Elliptic Integrals - 19.9 Inequalities

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
19.9#Ex1 ln ⁡ 4 ≤ K ⁡ ( k ) + ln ⁡ k ′ 4 complete-elliptic-integral-first-kind-K 𝑘 superscript 𝑘 ′ {\displaystyle{\displaystyle\ln 4\leq K\left(k\right)+\ln k^{\prime}}}
\ln@@{4} \leq \compellintKk@{k}+\ln@@{k^{\prime}}		 

ln(4) <= EllipticK(k)+ ln(sqrt(1 - (k)^(2)))

Log[4] <= EllipticK[(k)^2]+ Log[Sqrt[1 - (k)^(2)]]
Failure Failure Error
Failed [3 / 3]
Result: LessEqual[1.3862943611198906, Indeterminate]
Test Values: {Rule[k, 1]}


Result: LessEqual[1.3862943611198906, Complex[1.392181321740353, 0.49253850304507485]]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.9#Ex1 K ⁡ ( k ) + ln ⁡ k ′ ≤ π / 2 complete-elliptic-integral-first-kind-K 𝑘 superscript 𝑘 ′ 𝜋 2 {\displaystyle{\displaystyle K\left(k\right)+\ln k^{\prime}\leq\pi/2}}
\compellintKk@{k}+\ln@@{k^{\prime}} \leq \pi/2		 

EllipticK(k)+ ln(sqrt(1 - (k)^(2))) <= Pi/2

EllipticK[(k)^2]+ Log[Sqrt[1 - (k)^(2)]] <= Pi/2
Failure Failure Error
Failed [3 / 3]
Result: LessEqual[Indeterminate, 1.5707963267948966]
Test Values: {Rule[k, 1]}


Result: LessEqual[Complex[1.392181321740353, 0.49253850304507485], 1.5707963267948966]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.9#Ex2 1 ≤ E ⁡ ( k ) 1 complete-elliptic-integral-second-kind-E 𝑘 {\displaystyle{\displaystyle 1\leq E\left(k\right)}}
1 \leq \compellintEk@{k}		 

1 <= EllipticE(k)

1 <= EllipticE[(k)^2]
Failure Failure Successful [Tested: 3]
Failed [2 / 3]
Result: LessEqual[1.0, Complex[0.40629888645996043, 1.343854231387098]]
Test Values: {Rule[k, 2]}


Result: LessEqual[1.0, Complex[0.2655964076372759, 2.498348127732516]]
Test Values: {Rule[k, 3]}

19.9#Ex2 E ⁡ ( k ) ≤ π / 2 complete-elliptic-integral-second-kind-E 𝑘 𝜋 2 {\displaystyle{\displaystyle E\left(k\right)\leq\pi/2}}
\compellintEk@{k} \leq \pi/2		 

EllipticE(k) <= Pi/2

EllipticE[(k)^2] <= Pi/2
Failure Failure Successful [Tested: 3]
Failed [2 / 3]
Result: LessEqual[Complex[0.40629888645996043, 1.343854231387098], 1.5707963267948966]
Test Values: {Rule[k, 2]}


Result: LessEqual[Complex[0.2655964076372759, 2.498348127732516], 1.5707963267948966]
Test Values: {Rule[k, 3]}

19.9#Ex3 1 ≤ ( 2 / π ) ⁢ 1 - α 2 ⁢ Π ⁡ ( α 2 , k ) ≤ 1 / k ′ 1 2 𝜋 1 superscript 𝛼 2 complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 𝑘 1 superscript 𝑘 ′ {\displaystyle{\displaystyle 1\leq(2/\pi)\sqrt{1-\alpha^{2}}\Pi\left(\alpha^{2% },k\right)\leq 1/k^{\prime}}}
1 \leq (2/\pi)\sqrt{1-\alpha^{2}}\compellintPik@{\alpha^{2}}{k}\leq 1/k^{\prime}		 α 2 < 1 superscript 𝛼 2 1 {\displaystyle{\displaystyle\alpha^{2}<1}} 
1 <= (2/Pi)*sqrt(1 - (alpha)^(2))*EllipticPi((alpha)^(2), k) <= 1/(sqrt(1 - (k)^(2)))

1 <= (2/Pi)*Sqrt[1 - \[Alpha]^(2)]*EllipticPi[\[Alpha]^(2), (k)^2] <= 1/(Sqrt[1 - (k)^(2)])
Failure Failure Error
Failed [3 / 3]
Result: LessEqual[1.0, DirectedInfinity[], DirectedInfinity[]]
Test Values: {Rule[k, 1], Rule[α, 0.5]}


Result: LessEqual[1.0, Complex[0.4804983499812288, -0.6957733039705274], Complex[0.0, -0.5773502691896258]]
Test Values: {Rule[k, 2], Rule[α, 0.5]}

... skip entries to safe data
19.9.E2 1 + k ′ 2 8 < K ⁡ ( k ) ln ⁡ ( 4 / k ′ ) 1 superscript superscript 𝑘 ′ 2 8 complete-elliptic-integral-first-kind-K 𝑘 4 superscript 𝑘 ′ {\displaystyle{\displaystyle 1+\frac{{k^{\prime}}^{2}}{8}<\frac{K\left(k\right% )}{\ln\left(4/k^{\prime}\right)}}}
1+\frac{{k^{\prime}}^{2}}{8} < \frac{\compellintKk@{k}}{\ln@{4/k^{\prime}}}		 

