Elliptic Integrals - 19.25 Relations to Other Functions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
19.25#Ex1 K ⁡ ( k ) = R F ⁡ ( 0 , k ′ 2 , 1 ) complete-elliptic-integral-first-kind-K 𝑘 Carlson-integral-RF 0 superscript superscript 𝑘 ′ 2 1 {\displaystyle{\displaystyle K\left(k\right)=R_{F}\left(0,{k^{\prime}}^{2},1% \right)}}
\compellintKk@{k} = \CarlsonsymellintRF@{0}{{k^{\prime}}^{2}}{1}		 

EllipticK(k) = 0.5*int(1/(sqrt(t+0)*sqrt(t+1 - (k)^(2))*sqrt(t+1)), t = 0..infinity)

EllipticK[(k)^2] == EllipticF[ArcCos[Sqrt[0/1]],(1-1 - (k)^(2))/(1-0)]/Sqrt[1-0]
Failure Failure Error
Failed [3 / 3]
Result: DirectedInfinity[]
Test Values: {Rule[k, 1]}


Result: Complex[-0.16657773258291342, -1.0782578237498217]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.25#Ex2 E ⁡ ( k ) = 2 ⁢ R G ⁡ ( 0 , k ′ 2 , 1 ) complete-elliptic-integral-second-kind-E 𝑘 2 Carlson-integral-RG 0 superscript superscript 𝑘 ′ 2 1 {\displaystyle{\displaystyle E\left(k\right)=2R_{G}\left(0,{k^{\prime}}^{2},1% \right)}}
\compellintEk@{k} = 2\CarlsonsymellintRG@{0}{{k^{\prime}}^{2}}{1}		 

Error

EllipticE[(k)^2] == 2*Sqrt[1-0]*(EllipticE[ArcCos[Sqrt[0/1]],(1-1 - (k)^(2))/(1-0)]+(Cot[ArcCos[Sqrt[0/1]]])^2*EllipticF[ArcCos[Sqrt[0/1]],(1-1 - (k)^(2))/(1-0)]+Cot[ArcCos[Sqrt[0/1]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/1]]]^2])
Missing Macro Error Failure -
Failed [3 / 3]
Result: -2.820197789027711
Test Values: {Rule[k, 1]}


Result: Complex[-4.864068276731299, 1.343854231387098]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.25#Ex3 E ⁡ ( k ) = 1 3 ⁢ k ′ 2 ⁢ ( R D ⁡ ( 0 , k ′ 2 , 1 ) + R D ⁡ ( 0 , 1 , k ′ 2 ) ) complete-elliptic-integral-second-kind-E 𝑘 1 3 superscript superscript 𝑘 ′ 2 Carlson-integral-RD 0 superscript superscript 𝑘 ′ 2 1 Carlson-integral-RD 0 1 superscript superscript 𝑘 ′ 2 {\displaystyle{\displaystyle E\left(k\right)=\tfrac{1}{3}{k^{\prime}}^{2}\left% (R_{D}\left(0,{k^{\prime}}^{2},1\right)+R_{D}\left(0,1,{k^{\prime}}^{2}\right)% \right)}}
\compellintEk@{k} = \tfrac{1}{3}{k^{\prime}}^{2}\left(\CarlsonsymellintRD@{0}{{k^{\prime}}^{2}}{1}+\CarlsonsymellintRD@{0}{1}{{k^{\prime}}^{2}}\right)		 

Error

EllipticE[(k)^2] == Divide[1,3]*1 - (k)^(2)*(3*(EllipticF[ArcCos[Sqrt[0/1]],(1-1 - (k)^(2))/(1-0)]-EllipticE[ArcCos[Sqrt[0/1]],(1-1 - (k)^(2))/(1-0)])/((1-1 - (k)^(2))*(1-0)^(1/2))+ 3*(EllipticF[ArcCos[Sqrt[0/1 - (k)^(2)]],(1 - (k)^(2)-1)/(1 - (k)^(2)-0)]-EllipticE[ArcCos[Sqrt[0/1 - (k)^(2)]],(1 - (k)^(2)-1)/(1 - (k)^(2)-0)])/((1 - (k)^(2)-1)*(1 - (k)^(2)-0)^(1/2)))
Missing Macro Error Failure -
Failed [3 / 3]
Result: DirectedInfinity[]
Test Values: {Rule[k, 1]}


Result: Complex[7.885081986624734, -2.293856789051463]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.25#Ex4 K ⁡ ( k ) - E ⁡ ( k ) = k 2 ⁢ D ⁡ ( k ) complete-elliptic-integral-first-kind-K 𝑘 complete-elliptic-integral-second-kind-E 𝑘 superscript 𝑘 2 complete-elliptic-integral-D 𝑘 {\displaystyle{\displaystyle K\left(k\right)-E\left(k\right)=k^{2}D\left(k% \right)}}
\compellintKk@{k}-\compellintEk@{k} = k^{2}\compellintDk@{k}		 

EllipticK(k)- EllipticE(k) = (k)^(2)* (EllipticK(k) - EllipticE(k))/(k)^2

EllipticK[(k)^2]- EllipticE[(k)^2] == (k)^(2)* Divide[EllipticK[(k)^2] - EllipticE[(k)^2], (k)^4]
Successful Failure -
Failed [3 / 3]
Result: Indeterminate
Test Values: {Rule[k, 1]}


Result: Complex[0.3274322182097533, -1.81658404135269]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.25#Ex4 k 2 ⁢ D ⁡ ( k ) = 1 3 ⁢ k 2 ⁢ R D ⁡ ( 0 , k ′ 2 , 1 ) superscript 𝑘 2 complete-elliptic-integral-D 𝑘 1 3 superscript 𝑘 2 Carlson-integral-RD 0 superscript superscript 𝑘 ′ 2 1 {\displaystyle{\displaystyle k^{2}D\left(k\right)=\tfrac{1}{3}k^{2}R_{D}\left(% 0,{k^{\prime}}^{2},1\right)}}
k^{2}\compellintDk@{k} = \tfrac{1}{3}k^{2}\CarlsonsymellintRD@{0}{{k^{\prime}}^{2}}{1}		 

