result: 4.370079726
Test Values: {phi = 2}

 Successful [Tested: 10]
19.6#Ex24 Π ⁡ ( ϕ , α 2 , 0 ) = R C ⁡ ( c - 1 , c - α 2 ) elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 0 Carlson-integral-RC 𝑐 1 𝑐 superscript 𝛼 2 {\displaystyle{\displaystyle\Pi\left(\phi,\alpha^{2},0\right)=R_{C}\left(c-1,c% -\alpha^{2}\right)}}
\incellintPik@{\phi}{\alpha^{2}}{0} = \CarlsonellintRC@{c-1}{c-\alpha^{2}}		 

Error
 
EllipticPi[\[Alpha]^(2), \[Phi],(0)^2] == 1/Sqrt[c - \[Alpha]^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-(c - 1)/(c - \[Alpha]^(2))]
 Missing Macro Error Failure -
Failed [180 / 180]
Result: Complex[0.4032669574270382, 0.8997227991212673]
Test Values: {Rule[c, -1.5], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

Result: Complex[-0.17167863497284278, 0.9673069947694621]
Test Values: {Rule[c, -1.5], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.6#Ex25 Π ⁡ ( ϕ , α 2 , 1 ) = 1 1 - α 2 ⁢ ( R C ⁡ ( c , c - 1 ) - α 2 ⁢ R C ⁡ ( c , c - α 2 ) ) elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 1 1 1 superscript 𝛼 2 Carlson-integral-RC 𝑐 𝑐 1 superscript 𝛼 2 Carlson-integral-RC 𝑐 𝑐 superscript 𝛼 2 {\displaystyle{\displaystyle\Pi\left(\phi,\alpha^{2},1\right)=\frac{1}{1-% \alpha^{2}}\left(R_{C}\left(c,c-1\right)-\alpha^{2}R_{C}\left(c,c-\alpha^{2}% \right)\right)}}
\incellintPik@{\phi}{\alpha^{2}}{1} = \frac{1}{1-\alpha^{2}}\left(\CarlsonellintRC@{c}{c-1}-\alpha^{2}\CarlsonellintRC@{c}{c-\alpha^{2}}\right)		 

Error
 
EllipticPi[\[Alpha]^(2), \[Phi],(1)^2] == Divide[1,1 - \[Alpha]^(2)]*(1/Sqrt[c - 1]*Hypergeometric2F1[1/2,1/2,3/2,1-(c)/(c - 1)]- \[Alpha]^(2)* 1/Sqrt[c - \[Alpha]^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-(c)/(c - \[Alpha]^(2))])
 Missing Macro Error Failure -
Failed [180 / 180]
Result: Complex[0.39392267303966433, 0.8870442763896845]
Test Values: {Rule[c, -1.5], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

Result: Complex[-0.15564928813724596, 0.9274825692848638]
Test Values: {Rule[c, -1.5], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.6#Ex26 Π ⁡ ( ϕ , 1 , 1 ) = 1 2 ⁢ ( R C ⁡ ( c , c - 1 ) + c ⁢ ( c - 1 ) - 1 ) elliptic-integral-third-kind-Pi italic-ϕ 1 1 1 2 Carlson-integral-RC 𝑐 𝑐 1 𝑐 superscript 𝑐 1 1 {\displaystyle{\displaystyle\Pi\left(\phi,1,1\right)=\tfrac{1}{2}(R_{C}\left(c% ,c-1\right)+\sqrt{c}(c-1)^{-1})}}
\incellintPik@{\phi}{1}{1} = \tfrac{1}{2}(\CarlsonellintRC@{c}{c-1}+\sqrt{c}(c-1)^{-1})		 

Error
 
EllipticPi[1, \[Phi],(1)^2] == Divide[1,2]*(1/Sqrt[c - 1]*Hypergeometric2F1[1/2,1/2,3/2,1-(c)/(c - 1)]+Sqrt[c]*(c - 1)^(- 1))
 Missing Macro Error Failure -
Failed [60 / 60]
Result: Complex[0.42461599644771203, 0.9033982135739806]
Test Values: {Rule[c, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

Result: Complex[-0.19222674503116347, 1.0138365568937844]
Test Values: {Rule[c, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.6#Ex27 Π ⁡ ( ϕ , 0 , k ) = F ⁡ ( ϕ , k ) elliptic-integral-third-kind-Pi italic-ϕ 0 𝑘 elliptic-integral-first-kind-F italic-ϕ 𝑘 {\displaystyle{\displaystyle\Pi\left(\phi,0,k\right)=F\left(\phi,k\right)}}
\incellintPik@{\phi}{0}{k} = \incellintFk@{\phi}{k}		 

EllipticPi(sin(phi), 0, k) = EllipticF(sin(phi), k)
 
EllipticPi[0, \[Phi],(k)^2] == EllipticF[\[Phi], (k)^2]
 Successful Successful - Successful [Tested: 30]
19.6#Ex28 Π ⁡ ( ϕ , k 2 , k ) = 1 k ′ 2 ⁢ ( E ⁡ ( ϕ , k ) - k 2 Δ ⁢ sin ⁡ ϕ ⁢ cos ⁡ ϕ ) elliptic-integral-third-kind-Pi italic-ϕ superscript 𝑘 2 𝑘 1 superscript superscript 𝑘 ′ 2 elliptic-integral-second-kind-E italic-ϕ 𝑘 superscript 𝑘 2 Δ italic-ϕ italic-ϕ {\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)}}
\incellintPik@{\phi}{k^{2}}{k} = \frac{1}{{k^{\prime}}^{2}}\left(\incellintEk@{\phi}{k}-\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
Failed [300 / 300]
Result: Float(infinity)+Float(infinity)*I
Test Values: {Delta = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, k = 1}