1 +(1 - (k)^(2))/(8) < (EllipticK(k))/(ln(4/(sqrt(1 - (k)^(2)))))

1 +Divide[1 - (k)^(2),8] < Divide[EllipticK[(k)^2],Log[4/(Sqrt[1 - (k)^(2)])]]
Failure Failure Error
Failed [3 / 3]
Result: Less[1.0, Indeterminate]
Test Values: {Rule[k, 1]}


Result: Less[0.625, Complex[0.7573351019929213, 0.13305010797062605]]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.9.E2 K ⁡ ( k ) ln ⁡ ( 4 / k ′ ) < 1 + k ′ 2 4 complete-elliptic-integral-first-kind-K 𝑘 4 superscript 𝑘 ′ 1 superscript superscript 𝑘 ′ 2 4 {\displaystyle{\displaystyle\frac{K\left(k\right)}{\ln\left(4/k^{\prime}\right% )}<1+\frac{{k^{\prime}}^{2}}{4}}}
\frac{\compellintKk@{k}}{\ln@{4/k^{\prime}}} < 1+\frac{{k^{\prime}}^{2}}{4}		 

(EllipticK(k))/(ln(4/(sqrt(1 - (k)^(2))))) < 1 +(1 - (k)^(2))/(4)

Divide[EllipticK[(k)^2],Log[4/(Sqrt[1 - (k)^(2)])]] < 1 +Divide[1 - (k)^(2),4]
Failure Failure Error
Failed [3 / 3]
Result: Less[Indeterminate, 1.0]
Test Values: {Rule[k, 1]}


Result: Less[Complex[0.7573351019929213, 0.13305010797062605], 0.25]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.9.E3 9 + k 2 ⁢ k ′ 2 8 < ( 8 + k 2 ) ⁢ K ⁡ ( k ) ln ⁡ ( 4 / k ′ ) 9 superscript 𝑘 2 superscript superscript 𝑘 ′ 2 8 8 superscript 𝑘 2 complete-elliptic-integral-first-kind-K 𝑘 4 superscript 𝑘 ′ {\displaystyle{\displaystyle 9+\frac{k^{2}{k^{\prime}}^{2}}{8}<\frac{(8+k^{2})% K\left(k\right)}{\ln\left(4/k^{\prime}\right)}}}
9+\frac{k^{2}{k^{\prime}}^{2}}{8} < \frac{(8+k^{2})\compellintKk@{k}}{\ln@{4/k^{\prime}}}		 

9 +((k)^(2)*1 - (k)^(2))/(8) < ((8 + (k)^(2))*EllipticK(k))/(ln(4/(sqrt(1 - (k)^(2)))))

9 +Divide[(k)^(2)*1 - (k)^(2),8] < Divide[(8 + (k)^(2))*EllipticK[(k)^2],Log[4/(Sqrt[1 - (k)^(2)])]]
Failure Failure Error
Failed [3 / 3]
Result: Less[9.0, Indeterminate]
Test Values: {Rule[k, 1]}


Result: Less[9.0, Complex[9.088021223915057, 1.5966012956475137]]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.9.E3 ( 8 + k 2 ) ⁢ K ⁡ ( k ) ln ⁡ ( 4 / k ′ ) < 9.096 8 superscript 𝑘 2 complete-elliptic-integral-first-kind-K 𝑘 4 superscript 𝑘 ′ 9.096 {\displaystyle{\displaystyle\frac{(8+k^{2})K\left(k\right)}{\ln\left(4/k^{% \prime}\right)}<9.096}}
\frac{(8+k^{2})\compellintKk@{k}}{\ln@{4/k^{\prime}}} < 9.096		 

((8 + (k)^(2))*EllipticK(k))/(ln(4/(sqrt(1 - (k)^(2))))) < 9.096

Divide[(8 + (k)^(2))*EllipticK[(k)^2],Log[4/(Sqrt[1 - (k)^(2)])]] < 9.096
Failure Failure Error
Failed [3 / 3]
Result: Less[Indeterminate, 9.096]
Test Values: {Rule[k, 1]}


Result: Less[Complex[9.088021223915057, 1.5966012956475137], 9.096]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.9.E4 ( 1 + k ′ 3 / 2 2 ) 2 / 3 ≤ 2 π ⁢ E ⁡ ( k ) superscript 1 superscript superscript 𝑘 ′ 3 2 2 2 3 2 𝜋 complete-elliptic-integral-second-kind-E 𝑘 {\displaystyle{\displaystyle\left(\frac{1+{k^{\prime}}^{3/2}}{2}\right)^{2/3}% \leq\frac{2}{\pi}E\left(k\right)}}
\left(\frac{1+{k^{\prime}}^{3/2}}{2}\right)^{2/3} \leq \frac{2}{\pi}\compellintEk@{k}		 

((1 +(sqrt(1 - (k)^(2)))^(3/2))/(2))^(2/3) <= (2)/(Pi)*EllipticE(k)