Error

(k)^(2)* Divide[EllipticK[(k)^2] - EllipticE[(k)^2], (k)^4] == Divide[1,3]*(k)^(2)* 3*(EllipticF[ArcCos[Sqrt[0/1]],(1-1 - (k)^(2))/(1-0)]-EllipticE[ArcCos[Sqrt[0/1]],(1-1 - (k)^(2))/(1-0)])/((1-1 - (k)^(2))*(1-0)^(1/2))
Missing Macro Error Failure -
Failed [3 / 3]
Result: DirectedInfinity[]
Test Values: {Rule[k, 1]}


Result: Complex[-1.5165865988698335, -0.6055280137842299]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.25#Ex5 E ⁡ ( k ) - k ′ 2 ⁢ K ⁡ ( k ) = 1 3 ⁢ k 2 ⁢ k ′ 2 ⁢ R D ⁡ ( 0 , 1 , k ′ 2 ) complete-elliptic-integral-second-kind-E 𝑘 superscript superscript 𝑘 ′ 2 complete-elliptic-integral-first-kind-K 𝑘 1 3 superscript 𝑘 2 superscript superscript 𝑘 ′ 2 Carlson-integral-RD 0 1 superscript superscript 𝑘 ′ 2 {\displaystyle{\displaystyle E\left(k\right)-{k^{\prime}}^{2}K\left(k\right)=% \tfrac{1}{3}k^{2}{k^{\prime}}^{2}R_{D}\left(0,1,{k^{\prime}}^{2}\right)}}
\compellintEk@{k}-{k^{\prime}}^{2}\compellintKk@{k} = \tfrac{1}{3}k^{2}{k^{\prime}}^{2}\CarlsonsymellintRD@{0}{1}{{k^{\prime}}^{2}}		 

Error

EllipticE[(k)^2]-1 - (k)^(2)*EllipticK[(k)^2] == Divide[1,3]*(k)^(2)*1 - (k)^(2)*3*(EllipticF[ArcCos[Sqrt[0/1 - (k)^(2)]],(1 - (k)^(2)-1)/(1 - (k)^(2)-0)]-EllipticE[ArcCos[Sqrt[0/1 - (k)^(2)]],(1 - (k)^(2)-1)/(1 - (k)^(2)-0)])/((1 - (k)^(2)-1)*(1 - (k)^(2)-0)^(1/2))
Missing Macro Error Failure -
Failed [3 / 3]
Result: Indeterminate
Test Values: {Rule[k, 1]}


Result: Complex[-2.3636107378197124, 2.0191745059478237]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.25.E2 Π ⁡ ( α 2 , k ) - K ⁡ ( k ) = 1 3 ⁢ α 2 ⁢ R J ⁡ ( 0 , k ′ 2 , 1 , 1 - α 2 ) complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 𝑘 complete-elliptic-integral-first-kind-K 𝑘 1 3 superscript 𝛼 2 Carlson-integral-RJ 0 superscript superscript 𝑘 ′ 2 1 1 superscript 𝛼 2 {\displaystyle{\displaystyle\Pi\left(\alpha^{2},k\right)-K\left(k\right)=% \tfrac{1}{3}\alpha^{2}R_{J}\left(0,{k^{\prime}}^{2},1,1-\alpha^{2}\right)}}
\compellintPik@{\alpha^{2}}{k}-\compellintKk@{k} = \tfrac{1}{3}\alpha^{2}\CarlsonsymellintRJ@{0}{{k^{\prime}}^{2}}{1}{1-\alpha^{2}}		 

Error

EllipticPi[\[Alpha]^(2), (k)^2]- EllipticK[(k)^2] == Divide[1,3]*\[Alpha]^(2)* 3*(1-0)/(1-1 - \[Alpha]^(2))*(EllipticPi[(1-1 - \[Alpha]^(2))/(1-0),ArcCos[Sqrt[0/1]],(1-1 - (k)^(2))/(1-0)]-EllipticF[ArcCos[Sqrt[0/1]],(1-1 - (k)^(2))/(1-0)])/Sqrt[1-0]
Missing Macro Error Failure -
Failed [9 / 9]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[α, 1.5]}


Result: Complex[-1.5241433161083033, 0.5547659663605348]
Test Values: {Rule[k, 2], Rule[α, 1.5]}

... skip entries to safe data
19.25.E4 Π ⁡ ( α 2 , k ) = - 1 3 ⁢ ( k 2 / α 2 ) ⁢ R J ⁡ ( 0 , 1 - k 2 , 1 , 1 - ( k 2 / α 2 ) ) complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 𝑘 1 3 superscript 𝑘 2 superscript 𝛼 2 Carlson-integral-RJ 0 1 superscript 𝑘 2 1 1 superscript 𝑘 2 superscript 𝛼 2 {\displaystyle{\displaystyle\Pi\left(\alpha^{2},k\right)=-\tfrac{1}{3}(k^{2}/% \alpha^{2})R_{J}\left(0,1-k^{2},1,1-(k^{2}/\alpha^{2})\right)}}
\compellintPik@{\alpha^{2}}{k} = -\tfrac{1}{3}(k^{2}/\alpha^{2})\CarlsonsymellintRJ@{0}{1-k^{2}}{1}{1-(k^{2}/\alpha^{2})}		 - ∞ < k 2 , k 2 < 1 , 1 < α 2 formulae-sequence superscript 𝑘 2 formulae-sequence superscript 𝑘 2 1 1 superscript 𝛼 2 {\displaystyle{\displaystyle-\infty<k^{2},k^{2}<1,1<\alpha^{2}}} 
Error

