19.9

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
19.9#Ex1	${\displaystyle{\displaystyle\ln 4\leq K\left(k\right)+\ln k^{\prime}}}$ \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	${\displaystyle{\displaystyle K\left(k\right)+\ln k^{\prime}\leq\pi/2}}$ \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	${\displaystyle{\displaystyle 1\leq E\left(k\right)}}$ 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	${\displaystyle{\displaystyle E\left(k\right)\leq\pi/2}}$ \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	${\displaystyle{\displaystyle 1\leq(2/\pi)\sqrt{1-\alpha^{2}}\Pi\left(\alpha^{2},k\right)\leq 1/k^{\prime}}}$ 1 \leq (2/\pi)\sqrt{1-\alpha^{2}}\compellintPik@{\alpha^{2}}{k}\leq 1/k^{\prime}	${\displaystyle{\displaystyle\alpha^{2}<1}}$	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	${\displaystyle{\displaystyle 1+\frac{{k^{\prime}}^{2}}{8}<\frac{K\left(k\right)}{\ln\left(4/k^{\prime}\right)}}}$ 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	${\displaystyle{\displaystyle\frac{K\left(k\right)}{\ln\left(4/k^{\prime}\right)}<1+\frac{{k^{\prime}}^{2}}{4}}}$ \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	${\displaystyle{\displaystyle 9+\frac{k^{2}{k^{\prime}}^{2}}{8}<\frac{(8+k^{2})K\left(k\right)}{\ln\left(4/k^{\prime}\right)}}}$ 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	${\displaystyle{\displaystyle\frac{(8+k^{2})K\left(k\right)}{\ln\left(4/k^{\prime}\right)}<9.096}}$ \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	${\displaystyle{\displaystyle\left(\frac{1+{k^{\prime}}^{3/2}}{2}\right)^{2/3}\leq\frac{2}{\pi}E\left(k\right)}}$ \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	${\displaystyle{\displaystyle\frac{2}{\pi}E\left(k\right)\leq\left(\frac{1+{k^{\prime}}^{2}}{2}\right)^{1/2}}}$ \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	${\displaystyle{\displaystyle\ln\frac{(1+\sqrt{k^{\prime}})^{2}}{k}<\frac{\pi{K^{\prime}}\left(k\right)}{2K\left(k\right)}}}$ \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	${\displaystyle{\displaystyle\frac{\pi{K^{\prime}}\left(k\right)}{2K\left(k\right)}<\ln\frac{2(1+k^{\prime})}{k}}}$ \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	${\displaystyle{\displaystyle(1-\tfrac{3}{4}k^{2})^{-1/2}<\frac{4}{\pi k^{2}}(K\left(k\right)-E\left(k\right))}}$ (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	${\displaystyle{\displaystyle\frac{4}{\pi k^{2}}(K\left(k\right)-E\left(k\right))<(k^{\prime})^{-3/4}}}$ \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	${\displaystyle{\displaystyle(1-\tfrac{1}{4}k^{2})^{-1/2}<\frac{4}{\pi k^{2}}(E\left(k\right)-{k^{\prime}}^{2}K\left(k\right))}}$ (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	${\displaystyle{\displaystyle k^{\prime}<\frac{E\left(k\right)}{K\left(k\right)}}}$ 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	${\displaystyle{\displaystyle\frac{E\left(k\right)}{K\left(k\right)}<\left(\frac{1+k^{\prime}}{2}\right)^{2}}}$ \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	${\displaystyle{\displaystyle L(a,b)=4aE\left(k\right)}}$ L(a,b) = 4a\compellintEk@{k}	${\displaystyle{\displaystyle k^{2}=1-(b^{2}/a^{2}),a>b}}$	L(a , b) = 4*a*EllipticE(k)	L[a , b] == 4*a*EllipticE[(k)^2]	Error	Failure	-	Error
19.9.E11	${\displaystyle{\displaystyle\phi\leq F\left(\phi,k\right)}}$ \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	${\displaystyle{\displaystyle E\left(\phi,k\right)\leq\phi}}$ \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	${\displaystyle{\displaystyle\Pi\left(\phi,\alpha^{2},0\right)\leq\Pi\left(\phi,\alpha^{2},k\right)}}$ \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	${\displaystyle{\displaystyle\frac{3}{1+\Delta+\cos\phi}<\frac{F\left(\phi,k\right)}{\sin\phi}}}$ \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	${\displaystyle{\displaystyle\frac{F\left(\phi,k\right)}{\sin\phi}<\frac{1}{(\Delta\cos\phi)^{1/3}}}}$ \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	${\displaystyle{\displaystyle 1<F\left(\phi,k\right)\bigg{/}\left((\sin\phi)\ln\left(\frac{4}{\Delta+\cos\phi}\right)\right)}}$ 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	${\displaystyle{\displaystyle F\left(\phi,k\right)\bigg{/}\left((\sin\phi)\ln\left(\frac{4}{\Delta+\cos\phi}\right)\right)<\frac{4}{2+(1+k^{2}){\sin^{2}}\phi}}}$ \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	${\displaystyle{\displaystyle F\left(\phi,k\right)=\frac{2}{\pi}K\left(k^{\prime}\right)\ln\left(\frac{4}{\Delta+\cos\phi}\right)-\theta\Delta^{2}}}$ \incellintFk@{\phi}{k} = \frac{2}{\pi}\compellintKk@{k^{\prime}}\ln@{\frac{4}{\Delta+\cos@@{\phi}}}-\theta\Delta^{2}	${\displaystyle{\displaystyle(\sin\phi)/8<\theta,\theta<(\ln 2)/(k^{2}\sin\phi)}}$	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	${\displaystyle{\displaystyle L\leq F\left(\phi,k\right)}}$ 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	${\displaystyle{\displaystyle F\left(\phi,k\right)\leq\sqrt{UL}}}$ \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	${\displaystyle{\displaystyle\sqrt{UL}\leq\tfrac{1}{2}(U+L)}}$ \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	${\displaystyle{\displaystyle\tfrac{1}{2}(U+L)\leq U}}$ \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	${\displaystyle{\displaystyle L=(1/\sigma)\operatorname{arctanh}\left(\sigma\sin\phi\right)}}$ L = (1/\sigma)\atanh@{\sigma\sin@@{\phi}}	${\displaystyle{\displaystyle\sigma=\sqrt{(1+k^{2})/2}}}$	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	${\displaystyle{\displaystyle U=\tfrac{1}{2}\operatorname{arctanh}\left(\sin\phi\right)+\tfrac{1}{2}k^{-1}\operatorname{arctanh}\left(k\sin\phi\right)}}$ 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