(Divide[1 +(Sqrt[1 - (k)^(2)])^(3/2),2])^(2/3) <= Divide[2,Pi]*EllipticE[(k)^2]
Failure Failure Successful [Tested: 3]
Failed [2 / 3]
Result: LessEqual[Complex[0.2518251425072316, 0.8700591952646104], Complex[0.2586579046113418, 0.8555241748808654]]
Test Values: {Rule[k, 2]}


Result: LessEqual[Complex[0.1858923839966674, 1.6059081831429025], Complex[0.16908392457168991, 1.5904978163720476]]
Test Values: {Rule[k, 3]}

19.9.E4 2 π ⁢ E ⁡ ( k ) ≤ ( 1 + k ′ 2 2 ) 1 / 2 2 𝜋 complete-elliptic-integral-second-kind-E 𝑘 superscript 1 superscript superscript 𝑘 ′ 2 2 1 2 {\displaystyle{\displaystyle\frac{2}{\pi}E\left(k\right)\leq\left(\frac{1+{k^{% \prime}}^{2}}{2}\right)^{1/2}}}
\frac{2}{\pi}\compellintEk@{k} \leq \left(\frac{1+{k^{\prime}}^{2}}{2}\right)^{1/2}		 

(2)/(Pi)*EllipticE(k) <= ((1 +1 - (k)^(2))/(2))^(1/2)

Divide[2,Pi]*EllipticE[(k)^2] <= (Divide[1 +1 - (k)^(2),2])^(1/2)
Failure Failure Successful [Tested: 3]
Failed [2 / 3]
Result: LessEqual[Complex[0.2586579046113418, 0.8555241748808654], Complex[0.0, 1.0]]
Test Values: {Rule[k, 2]}


Result: LessEqual[Complex[0.16908392457168991, 1.5904978163720476], Complex[0.0, 1.8708286933869707]]
Test Values: {Rule[k, 3]}

19.9.E5 ln ⁡ ( 1 + k ′ ) 2 k < π ⁢ K ′ ⁡ ( k ) 2 ⁢ K ⁡ ( k ) superscript 1 superscript 𝑘 ′ 2 𝑘 𝜋 complementary-complete-elliptic-integral-first-kind-K 𝑘 2 complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle\ln\frac{(1+\sqrt{k^{\prime}})^{2}}{k}<\frac{\pi{K% ^{\prime}}\left(k\right)}{2K\left(k\right)}}}
\ln@@{\frac{(1+\sqrt{k^{\prime}})^{2}}{k}} < \frac{\pi\ccompellintKk@{k}}{2\compellintKk@{k}}		 

ln(((1 +sqrt(sqrt(1 - (k)^(2))))^(2))/(k)) < (Pi*EllipticCK(k))/(2*EllipticK(k))

Log[Divide[(1 +Sqrt[Sqrt[1 - (k)^(2)]])^(2),k]] < Divide[Pi*EllipticK[1-(k)^2],2*EllipticK[(k)^2]]
Failure Failure Error
Failed [3 / 3]
Result: False
Test Values: {Rule[k, 1]}


Result: Less[Complex[0.8314429455293103, 0.8983332083070389], Complex[0.762166367418117, 0.9750101446769989]]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.9.E5 π ⁢ K ′ ⁡ ( k ) 2 ⁢ K ⁡ ( k ) < ln ⁡ 2 ⁢ ( 1 + k ′ ) k 𝜋 complementary-complete-elliptic-integral-first-kind-K 𝑘 2 complete-elliptic-integral-first-kind-K 𝑘 2 1 superscript 𝑘 ′ 𝑘 {\displaystyle{\displaystyle\frac{\pi{K^{\prime}}\left(k\right)}{2K\left(k% \right)}<\ln\frac{2(1+k^{\prime})}{k}}}
\frac{\pi\ccompellintKk@{k}}{2\compellintKk@{k}} < \ln@@{\frac{2(1+k^{\prime})}{k}}		 

(Pi*EllipticCK(k))/(2*EllipticK(k)) < ln((2*(1 +sqrt(1 - (k)^(2))))/(k))

Divide[Pi*EllipticK[1-(k)^2],2*EllipticK[(k)^2]] < Log[Divide[2*(1 +Sqrt[1 - (k)^(2)]),k]]
Failure Failure Error
Failed [2 / 3]
Result: Less[Complex[0.762166367418117, 0.9750101446769989], Complex[0.6931471805599452, 1.0471975511965976]]
Test Values: {Rule[k, 2]}


Result: Less[Complex[0.7130154358988758, 1.1147297033963086], Complex[0.6931471805599453, 1.2309594173407747]]
Test Values: {Rule[k, 3]}

19.9.E6 ( 1 - 3 4 ⁢ k 2 ) - 1 / 2 < 4 π ⁢ k 2 ⁢ ( K ⁡ ( k ) - E ⁡ ( k ) ) superscript 1 3 4 superscript 𝑘 2 1 2 4 𝜋 superscript 𝑘 2 complete-elliptic-integral-first-kind-K 𝑘 complete-elliptic-integral-second-kind-E 𝑘 {\displaystyle{\displaystyle(1-\tfrac{3}{4}k^{2})^{-1/2}<\frac{4}{\pi k^{2}}(K% \left(k\right)-E\left(k\right))}}
(1-\tfrac{3}{4}k^{2})^{-1/2} < \frac{4}{\pi k^{2}}(\compellintKk@{k}-\compellintEk@{k})		 