EllipticPi[\[Alpha]^(2), (k)^2] == -Divide[1,3]*((k)^(2)/\[Alpha]^(2))*3*(1-0)/(1-1 -((k)^(2)/\[Alpha]^(2)))*(EllipticPi[(1-1 -((k)^(2)/\[Alpha]^(2)))/(1-0),ArcCos[Sqrt[0/1]],(1-1 - (k)^(2))/(1-0)]-EllipticF[ArcCos[Sqrt[0/1]],(1-1 - (k)^(2))/(1-0)])/Sqrt[1-0]
Missing Macro Error Failure - Skip - No test values generated
19.25.E5 F ⁡ ( ϕ , k ) = R F ⁡ ( c - 1 , c - k 2 , c ) elliptic-integral-first-kind-F italic-ϕ 𝑘 Carlson-integral-RF 𝑐 1 𝑐 superscript 𝑘 2 𝑐 {\displaystyle{\displaystyle F\left(\phi,k\right)=R_{F}\left(c-1,c-k^{2},c% \right)}}
\incellintFk@{\phi}{k} = \CarlsonsymellintRF@{c-1}{c-k^{2}}{c}		 

EllipticF(sin(phi), k) = 0.5*int(1/(sqrt(t+c - 1)*sqrt(t+c - (k)^(2))*sqrt(t+c)), t = 0..infinity)

EllipticF[\[Phi], (k)^2] == EllipticF[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)]/Sqrt[c-c - 1]
Failure Failure
Failed [180 / 180]
Result: Float(undefined)+Float(undefined)*I
Test Values: {c = -3/2, phi = 1/2*3^(1/2)+1/2*I, k = 1}


Result: 3.854689052+3.461698034*I
Test Values: {c = -3/2, phi = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [180 / 180]
Result: Complex[2.0026000841930385, 1.2187088711714384]
Test Values: {Rule[c, -1.5], Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[1.4748265293714395, 0.7583435972865697]
Test Values: {Rule[c, -1.5], Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.25.E6 ∂ ⁡ F ⁡ ( ϕ , k ) ∂ ⁡ k = 1 3 ⁢ k ⁢ R D ⁡ ( c - 1 , c , c - k 2 ) partial-derivative elliptic-integral-first-kind-F italic-ϕ 𝑘 𝑘 1 3 𝑘 Carlson-integral-RD 𝑐 1 𝑐 𝑐 superscript 𝑘 2 {\displaystyle{\displaystyle\frac{\partial F\left(\phi,k\right)}{\partial k}=% \tfrac{1}{3}kR_{D}\left(c-1,c,c-k^{2}\right)}}
\pderiv{\incellintFk@{\phi}{k}}{k} = \tfrac{1}{3}k\CarlsonsymellintRD@{c-1}{c}{c-k^{2}}		 

Error

D[EllipticF[\[Phi], (k)^2], k] == Divide[1,3]*k*3*(EllipticF[ArcCos[Sqrt[c - 1/c - (k)^(2)]],(c - (k)^(2)-c)/(c - (k)^(2)-c - 1)]-EllipticE[ArcCos[Sqrt[c - 1/c - (k)^(2)]],(c - (k)^(2)-c)/(c - (k)^(2)-c - 1)])/((c - (k)^(2)-c)*(c - (k)^(2)-c - 1)^(1/2))
Missing Macro Error Failure -
Failed [180 / 180]
Result: Indeterminate
Test Values: {Rule[c, -1.5], Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-0.4045300788217367, 0.4404710702025501]
Test Values: {Rule[c, -1.5], Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.25.E7 E ⁡ ( ϕ , k ) = 2 ⁢ R G ⁡ ( c - 1 , c - k 2 , c ) - ( c - 1 ) ⁢ R F ⁡ ( c - 1 , c - k 2 , c ) - ( c - 1 ) ⁢ ( c - k 2 ) / c elliptic-integral-second-kind-E italic-ϕ 𝑘 2 Carlson-integral-RG 𝑐 1 𝑐 superscript 𝑘 2 𝑐 𝑐 1 Carlson-integral-RF 𝑐 1 𝑐 superscript 𝑘 2 𝑐 𝑐 1 𝑐 superscript 𝑘 2 𝑐 {\displaystyle{\displaystyle E\left(\phi,k\right)=2R_{G}\left(c-1,c-k^{2},c% \right)-(c-1)R_{F}\left(c-1,c-k^{2},c\right)-\sqrt{(c-1)(c-k^{2})/c}}}
\incellintEk@{\phi}{k} = 2\CarlsonsymellintRG@{c-1}{c-k^{2}}{c}-(c-1)\CarlsonsymellintRF@{c-1}{c-k^{2}}{c}-\sqrt{(c-1)(c-k^{2})/c}		 

Error

EllipticE[\[Phi], (k)^2] == 2*Sqrt[c-c - 1]*(EllipticE[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)]+(Cot[ArcCos[Sqrt[c - 1/c]]])^2*EllipticF[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)]+Cot[ArcCos[Sqrt[c - 1/c]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[c - 1/c]]]^2])-(c - 1)*EllipticF[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)]/Sqrt[c-c - 1]-Sqrt[(c - 1)*(c - (k)^(2))/c]
Missing Macro Error Failure -
Failed [180 / 180]
Result: Complex[5.787775994567906, 4.022803158659452]
Test Values: {Rule[c, -1.5], Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[6.805668366738806, 3.968311704298834]
Test Values: {Rule[c, -1.5], Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.25.E9 E ⁡ ( ϕ , k ) = R F ⁡ ( c - 1 , c - k 2 , c ) - 1 3 ⁢ k 2 ⁢ R D ⁡ ( c - 1 , c - k 2 , c ) elliptic-integral-second-kind-E italic-ϕ 𝑘 Carlson-integral-RF 𝑐 1 𝑐 superscript 𝑘 2 𝑐 1 3 superscript 𝑘 2 Carlson-integral-RD 𝑐 1 𝑐 superscript 𝑘 2 𝑐 {\displaystyle{\displaystyle E\left(\phi,k\right)=R_{F}\left(c-1,c-k^{2},c% \right)-\tfrac{1}{3}k^{2}R_{D}\left(c-1,c-k^{2},c\right)}}
\incellintEk@{\phi}{k} = \CarlsonsymellintRF@{c-1}{c-k^{2}}{c}-\tfrac{1}{3}k^{2}\CarlsonsymellintRD@{c-1}{c-k^{2}}{c}		 