Result: -.4574406724+1.116997071*I
Test Values: {Delta = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, k = 2}

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

Result: Complex[-0.8161437733664769, 0.6845645198965172]
Test Values: {Rule[k, 2], Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.6#Ex29 Π ⁡ ( ϕ , 1 , k ) = F ⁡ ( ϕ , k ) - 1 k ′ 2 ⁢ ( E ⁡ ( ϕ , k ) - Δ ⁢ tan ⁡ ϕ ) elliptic-integral-third-kind-Pi italic-ϕ 1 𝑘 elliptic-integral-first-kind-F italic-ϕ 𝑘 1 superscript superscript 𝑘 ′ 2 elliptic-integral-second-kind-E italic-ϕ 𝑘 Δ italic-ϕ {\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)}}
\incellintPik@{\phi}{1}{k} = \incellintFk@{\phi}{k}-\frac{1}{{k^{\prime}}^{2}}(\incellintEk@{\phi}{k}-\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
Failed [300 / 300]
Result: Float(infinity)+Float(infinity)*I
Test Values: {Delta = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, k = 1}

Result: -.5381374542+.4861981155*I
Test Values: {Delta = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, k = 2}

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

Result: Complex[-0.12805668293605252, 0.0652384492706456]
Test Values: {Rule[k, 2], Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.6#Ex30 Π ⁡ ( 1 2 ⁢ π , α 2 , k ) = Π ⁡ ( α 2 , k ) elliptic-integral-third-kind-Pi 1 2 𝜋 superscript 𝛼 2 𝑘 complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 𝑘 {\displaystyle{\displaystyle\Pi\left(\tfrac{1}{2}\pi,\alpha^{2},k\right)=\Pi% \left(\alpha^{2},k\right)}}
\incellintPik@{\tfrac{1}{2}\pi}{\alpha^{2}}{k} = \compellintPik@{\alpha^{2}}{k}		 

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 - Successful [Tested: 9]
19.6#Ex31 lim ϕ → 0 ⁡ Π ⁡ ( ϕ , α 2 , k ) / ϕ = 1 subscript → italic-ϕ 0 elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 𝑘 italic-ϕ 1 {\displaystyle{\displaystyle\lim_{\phi\to 0}\ifrac{\Pi\left(\phi,\alpha^{2},k% \right)}{\phi}=1}}
\lim_{\phi\to 0}\ifrac{\incellintPik@{\phi}{\alpha^{2}}{k}}{\phi} = 1		 

limit((EllipticPi(sin(phi), (alpha)^(2), k))/(phi), phi = 0) = 1
 
Limit[Divide[EllipticPi[\[Alpha]^(2), \[Phi],(k)^2],\[Phi]], \[Phi] -> 0, GenerateConditions->None] == 1
 Successful Successful - Successful [Tested: 9]
19.6#Ex32 R C ⁡ ( x , x ) = x - 1 / 2 Carlson-integral-RC 𝑥 𝑥 superscript 𝑥 1 2 {\displaystyle{\displaystyle R_{C}\left(x,x\right)=x^{-1/2}}}
\CarlsonellintRC@{x}{x} = x^{-1/2}		 

Error
 
1/Sqrt[x]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(x)] == (x)^(- 1/2)
 Missing Macro Error Successful - Successful [Tested: 3]
19.6#Ex33 R C ⁡ ( λ ⁢ x , λ ⁢ y ) = λ - 1 / 2 ⁢ R C ⁡ ( x , y ) Carlson-integral-RC 𝜆 𝑥 𝜆 𝑦 superscript 𝜆 1 2 Carlson-integral-RC 𝑥 𝑦 {\displaystyle{\displaystyle R_{C}\left(\lambda x,\lambda y\right)=\lambda^{-1% /2}R_{C}\left(x,y\right)}}
\CarlsonellintRC@{\lambda x}{\lambda y} = \lambda^{-1/2}\CarlsonellintRC@{x}{y}		 

Error
 
1/Sqrt[\[Lambda]*y]*Hypergeometric2F1[1/2,1/2,3/2,1-(\[Lambda]*x)/(\[Lambda]*y)] == \[Lambda]^(- 1/2)* 1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)]
 Missing Macro Error Failure -
Failed [75 / 180]
Result: Complex[2.0541315094196904, 2.1051836996148214]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

Result: Complex[2.941079989400646, 0.036099349881403064]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.6#Ex35 R C ⁡ ( 0 , y ) = 1 2 ⁢ π ⁢ y - 1 / 2 Carlson-integral-RC 0 𝑦 1 2 𝜋 superscript 𝑦 1 2 {\displaystyle{\displaystyle R_{C}\left(0,y\right)=\tfrac{1}{2}\pi y^{-1/2}}}
\CarlsonellintRC@{0}{y} = \tfrac{1}{2}\pi y^{-1/2}		 | ph ⁡ y | < π phase 𝑦 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}y|<\pi}} 
Error
 
1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(0)/(y)] == Divide[1,2]*Pi*(y)^(- 1/2)
 Missing Macro Error Successful - Successful [Tested: 3]
19.6#Ex36 R C ⁡ ( 0 , y ) = 0 Carlson-integral-RC 0 𝑦 0 {\displaystyle{\displaystyle R_{C}\left(0,y\right)=0}}
\CarlsonellintRC@{0}{y} = 0		 y < 0 𝑦 0 {\displaystyle{\displaystyle y<0}} 
Error
 
1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(0)/(y)] == 0
 Missing Macro Error Failure -
Failed [3 / 3]
Result: Complex[0.0, -1.2825498301618643]
Test Values: {Rule[y, -1.5]}

Result: Complex[0.0, -2.221441469079183]
Test Values: {Rule[y, -0.5]}

... skip entries to safe data
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