(1 -(3)/(4)*(k)^(2))^(- 1/2) < (4)/(Pi*(k)^(2))*(EllipticK(k)- EllipticE(k))

(1 -Divide[3,4]*(k)^(2))^(- 1/2) < Divide[4,Pi*(k)^(2)]*(EllipticK[(k)^2]- EllipticE[(k)^2])
Failure Failure Error
Failed [3 / 3]
Result: Less[2.0, DirectedInfinity[]]
Test Values: {Rule[k, 1]}


Result: Less[Complex[0.0, -0.7071067811865475], Complex[0.13896654948167025, -0.7709822125950203]]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.9.E6 4 π ⁢ k 2 ⁢ ( K ⁡ ( k ) - E ⁡ ( k ) ) < ( k ′ ) - 3 / 4 4 𝜋 superscript 𝑘 2 complete-elliptic-integral-first-kind-K 𝑘 complete-elliptic-integral-second-kind-E 𝑘 superscript superscript 𝑘 ′ 3 4 {\displaystyle{\displaystyle\frac{4}{\pi k^{2}}(K\left(k\right)-E\left(k\right% ))<(k^{\prime})^{-3/4}}}
\frac{4}{\pi k^{2}}(\compellintKk@{k}-\compellintEk@{k}) < (k^{\prime})^{-3/4}		 

(4)/(Pi*(k)^(2))*(EllipticK(k)- EllipticE(k)) < (sqrt(1 - (k)^(2)))^(- 3/4)

Divide[4,Pi*(k)^(2)]*(EllipticK[(k)^2]- EllipticE[(k)^2]) < (Sqrt[1 - (k)^(2)])^(- 3/4)
Failure Failure Error
Failed [3 / 3]
Result: Less[DirectedInfinity[], DirectedInfinity[]]
Test Values: {Rule[k, 1]}


Result: Less[Complex[0.13896654948167025, -0.7709822125950203], Complex[0.2534656958546175, -0.6119203205285516]]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.9.E7 ( 1 - 1 4 ⁢ k 2 ) - 1 / 2 < 4 π ⁢ k 2 ⁢ ( E ⁡ ( k ) - k ′ 2 ⁢ K ⁡ ( k ) ) superscript 1 1 4 superscript 𝑘 2 1 2 4 𝜋 superscript 𝑘 2 complete-elliptic-integral-second-kind-E 𝑘 superscript superscript 𝑘 ′ 2 complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle(1-\tfrac{1}{4}k^{2})^{-1/2}<\frac{4}{\pi k^{2}}(E% \left(k\right)-{k^{\prime}}^{2}K\left(k\right))}}
(1-\tfrac{1}{4}k^{2})^{-1/2} < \frac{4}{\pi k^{2}}(\compellintEk@{k}-{k^{\prime}}^{2}\compellintKk@{k})		 

(1 -(1)/(4)*(k)^(2))^(- 1/2) < (4)/(Pi*(k)^(2))*(EllipticE(k)-1 - (k)^(2)*EllipticK(k))

(1 -Divide[1,4]*(k)^(2))^(- 1/2) < Divide[4,Pi*(k)^(2)]*(EllipticE[(k)^2]-1 - (k)^(2)*EllipticK[(k)^2])
Failure Failure Error
Failed [3 / 3]
Result: Less[1.1547005383792517, DirectedInfinity[]]
Test Values: {Rule[k, 1]}


Result: Less[DirectedInfinity[], Complex[-1.2621629410274844, 1.800642588058783]]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.9.E8 k ′ < E ⁡ ( k ) K ⁡ ( k ) superscript 𝑘 ′ complete-elliptic-integral-second-kind-E 𝑘 complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle k^{\prime}<\frac{E\left(k\right)}{K\left(k\right)% }}}
k^{\prime} < \frac{\compellintEk@{k}}{\compellintKk@{k}}		 

sqrt(1 - (k)^(2)) < (EllipticE(k))/(EllipticK(k))

Sqrt[1 - (k)^(2)] < Divide[EllipticE[(k)^2],EllipticK[(k)^2]]
Failure Failure Error
Failed [3 / 3]
Result: False
Test Values: {Rule[k, 1]}


Result: Less[Complex[0.0, 1.7320508075688772], Complex[-0.5907718728609501, 0.8386174564999851]]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.9.E8 E ⁡ ( k ) K ⁡ ( k ) < ( 1 + k ′ 2 ) 2 complete-elliptic-integral-second-kind-E 𝑘 complete-elliptic-integral-first-kind-K 𝑘 superscript 1 superscript 𝑘 ′ 2 2 {\displaystyle{\displaystyle\frac{E\left(k\right)}{K\left(k\right)}<\left(% \frac{1+k^{\prime}}{2}\right)^{2}}}
\frac{\compellintEk@{k}}{\compellintKk@{k}} < \left(\frac{1+k^{\prime}}{2}\right)^{2}		 

(EllipticE(k))/(EllipticK(k)) < ((1 +sqrt(1 - (k)^(2)))/(2))^(2)

Divide[EllipticE[(k)^2],EllipticK[(k)^2]] < (Divide[1 +Sqrt[1 - (k)^(2)],2])^(2)
Failure Failure Error
Failed [2 / 3]
Result: Less[Complex[-0.5907718728609501, 0.8386174564999851], Complex[-0.4999999999999999, 0.8660254037844386]]
Test Values: {Rule[k, 2]}