Error

EllipticE[\[Phi], (k)^2] == EllipticF[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)]/Sqrt[c-c - 1]-Divide[1,3]*(k)^(2)* 3*(EllipticF[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)]-EllipticE[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)])/((c-c - (k)^(2))*(c-c - 1)^(1/2))
Missing Macro Error Failure -
Failed [180 / 180]
Result: Complex[3.5743811704478246, 0.7698502565730785]
Test Values: {Rule[c, -1.5], Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[3.9424508382496875, -1.017653751864599]
Test Values: {Rule[c, -1.5], Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.25.E10 E ⁡ ( ϕ , k ) = k ′ 2 ⁢ R F ⁡ ( c - 1 , c - k 2 , c ) + 1 3 ⁢ k 2 ⁢ k ′ 2 ⁢ R D ⁡ ( c - 1 , c , c - k 2 ) + k 2 ⁢ ( c - 1 ) / ( c ⁢ ( c - k 2 ) ) elliptic-integral-second-kind-E italic-ϕ 𝑘 superscript superscript 𝑘 ′ 2 Carlson-integral-RF 𝑐 1 𝑐 superscript 𝑘 2 𝑐 1 3 superscript 𝑘 2 superscript superscript 𝑘 ′ 2 Carlson-integral-RD 𝑐 1 𝑐 𝑐 superscript 𝑘 2 superscript 𝑘 2 𝑐 1 𝑐 𝑐 superscript 𝑘 2 {\displaystyle{\displaystyle E\left(\phi,k\right)={k^{\prime}}^{2}R_{F}\left(c% -1,c-k^{2},c\right)+\tfrac{1}{3}k^{2}{k^{\prime}}^{2}R_{D}\left(c-1,c,c-k^{2}% \right)+k^{2}\sqrt{(c-1)/(c(c-k^{2}))}}}
\incellintEk@{\phi}{k} = {k^{\prime}}^{2}\CarlsonsymellintRF@{c-1}{c-k^{2}}{c}+\tfrac{1}{3}k^{2}{k^{\prime}}^{2}\CarlsonsymellintRD@{c-1}{c}{c-k^{2}}+k^{2}\sqrt{(c-1)/(c(c-k^{2}))}		 c > k 2 𝑐 superscript 𝑘 2 {\displaystyle{\displaystyle c>k^{2}}} 
Error

EllipticE[\[Phi], (k)^2] == 1 - (k)^(2)*EllipticF[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)]/Sqrt[c-c - 1]+Divide[1,3]*(k)^(2)*1 - (k)^(2)*3*(EllipticF[ArcCos[Sqrt[c - 1/c - (k)^(2)]],(c - (k)^(2)-c)/(c - (k)^(2)-c - 1)]-EllipticE[ArcCos[Sqrt[c - 1/c - (k)^(2)]],(c - (k)^(2)-c)/(c - (k)^(2)-c - 1)])/((c - (k)^(2)-c)*(c - (k)^(2)-c - 1)^(1/2))+ (k)^(2)*Sqrt[(c - 1)/(c*(c - (k)^(2)))]
Missing Macro Error Failure -
Failed [20 / 20]
Result: Complex[-1.0687219916023158, 0.8637282710955538]
Test Values: {Rule[c, 1.5], Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-1.7724732696890155, 1.0672164584507502]
Test Values: {Rule[c, 1.5], Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.25.E11 E ⁡ ( ϕ , k ) = - 1 3 ⁢ k ′ 2 ⁢ R D ⁡ ( c - k 2 , c , c - 1 ) + ( c - k 2 ) / ( c ⁢ ( c - 1 ) ) elliptic-integral-second-kind-E italic-ϕ 𝑘 1 3 superscript superscript 𝑘 ′ 2 Carlson-integral-RD 𝑐 superscript 𝑘 2 𝑐 𝑐 1 𝑐 superscript 𝑘 2 𝑐 𝑐 1 {\displaystyle{\displaystyle E\left(\phi,k\right)=-\tfrac{1}{3}{k^{\prime}}^{2% }R_{D}\left(c-k^{2},c,c-1\right)+\sqrt{(c-k^{2})/(c(c-1))}}}
\incellintEk@{\phi}{k} = -\tfrac{1}{3}{k^{\prime}}^{2}\CarlsonsymellintRD@{c-k^{2}}{c}{c-1}+\sqrt{(c-k^{2})/(c(c-1))}		 ϕ ≠ 1 2 ⁢ π italic-ϕ 1 2 𝜋 {\displaystyle{\displaystyle\phi\neq\tfrac{1}{2}\pi}} 
Error

EllipticE[\[Phi], (k)^2] == -Divide[1,3]*1 - (k)^(2)*3*(EllipticF[ArcCos[Sqrt[c - (k)^(2)/c - 1]],(c - 1-c)/(c - 1-c - (k)^(2))]-EllipticE[ArcCos[Sqrt[c - (k)^(2)/c - 1]],(c - 1-c)/(c - 1-c - (k)^(2))])/((c - 1-c)*(c - 1-c - (k)^(2))^(1/2))+Sqrt[(c - (k)^(2))/(c*(c - 1))]
Missing Macro Error Failure -
Failed [180 / 180]
Result: Complex[3.6312701919621486, -1.3602272606820804]
Test Values: {Rule[c, -1.5], Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[0.7754142926962797, -0.6029933704091625]
Test Values: {Rule[c, -1.5], Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.25.E12 ∂ ⁡ E ⁡ ( ϕ , k ) ∂ ⁡ k = - 1 3 ⁢ k ⁢ R D ⁡ ( c - 1 , c - k 2 , c ) partial-derivative elliptic-integral-second-kind-E italic-ϕ 𝑘 𝑘 1 3 𝑘 Carlson-integral-RD 𝑐 1 𝑐 superscript 𝑘 2 𝑐 {\displaystyle{\displaystyle\frac{\partial E\left(\phi,k\right)}{\partial k}=-% \tfrac{1}{3}kR_{D}\left(c-1,c-k^{2},c\right)}}
\pderiv{\incellintEk@{\phi}{k}}{k} = -\tfrac{1}{3}k\CarlsonsymellintRD@{c-1}{c-k^{2}}{c}		 