Result: Less[Complex[-1.9604512687154212, 1.5690726247192568], Complex[-1.7500000000000004, 1.4142135623730951]]
Test Values: {Rule[k, 3]}

19.9.E9 L ⁢ ( a , b ) = 4 ⁢ a ⁢ E ⁡ ( k ) 𝐿 𝑎 𝑏 4 𝑎 complete-elliptic-integral-second-kind-E 𝑘 {\displaystyle{\displaystyle L(a,b)=4aE\left(k\right)}}
L(a,b) = 4a\compellintEk@{k}		 k 2 = 1 - ( b 2 / a 2 ) , a > b formulae-sequence superscript 𝑘 2 1 superscript 𝑏 2 superscript 𝑎 2 𝑎 𝑏 {\displaystyle{\displaystyle k^{2}=1-(b^{2}/a^{2}),a>b}} 
L(a , b) = 4*a*EllipticE(k)

L[a , b] == 4*a*EllipticE[(k)^2]
Error Failure - Error
19.9.E11 ϕ ≤ F ⁡ ( ϕ , k ) italic-ϕ elliptic-integral-first-kind-F italic-ϕ 𝑘 {\displaystyle{\displaystyle\phi\leq F\left(\phi,k\right)}}
\phi \leq \incellintFk@{\phi}{k}		 

phi <= EllipticF(sin(phi), k)

\[Phi] <= EllipticF[\[Phi], (k)^2]
Failure Failure
Failed [4 / 30]
Result: -1.500000000 <= -3.340677542
Test Values: {phi = -3/2, k = 1}


Result: -.5000000000 <= -.5222381033
Test Values: {phi = -1/2, k = 1}

... skip entries to safe data
Failed [28 / 30]
Result: LessEqual[Complex[0.43301270189221935, 0.24999999999999997], Complex[0.43180375739814203, 0.27142936483528934]]
Test Values: {Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: LessEqual[Complex[0.43301270189221935, 0.24999999999999997], Complex[0.3965687056216178, 0.33175091278780894]]
Test Values: {Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.9.E12 E ⁡ ( ϕ , k ) ≤ ϕ elliptic-integral-second-kind-E italic-ϕ 𝑘 italic-ϕ {\displaystyle{\displaystyle E\left(\phi,k\right)\leq\phi}}
\incellintEk@{\phi}{k} \leq \phi		 

EllipticE(sin(phi), k) <= phi
 
EllipticE[\[Phi], (k)^2] <= \[Phi]
 Failure Failure
Failed [4 / 30]
Result: -.9974949866 <= -1.500000000
Test Values: {phi = -3/2, k = 1}

Result: -.4794255386 <= -.5000000000
Test Values: {phi = -1/2, k = 1}

... skip entries to safe data
Failed [27 / 30]
Result: LessEqual[Complex[0.43278851685803155, 0.22929764467344024], Complex[0.43301270189221935, 0.24999999999999997]]
Test Values: {Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

Result: LessEqual[Complex[0.44208095936294645, 0.16535187593702125], Complex[0.43301270189221935, 0.24999999999999997]]
Test Values: {Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.9.E13 Π ⁡ ( ϕ , α 2 , 0 ) ≤ Π ⁡ ( ϕ , α 2 , k ) elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 0 elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 𝑘 {\displaystyle{\displaystyle\Pi\left(\phi,\alpha^{2},0\right)\leq\Pi\left(\phi% ,\alpha^{2},k\right)}}
\incellintPik@{\phi}{\alpha^{2}}{0} \leq \incellintPik@{\phi}{\alpha^{2}}{k}		 

EllipticPi(sin(phi), (alpha)^(2), 0) <= EllipticPi(sin(phi), (alpha)^(2), k)
 
EllipticPi[\[Alpha]^(2), \[Phi],(0)^2] <= EllipticPi[\[Alpha]^(2), \[Phi],(k)^2]
 Failure Failure
Failed [8 / 90]
Result: -.6351972518 <= -.6692391842
Test Values: {alpha = 3/2, phi = -1/2, k = 1}

Result: -.6351972518 <= -.9273807742
Test Values: {alpha = 3/2, phi = -1/2, k = 2}

... skip entries to safe data
Failed [84 / 90]
Result: LessEqual[Complex[0.4032669574270382, 0.3492210121777662], Complex[0.39392267303966433, 0.37152709024037445]]
Test Values: {Rule[k, 1], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

Result: LessEqual[Complex[0.4032669574270382, 0.3492210121777662], Complex[0.33490711362096304, 0.4200642464932446]]
Test Values: {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
19.9.E14 3 1 + Δ + cos ⁡ ϕ < F ⁡ ( ϕ , k ) sin ⁡ ϕ 3 1 Δ italic-ϕ elliptic-integral-first-kind-F italic-ϕ 𝑘 italic-ϕ {\displaystyle{\displaystyle\frac{3}{1+\Delta+\cos\phi}<\frac{F\left(\phi,k% \right)}{\sin\phi}}}
\frac{3}{1+\Delta+\cos@@{\phi}} < \frac{\incellintFk@{\phi}{k}}{\sin@@{\phi}}		 