Error

D[EllipticE[\[Phi], (k)^2], k] == -Divide[1,3]*k*3*(EllipticF[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)]-EllipticE[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)])/((c-c - (k)^(2))*(c-c - 1)^(1/2))
Missing Macro Error Failure -
Failed [180 / 180]
Result: Complex[1.571781086254786, -0.44885861459835996]
Test Values: {Rule[c, -1.5], Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[1.233812154439124, -0.8879986745755843]
Test Values: {Rule[c, -1.5], Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.25.E13 D ⁡ ( ϕ , k ) = 1 3 ⁢ R D ⁡ ( c - 1 , c - k 2 , c ) elliptic-integral-third-kind-D italic-ϕ 𝑘 1 3 Carlson-integral-RD 𝑐 1 𝑐 superscript 𝑘 2 𝑐 {\displaystyle{\displaystyle D\left(\phi,k\right)=\tfrac{1}{3}R_{D}\left(c-1,c% -k^{2},c\right)}}
\incellintDk@{\phi}{k} = \tfrac{1}{3}\CarlsonsymellintRD@{c-1}{c-k^{2}}{c}		 

Error

Divide[EllipticF[\[Phi], (k)^2] - EllipticE[\[Phi], (k)^2], (k)^4] == Divide[1,3]*3*(EllipticF[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)]-EllipticE[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)])/((c-c - (k)^(2))*(c-c - 1)^(1/2))
Missing Macro Error Failure -
Failed [180 / 180]
Result: Complex[-1.571781086254786, 0.44885861459835996]
Test Values: {Rule[c, -1.5], Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-0.6083725296430629, 0.41279951787826946]
Test Values: {Rule[c, -1.5], Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.25.E14 Π ⁡ ( ϕ , α 2 , k ) - F ⁡ ( ϕ , k ) = 1 3 ⁢ α 2 ⁢ R J ⁡ ( c - 1 , c - k 2 , c , c - α 2 ) elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 𝑘 elliptic-integral-first-kind-F italic-ϕ 𝑘 1 3 superscript 𝛼 2 Carlson-integral-RJ 𝑐 1 𝑐 superscript 𝑘 2 𝑐 𝑐 superscript 𝛼 2 {\displaystyle{\displaystyle\Pi\left(\phi,\alpha^{2},k\right)-F\left(\phi,k% \right)=\tfrac{1}{3}\alpha^{2}R_{J}\left(c-1,c-k^{2},c,c-\alpha^{2}\right)}}
\incellintPik@{\phi}{\alpha^{2}}{k}-\incellintFk@{\phi}{k} = \tfrac{1}{3}\alpha^{2}\CarlsonsymellintRJ@{c-1}{c-k^{2}}{c}{c-\alpha^{2}}		 

Error

EllipticPi[\[Alpha]^(2), \[Phi],(k)^2]- EllipticF[\[Phi], (k)^2] == Divide[1,3]*\[Alpha]^(2)* 3*(c-c - 1)/(c-c - \[Alpha]^(2))*(EllipticPi[(c-c - \[Alpha]^(2))/(c-c - 1),ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)]-EllipticF[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)])/Sqrt[c-c - 1]
Missing Macro Error Failure -
Failed [300 / 300]
Result: Complex[-0.9803588804354156, -0.9579910370435353]
Test Values: {Rule[c, -1.5], Rule[k, 1], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-0.6164275583611891, -0.384238714210872]
Test Values: {Rule[c, -1.5], Rule[k, 2], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.25.E16 Π ⁡ ( ϕ , α 2 , k ) = - 1 3 ⁢ ω 2 ⁢ R J ⁡ ( c - 1 , c - k 2 , c , c - ω 2 ) + ( c - 1 ) ⁢ ( c - k 2 ) ( α 2 - 1 ) ⁢ ( 1 - ω 2 ) ⁢ R C ⁡ ( c ⁢ ( α 2 - 1 ) ⁢ ( 1 - ω 2 ) , ( α 2 - c ) ⁢ ( c - ω 2 ) ) elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 𝑘 1 3 superscript 𝜔 2 Carlson-integral-RJ 𝑐 1 𝑐 superscript 𝑘 2 𝑐 𝑐 superscript 𝜔 2 𝑐 1 𝑐 superscript 𝑘 2 superscript 𝛼 2 1 1 superscript 𝜔 2 Carlson-integral-RC 𝑐 superscript 𝛼 2 1 1 superscript 𝜔 2 superscript 𝛼 2 𝑐 𝑐 superscript 𝜔 2 {\displaystyle{\displaystyle\Pi\left(\phi,\alpha^{2},k\right)=-\tfrac{1}{3}% \omega^{2}R_{J}\left(c-1,c-k^{2},c,c-\omega^{2}\right)+\sqrt{\frac{(c-1)(c-k^{% 2})}{(\alpha^{2}-1)(1-\omega^{2})}}\*R_{C}\left(c(\alpha^{2}-1)(1-\omega^{2}),% (\alpha^{2}-c)(c-\omega^{2})\right)}}
\incellintPik@{\phi}{\alpha^{2}}{k} = -\tfrac{1}{3}\omega^{2}\CarlsonsymellintRJ@{c-1}{c-k^{2}}{c}{c-\omega^{2}}+\sqrt{\frac{(c-1)(c-k^{2})}{(\alpha^{2}-1)(1-\omega^{2})}}\*\CarlsonellintRC@{c(\alpha^{2}-1)(1-\omega^{2})}{(\alpha^{2}-c)(c-\omega^{2})}		 ω 2 = k 2 / α 2 superscript 𝜔 2 superscript 𝑘 2 superscript 𝛼 2 {\displaystyle{\displaystyle\omega^{2}=k^{2}/\alpha^{2}}} 
Error