(3)/(1 + Delta + cos(phi)) < (EllipticF(sin(phi), k))/(sin(phi))
 
Divide[3,1 + \[CapitalDelta]+ Cos[\[Phi]]] < Divide[EllipticF[\[Phi], (k)^2],Sin[\[Phi]]]
 Failure Failure
Failed [16 / 300]
Result: 7.945282179 < 1.089299717
Test Values: {Delta = -3/2, phi = -1/2, k = 1}

Result: 7.945282179 < 1.412977582
Test Values: {Delta = -3/2, phi = -1/2, k = 2}

... skip entries to safe data
Failed [284 / 300]
Result: Less[Complex[1.261572446843062, -0.07667841479591199], Complex[1.0384958486950706, 0.07695378095553612]]
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: Less[Complex[1.261572446843062, -0.07667841479591199], Complex[1.0325857379409573, 0.21946385233164167]]
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
19.9.E14 F ⁡ ( ϕ , k ) sin ⁡ ϕ < 1 ( Δ ⁢ cos ⁡ ϕ ) 1 / 3 elliptic-integral-first-kind-F italic-ϕ 𝑘 italic-ϕ 1 superscript Δ italic-ϕ 1 3 {\displaystyle{\displaystyle\frac{F\left(\phi,k\right)}{\sin\phi}<\frac{1}{(% \Delta\cos\phi)^{1/3}}}}
\frac{\incellintFk@{\phi}{k}}{\sin@@{\phi}} < \frac{1}{(\Delta\cos@@{\phi})^{1/3}}		 

(EllipticF(sin(phi), k))/(sin(phi)) < (1)/((Delta*cos(phi))^(1/3))
 
Divide[EllipticF[\[Phi], (k)^2],Sin[\[Phi]]] < Divide[1,(\[CapitalDelta]*Cos[\[Phi]])^(1/3)]
 Failure Failure
Failed [20 / 300]
Result: 1.675417084 < 1.170093898
Test Values: {Delta = -3/2, phi = -2, k = 1}

Result: 1.675417084 < 1.170093898
Test Values: {Delta = -3/2, phi = 2, k = 1}

... skip entries to safe data
Failed [298 / 300]
Result: Less[Complex[1.0384958486950706, 0.07695378095553612], Complex[1.2731409874856745, -0.17545913345292982]]
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: Less[Complex[1.0325857379409573, 0.21946385233164167], Complex[1.2731409874856745, -0.17545913345292982]]
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
19.9.E15 1 < F ⁡ ( ϕ , k ) / ( ( sin ⁡ ϕ ) ⁢ ln ⁡ ( 4 Δ + cos ⁡ ϕ ) ) 1 elliptic-integral-first-kind-F italic-ϕ 𝑘 italic-ϕ 4 Δ italic-ϕ {\displaystyle{\displaystyle 1<F\left(\phi,k\right)\bigg{/}\left((\sin\phi)\ln% \left(\frac{4}{\Delta+\cos\phi}\right)\right)}}
1 < \incellintFk@{\phi}{k}\bigg{/}\left((\sin@@{\phi})\ln@{\frac{4}{\Delta+\cos@@{\phi}}}\right)		 

1 < EllipticF(sin(phi), k)/((sin(phi))*ln((4)/(Delta + cos(phi))))
 
1 < EllipticF[\[Phi], (k)^2]/((Sin[\[Phi]])*Log[Divide[4,\[CapitalDelta]+ Cos[\[Phi]]]])
 Failure Failure
Failed [6 / 300]
Result: 1. < .4615167558
Test Values: {Delta = -1/2, phi = -1/2, k = 1}

Result: 1. < .5986532627
Test Values: {Delta = -1/2, phi = -1/2, k = 2}

... skip entries to safe data
Failed [288 / 300]
Result: Less[1.0, Complex[0.9573719244599448, 0.16621131448588694]]
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: Less[1.0, Complex[0.9388814261604885, 0.2980132161872323]]
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
19.9.E15 F ⁡ ( ϕ , k ) / ( ( sin ⁡ ϕ ) ⁢ ln ⁡ ( 4 Δ + cos ⁡ ϕ ) ) < 4 2 + ( 1 + k 2 ) ⁢ sin 2 ⁡ ϕ elliptic-integral-first-kind-F italic-ϕ 𝑘 italic-ϕ 4 Δ italic-ϕ 4 2 1 superscript 𝑘 2 2 italic-ϕ {\displaystyle{\displaystyle F\left(\phi,k\right)\bigg{/}\left((\sin\phi)\ln% \left(\frac{4}{\Delta+\cos\phi}\right)\right)<\frac{4}{2+(1+k^{2}){\sin^{2}}% \phi}}}
\incellintFk@{\phi}{k}\bigg{/}\left((\sin@@{\phi})\ln@{\frac{4}{\Delta+\cos@@{\phi}}}\right) < \frac{4}{2+(1+k^{2})\sin^{2}@@{\phi}}		 

EllipticF(sin(phi), k)/((sin(phi))*ln((4)/(Delta + cos(phi)))) < (4)/(2 +(1 + (k)^(2))*(sin(phi))^(2))
 