EllipticPi[\[Alpha]^(2), \[Phi],(k)^2] == -Divide[1,3]*\[Omega]^(2)* 3*(c-c - 1)/(c-c - \[Omega]^(2))*(EllipticPi[(c-c - \[Omega]^(2))/(c-c - 1),ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)]-EllipticF[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)])/Sqrt[c-c - 1]+Sqrt[Divide[(c - 1)*(c - (k)^(2)),(\[Alpha]^(2)- 1)*(1 - \[Omega]^(2))]]* 1/Sqrt[(\[Alpha]^(2)- c)*(c - \[Omega]^(2))]*Hypergeometric2F1[1/2,1/2,3/2,1-(c*(\[Alpha]^(2)- 1)*(1 - \[Omega]^(2)))/((\[Alpha]^(2)- c)*(c - \[Omega]^(2)))]
Missing Macro Error Aborted -
Failed [300 / 300]
Result: Complex[-0.11631142199526823, 0.9703799109463437]
Test Values: {Rule[c, -1.5], Rule[k, 3], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ω, -2]}


Result: Complex[-0.11631142199526823, 0.9703799109463437]
Test Values: {Rule[c, -1.5], Rule[k, 3], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ω, 2]}

... skip entries to safe data
19.25.E17 F ⁡ ( ϕ , k ) = R F ⁡ ( x , y , z ) elliptic-integral-first-kind-F italic-ϕ 𝑘 Carlson-integral-RF 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle F\left(\phi,k\right)=R_{F}\left(x,y,z\right)}}
\incellintFk@{\phi}{k} = \CarlsonsymellintRF@{x}{y}{z}		 

EllipticF(sin(phi), k) = 0.5*int(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+x + y*I)), t = 0..infinity)

EllipticF[\[Phi], (k)^2] == EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]
Aborted Failure
Failed [300 / 300]
Result: 2.547570015-.6488873983*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2, k = 1}


Result: 2.209888328-.6080126261*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.5939484671297026, -0.40701440305540804]
Test Values: {Rule[k, 1], Rule[x, 1.5], Rule[y, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[0.5587134153531784, -0.34669285510288844]
Test Values: {Rule[k, 2], Rule[x, 1.5], Rule[y, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.25.E18 ( x , y , z ) = ( c - 1 , c - k 2 , c ) 𝑥 𝑦 𝑧 𝑐 1 𝑐 superscript 𝑘 2 𝑐 {\displaystyle{\displaystyle(x,y,z)=(c-1,c-k^{2},c)}}
(x,y,z) = (c-1,c-k^{2},c)		 

(x , y ,(x + y*I)) = (c - 1 , c - (k)^(2), c)
 
(x , y ,(x + y*I)) == (c - 1 , c - (k)^(2), c)
 Skipped - no semantic math Skipped - no semantic math - -
19.25#Ex6 ϕ = arccos ⁡ x / z italic-ϕ 𝑥 𝑧 {\displaystyle{\displaystyle\phi=\operatorname{arccos}\sqrt{\ifrac{x}{z}}}}
\phi = \acos@@{\sqrt{\ifrac{x}{z}}}		 

phi = arccos(sqrt((x)/(x + y*I)))

\[Phi] == ArcCos[Sqrt[Divide[x,x + y*I]]]
 Failure Failure
Failed [180 / 180]
Result: .806272406e-1+.9406867936*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2}

Result: .806272406e-1+.593132064e-1*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, y = 3/2}

... skip entries to safe data
Failed [180 / 180]
Result: Complex[-0.35238546150522904, 0.6906867935097715]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

Result: Complex[-1.0353981633974483, 0.8736994954019909]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.25#Ex6 arccos ⁡ x / z = arcsin ⁡ ( z - x ) / z 𝑥 𝑧 𝑧 𝑥 𝑧 {\displaystyle{\displaystyle\operatorname{arccos}\sqrt{\ifrac{x}{z}}=% \operatorname{arcsin}\sqrt{\ifrac{(z-x)}{z}}}}
\acos@@{\sqrt{\ifrac{x}{z}}} = \asin@@{\sqrt{\ifrac{(z-x)}{z}}}		 

arccos(sqrt((x)/(x + y*I))) = arcsin(sqrt(((x + y*I)- x)/(x + y*I)))
 
ArcCos[Sqrt[Divide[x,x + y*I]]] == ArcSin[Sqrt[Divide[(x + y*I)- x,x + y*I]]]
 Failure Failure Successful [Tested: 18] Successful [Tested: 18]
19.25#Ex7 k = z - y z - x 𝑘 𝑧 𝑦 𝑧 𝑥 {\displaystyle{\displaystyle k=\sqrt{\frac{z-y}{z-x}}}}
k = \sqrt{\frac{z-y}{z-x}}		 

k = sqrt(((x + y*I)- y)/((x + y*I)- x))
 
k == Sqrt[Divide[(x + y*I)- y,(x + y*I)- x]]
 Skipped - no semantic math Skipped - no semantic math - -
19.25#Ex8 α 2 = z - p z - x superscript 𝛼 2 𝑧 𝑝 𝑧 𝑥 {\displaystyle{\displaystyle\alpha^{2}=\frac{z-p}{z-x}}}
\alpha^{2} = \frac{z-p}{z-x}		 

(alpha)^(2) = ((x + y*I)- p)/((x + y*I)- x)
 
\[Alpha]^(2) == Divide[(x + y*I)- p,(x + y*I)- x]
 Skipped - no semantic math Skipped - no semantic math - -
19.25.E24 ( z - x ) 1 / 2 ⁢ R F ⁡ ( x , y , z ) = F ⁡ ( ϕ , k ) superscript 𝑧 𝑥 1 2 Carlson-integral-RF 𝑥 𝑦 𝑧 elliptic-integral-first-kind-F italic-ϕ 𝑘 {\displaystyle{\displaystyle(z-x)^{1/2}R_{F}\left(x,y,z\right)=F\left(\phi,k% \right)}}
(z-x)^{1/2}\CarlsonsymellintRF@{x}{y}{z} = \incellintFk@{\phi}{k}		 