EllipticF[\[Phi], (k)^2]/((Sin[\[Phi]])*Log[Divide[4,\[CapitalDelta]+ Cos[\[Phi]]]]) < Divide[4,2 +(1 + (k)^(2))*(Sin[\[Phi]])^(2)]
 Failure Failure
Failed [20 / 300]
Result: 3.582850518 < 1.002508151
Test Values: {Delta = 3/2, phi = -3/2, k = 1}

Result: 3.582850518 < 1.002508151
Test Values: {Delta = 3/2, phi = 3/2, k = 1}

... skip entries to safe data
Failed [296 / 300]
Result: Less[Complex[0.9573719244599448, 0.16621131448588694], Complex[1.7102149955099495, -0.29913282294542826]]
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: Less[Complex[0.9388814261604885, 0.2980132161872323], Complex[1.3149325512421652, -0.4880625346303866]]
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
19.9.E16 F ⁡ ( ϕ , k ) = 2 π ⁢ K ⁡ ( k ′ ) ⁢ ln ⁡ ( 4 Δ + cos ⁡ ϕ ) - θ ⁢ Δ 2 elliptic-integral-first-kind-F italic-ϕ 𝑘 2 𝜋 complete-elliptic-integral-first-kind-K superscript 𝑘 ′ 4 Δ italic-ϕ 𝜃 superscript Δ 2 {\displaystyle{\displaystyle F\left(\phi,k\right)=\frac{2}{\pi}K\left(k^{% \prime}\right)\ln\left(\frac{4}{\Delta+\cos\phi}\right)-\theta\Delta^{2}}}
\incellintFk@{\phi}{k} = \frac{2}{\pi}\compellintKk@{k^{\prime}}\ln@{\frac{4}{\Delta+\cos@@{\phi}}}-\theta\Delta^{2}		 ( sin ⁡ ϕ ) / 8 < θ , θ < ( ln ⁡ 2 ) / ( k 2 ⁢ sin ⁡ ϕ ) formulae-sequence italic-ϕ 8 𝜃 𝜃 2 superscript 𝑘 2 italic-ϕ {\displaystyle{\displaystyle(\sin\phi)/8<\theta,\theta<(\ln 2)/(k^{2}\sin\phi)}} 
EllipticF(sin(phi), k) = (2)/(Pi)*EllipticK(sqrt(1 - (k)^(2)))*ln((4)/(Delta + cos(phi)))- theta*(Delta)^(2)
 
EllipticF[\[Phi], (k)^2] == Divide[2,Pi]*EllipticK[(Sqrt[1 - (k)^(2)])^2]*Log[Divide[4,\[CapitalDelta]+ Cos[\[Phi]]]]- \[Theta]*\[CapitalDelta]^(2)
 Failure Failure
Failed [30 / 30]
Result: 2.264395299+.9232968251*I
Test Values: {Delta = 1/2*3^(1/2)+1/2*I, phi = 3/2, theta = 1/2, k = 1}

Result: -.185868314e-1+.7122804653*I
Test Values: {Delta = 1/2*3^(1/2)+1/2*I, phi = 1/2, theta = 1/2, k = 1}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[1.4412941413043292, 0.5689187621917111]
Test Values: {Rule[k, 1], Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, 0.5], Rule[ϕ, 1.5]}

Result: Complex[-0.5132046492108906, 0.2967418012807382]
Test Values: {Rule[k, 1], Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, 0.5], Rule[ϕ, 0.5]}

... skip entries to safe data
19.9.E17 L ≤ F ⁡ ( ϕ , k ) 𝐿 elliptic-integral-first-kind-F italic-ϕ 𝑘 {\displaystyle{\displaystyle L\leq F\left(\phi,k\right)}}
L \leq \incellintFk@{\phi}{k}		 

L <= EllipticF(sin(phi), k)
 
L <= EllipticF[\[Phi], (k)^2]
 Failure Failure
Failed [24 / 300]
Result: -1.500000000 <= -3.340677542
Test Values: {L = -3/2, phi = -3/2, k = 1}

Result: -1.500000000 <= -1.523452443
Test Values: {L = -3/2, phi = -2, k = 1}

... skip entries to safe data
Failed [288 / 300]
Result: LessEqual[Complex[0.43301270189221935, 0.24999999999999997], Complex[0.43180375739814203, 0.27142936483528934]]
Test Values: {Rule[k, 1], Rule[L, 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: LessEqual[Complex[0.43301270189221935, 0.24999999999999997], Complex[0.3965687056216178, 0.33175091278780894]]
Test Values: {Rule[k, 2], Rule[L, 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
19.9.E17 F ⁡ ( ϕ , k ) ≤ U ⁢ L elliptic-integral-first-kind-F italic-ϕ 𝑘 𝑈 𝐿 {\displaystyle{\displaystyle F\left(\phi,k\right)\leq\sqrt{UL}}}
\incellintFk@{\phi}{k} \leq \sqrt{UL}		 

EllipticF(sin(phi), k) <= sqrt(U*L)
 