((x + y*I)- x)^(1/2)* 0.5*int(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+x + y*I)), t = 0..infinity) = EllipticF(sin(phi), k)
 
((x + y*I)- x)^(1/2)* EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x] == EllipticF[\[Phi], (k)^2]
 Aborted Failure
Failed [300 / 300]
Result: -1.167656510+1.966567574*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2, k = 1}

Result: -.8299748231+1.925692802*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.015324342917649614, 0.4565416109140732]
Test Values: {Rule[k, 1], Rule[x, 1.5], Rule[y, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

Result: Complex[0.050559394694173865, 0.3962200629615536]
Test Values: {Rule[k, 2], Rule[x, 1.5], Rule[y, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.25.E25 ( z - x ) 3 / 2 ⁢ R D ⁡ ( x , y , z ) = ( 3 / k 2 ) ⁢ ( F ⁡ ( ϕ , k ) - E ⁡ ( ϕ , k ) ) superscript 𝑧 𝑥 3 2 Carlson-integral-RD 𝑥 𝑦 𝑧 3 superscript 𝑘 2 elliptic-integral-first-kind-F italic-ϕ 𝑘 elliptic-integral-second-kind-E italic-ϕ 𝑘 {\displaystyle{\displaystyle(z-x)^{3/2}R_{D}\left(x,y,z\right)=(3/k^{2})(F% \left(\phi,k\right)-E\left(\phi,k\right))}}
(z-x)^{3/2}\CarlsonsymellintRD@{x}{y}{z} = (3/k^{2})(\incellintFk@{\phi}{k}-\incellintEk@{\phi}{k})		 

Error
 
((x + y*I)- x)^(3/2)* 3*(EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticE[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/((x + y*I-y)*(x + y*I-x)^(1/2)) == (3/(k)^(2))*(EllipticF[\[Phi], (k)^2]- EllipticE[\[Phi], (k)^2])
 Missing Macro Error Failure -
Failed [300 / 300]
Result: Complex[-0.9041684186949032, 0.18989946051507803]
Test Values: {Rule[k, 1], Rule[x, 1.5], Rule[y, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

Result: Complex[-0.8729885067685752, 0.19149534336253457]
Test Values: {Rule[k, 2], Rule[x, 1.5], Rule[y, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.25.E26 ( z - x ) 3 / 2 ⁢ R J ⁡ ( x , y , z , p ) = ( 3 / α 2 ) ⁢ ( Π ⁡ ( ϕ , α 2 , k ) - F ⁡ ( ϕ , k ) ) superscript 𝑧 𝑥 3 2 Carlson-integral-RJ 𝑥 𝑦 𝑧 𝑝 3 superscript 𝛼 2 elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 𝑘 elliptic-integral-first-kind-F italic-ϕ 𝑘 {\displaystyle{\displaystyle(z-x)^{3/2}R_{J}\left(x,y,z,p\right)=(3/\alpha^{2}% ){(\Pi\left(\phi,\alpha^{2},k\right)-F\left(\phi,k\right))}}}
(z-x)^{3/2}\CarlsonsymellintRJ@{x}{y}{z}{p} = (3/\alpha^{2}){(\incellintPik@{\phi}{\alpha^{2}}{k}-\incellintFk@{\phi}{k})}		 

Error
 
((x + y*I)- x)^(3/2)* 3*(x + y*I-x)/(x + y*I-p)*(EllipticPi[(x + y*I-p)/(x + y*I-x),ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/Sqrt[x + y*I-x] == (3/\[Alpha]^(2))*(EllipticPi[\[Alpha]^(2), \[Phi],(k)^2]- EllipticF[\[Phi], (k)^2])
 Missing Macro Error Failure -
Failed [300 / 300]
Result: Complex[-8.905365206673954*^-4, 0.6653826564189609]
Test Values: {Rule[k, 1], Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, -1.5], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

Result: Complex[0.030816807002235325, 0.6810951786851601]
Test Values: {Rule[k, 2], Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, -1.5], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.25.E27 2 ⁢ ( z - x ) - 1 / 2 ⁢ R G ⁡ ( x , y , z ) = E ⁡ ( ϕ , k ) + ( cot ⁡ ϕ ) 2 ⁢ F ⁡ ( ϕ , k ) + ( cot ⁡ ϕ ) ⁢ 1 - k 2 ⁢ sin 2 ⁡ ϕ 2 superscript 𝑧 𝑥 1 2 Carlson-integral-RG 𝑥 𝑦 𝑧 elliptic-integral-second-kind-E italic-ϕ 𝑘 superscript italic-ϕ 2 elliptic-integral-first-kind-F italic-ϕ 𝑘 italic-ϕ 1 superscript 𝑘 2 2 italic-ϕ {\displaystyle{\displaystyle 2(z-x)^{-1/2}R_{G}\left(x,y,z\right)=E\left(\phi,% k\right)+(\cot\phi)^{2}F\left(\phi,k\right)+(\cot\phi)\sqrt{1-k^{2}{\sin^{2}}% \phi}}}
2(z-x)^{-1/2}\CarlsonsymellintRG@{x}{y}{z} = \incellintEk@{\phi}{k}+(\cot@@{\phi})^{2}\incellintFk@{\phi}{k}+(\cot@@{\phi})\sqrt{1-k^{2}\sin^{2}@@{\phi}}		 