EllipticF[\[Phi], (k)^2] <= Sqrt[U*L]
 Failure Failure Successful [Tested: 300]
Failed [300 / 300]
Result: LessEqual[Complex[0.43180375739814203, 0.27142936483528934], Complex[0.43301270189221935, 0.24999999999999997]]
Test Values: {Rule[k, 1], Rule[L, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[U, 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: LessEqual[Complex[0.3965687056216178, 0.33175091278780894], Complex[0.43301270189221935, 0.24999999999999997]]
Test Values: {Rule[k, 2], Rule[L, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[U, 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
19.9.E17 U ⁢ L ≤ 1 2 ⁢ ( U + L ) 𝑈 𝐿 1 2 𝑈 𝐿 {\displaystyle{\displaystyle\sqrt{UL}\leq\tfrac{1}{2}(U+L)}}
\sqrt{UL} \leq \tfrac{1}{2}(U+L)		 

sqrt(U*L) <= (1)/(2)*(U + L)
 
Sqrt[U*L] <= Divide[1,2]*(U + L)
 Failure Failure
Failed [9 / 100]
Result: 1.500000000 <= -1.500000000
Test Values: {L = -3/2, U = -3/2}

Result: .8660254040 <= -1.
Test Values: {L = -3/2, U = -1/2}

... skip entries to safe data
Failed [91 / 100]
Result: LessEqual[Complex[0.43301270189221935, 0.24999999999999997], Complex[0.43301270189221935, 0.24999999999999997]]
Test Values: {Rule[L, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[U, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

Result: LessEqual[Complex[0.12940952255126037, 0.48296291314453416], Complex[0.09150635094610973, 0.34150635094610965]]
Test Values: {Rule[L, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[U, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.9.E17 1 2 ⁢ ( U + L ) ≤ U 1 2 𝑈 𝐿 𝑈 {\displaystyle{\displaystyle\tfrac{1}{2}(U+L)\leq U}}
\tfrac{1}{2}(U+L) \leq U		 

(1)/(2)*(U + L) <= U
 
Divide[1,2]*(U + L) <= U
 Failure Failure
Failed [15 / 100]
Result: -1.750000000 <= -2.
Test Values: {L = -3/2, U = -2}

Result: 0. <= -1.500000000
Test Values: {L = 3/2, U = -3/2}

... skip entries to safe data
Failed [79 / 100]
Result: LessEqual[Complex[0.43301270189221935, 0.24999999999999997], Complex[0.43301270189221935, 0.24999999999999997]]
Test Values: {Rule[L, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[U, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

Result: LessEqual[Complex[0.09150635094610973, 0.34150635094610965], Complex[-0.2499999999999999, 0.43301270189221935]]
Test Values: {Rule[L, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[U, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.9#Ex4 L = ( 1 / σ ) ⁢ arctanh ⁡ ( σ ⁢ sin ⁡ ϕ ) 𝐿 1 𝜎 hyperbolic-inverse-tangent 𝜎 italic-ϕ {\displaystyle{\displaystyle L=(1/\sigma)\operatorname{arctanh}\left(\sigma% \sin\phi\right)}}
L = (1/\sigma)\atanh@{\sigma\sin@@{\phi}}		 σ = ( 1 + k 2 ) / 2 𝜎 1 superscript 𝑘 2 2 {\displaystyle{\displaystyle\sigma=\sqrt{(1+k^{2})/2}}} 
L = (1/sigma)*arctanh(sigma*sin(phi))
 
L == (1/\[Sigma])*ArcTanh[\[Sigma]*Sin[\[Phi]]]
 Failure Failure
Failed [300 / 300]
Result: .1841715885+.458206673e-1*I
Test Values: {L = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, sigma = 1/2*3^(1/2)+1/2*I}

Result: -.197696883e-1+.4084290873*I
Test Values: {L = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, sigma = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.008169183554908921, 0.015254361571334585]
Test Values: {Rule[L, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[σ, 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[0.6990489693230986, -0.19299436497537428]
Test Values: {Rule[L, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[σ, 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[2, 3]], Pi]]]]}

... skip entries to safe data
19.9#Ex5 U = 1 2 ⁢ arctanh ⁡ ( sin ⁡ ϕ ) + 1 2 ⁢ k - 1 ⁢ arctanh ⁡ ( k ⁢ sin ⁡ ϕ ) 𝑈 1 2 hyperbolic-inverse-tangent italic-ϕ 1 2 superscript 𝑘 1 hyperbolic-inverse-tangent 𝑘 italic-ϕ {\displaystyle{\displaystyle U=\tfrac{1}{2}\operatorname{arctanh}\left(\sin% \phi\right)+\tfrac{1}{2}k^{-1}\operatorname{arctanh}\left(k\sin\phi\right)}}
U = \tfrac{1}{2}\atanh@{\sin@@{\phi}}+\tfrac{1}{2}k^{-1}\atanh@{k\sin@@{\phi}}		 

U = (1)/(2)*arctanh(sin(phi))+(1)/(2)*(k)^(- 1)* arctanh(k*sin(phi))
 
U == Divide[1,2]*ArcTanh[Sin[\[Phi]]]+Divide[1,2]*(k)^(- 1)* ArcTanh[k*Sin[\[Phi]]]
 Failure Failure
Failed [300 / 300]
Result: .451553750e-1-.1773780507*I
Test Values: {U = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, k = 1}

Result: .3250459090-.1674857034*I
Test Values: {U = 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: Complex[0.0012089444940770466, -0.021429364835289427]
Test Values: {Rule[k, 1], Rule[U, 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[0.04320077983427789, -0.07655275524887523]
Test Values: {Rule[k, 2], Rule[U, 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
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