Error
 
2*((x + y*I)- x)^(- 1/2)* Sqrt[x + y*I-x]*(EllipticE[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]+(Cot[ArcCos[Sqrt[x/x + y*I]]])^2*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]+Cot[ArcCos[Sqrt[x/x + y*I]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[x/x + y*I]]]^2]) == EllipticE[\[Phi], (k)^2]+(Cot[\[Phi]])^(2)* EllipticF[\[Phi], (k)^2]+(Cot[\[Phi]])*Sqrt[1 - (k)^(2)* (Sin[\[Phi]])^(2)]
 Missing Macro Error Failure -
Failed [300 / 300]
Result: Complex[-1.8997799949200251, -0.4031557744461449]
Test Values: {Rule[k, 1], Rule[x, 1.5], Rule[y, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

Result: Complex[-3.0701379688219372, -2.1411109504853227]
Test Values: {Rule[k, 2], Rule[x, 1.5], Rule[y, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.25#Ex9 Δ ⁢ ( n , d ) = k 2 Δ n d superscript 𝑘 2 {\displaystyle{\displaystyle\Delta(\mathrm{n,d})=k^{2}}}
\Delta(\mathrm{n,d}) = k^{2}		 

Delta(n , d) = (k)^(2)
 
\[CapitalDelta][n , d] == (k)^(2)
 Skipped - no semantic math Skipped - no semantic math - -
19.25#Ex10 Δ ⁢ ( d , c ) = k ′ 2 Δ d c superscript superscript 𝑘 ′ 2 {\displaystyle{\displaystyle\Delta(\mathrm{d,c})={k^{\prime}}^{2}}}
\Delta(\mathrm{d,c}) = {k^{\prime}}^{2}		 

Delta(d , c) = 1 - (k)^(2)
 
\[CapitalDelta][d , c] == 1 - (k)^(2)
 Skipped - no semantic math Skipped - no semantic math - -
19.25#Ex11 Δ ⁢ ( n , c ) = 1 Δ n c 1 {\displaystyle{\displaystyle\Delta(\mathrm{n,c})=1}}
\Delta(\mathrm{n,c}) = 1		 

Delta(n , c) = 1
 
\[CapitalDelta][n , c] == 1
 Skipped - no semantic math Skipped - no semantic math - -
19.25.E30 am ⁡ ( u , k ) = R C ⁡ ( cs 2 ⁡ ( u , k ) , ns 2 ⁡ ( u , k ) ) Jacobi-elliptic-amplitude 𝑢 𝑘 Carlson-integral-RC Jacobi-elliptic-cs 2 𝑢 𝑘 Jacobi-elliptic-ns 2 𝑢 𝑘 {\displaystyle{\displaystyle\operatorname{am}\left(u,k\right)=R_{C}\left({% \operatorname{cs}^{2}}\left(u,k\right),{\operatorname{ns}^{2}}\left(u,k\right)% \right)}}
\Jacobiamk@{u}{k} = \CarlsonellintRC@{\Jacobiellcsk^{2}@{u}{k}}{\Jacobiellnsk^{2}@{u}{k}}		 

Error
 
JacobiAmplitude[u, Power[k, 2]] == 1/Sqrt[(JacobiNS[u, (k)^2])^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-((JacobiCS[u, (k)^2])^(2))/((JacobiNS[u, (k)^2])^(2))]
 Missing Macro Error Aborted -
Failed [18 / 30]
Result: Complex[-0.5428587296705786, 0.8636075147962846]
Test Values: {Rule[k, 1], Rule[u, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

Result: Complex[-0.6732377468613371, 0.8494366739388763]
Test Values: {Rule[k, 2], Rule[u, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.25.E31 u = R F ⁡ ( p ⁣ s 2 ⁡ ( u , k ) , q ⁣ s 2 ⁡ ( u , k ) , r ⁣ s 2 ⁡ ( u , k ) ) 𝑢 Carlson-integral-RF abstract-Jacobi-elliptic p s 2 𝑢 𝑘 abstract-Jacobi-elliptic q s 2 𝑢 𝑘 abstract-Jacobi-elliptic r s 2 𝑢 𝑘 {\displaystyle{\displaystyle u=R_{F}\left({\operatorname{ps}^{2}}\left(u,k% \right),{\operatorname{qs}^{2}}\left(u,k\right),{\operatorname{rs}^{2}}\left(u% ,k\right)\right)}}
u = \CarlsonsymellintRF@{\genJacobiellk{p}{s}^{2}@{u}{k}}{\genJacobiellk{q}{s}^{2}@{u}{k}}{\genJacobiellk{r}{s}^{2}@{u}{k}}		 

u = 0.5*int(1/(sqrt(t+genJacobiellk(p)*(s)^(2)* u*k)*sqrt(t+genJacobiellk(q)*(s)^(2)* u*k)*sqrt(t+genJacobiellk(r)*(s)^(2)* u*k)), t = 0..infinity)
 
u == EllipticF[ArcCos[Sqrt[genJacobiellk[p]*(s)^(2)* u*k/genJacobiellk[r]*(s)^(2)* u*k]],(genJacobiellk[r]*(s)^(2)* u*k-genJacobiellk[q]*(s)^(2)* u*k)/(genJacobiellk[r]*(s)^(2)* u*k-genJacobiellk[p]*(s)^(2)* u*k)]/Sqrt[genJacobiellk[r]*(s)^(2)* u*k-genJacobiellk[p]*(s)^(2)* u*k]
 Aborted Failure Error
Failed [300 / 300]
Result: Plus[Complex[0.43301270189221935, 0.24999999999999997], Times[Complex[-0.78471422644353, -0.9906313764027224], Power[Times[Complex[-1.7426678688862403, -1.3308892896287465], genJacobiellk], Rational[-1, 2]]]]
Test Values: {Rule[k, 1], Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[q, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[r, -1.5], Rule[s, -1.5], Rule[u, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

Result: Plus[Complex[0.43301270189221935, 0.24999999999999997], Times[Complex[-0.3766936106342851, -1.225388931598258], Power[Times[Complex[-3.4853357377724805, -2.661778579257493], genJacobiellk], Rational[-1, 2]]]]
Test Values: {Rule[k, 2], Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[q, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[r, -1.5], Rule[s, -1.5], Rule[u, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
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