View source for 19.9

{{DISPLAYTITLE:Elliptic Integrals - 19.9 Inequalities}}

<div style="width: 100%; height: 75vh; overflow: auto;">
{| class="wikitable sortable" style="margin: 0;"
|-
! scope="col" style="position: sticky; top: 0;" | DLMF 
! scope="col" style="position: sticky; top: 0;" | Formula 
! scope="col" style="position: sticky; top: 0;" | Constraints 
! scope="col" style="position: sticky; top: 0;" | Maple
! scope="col" style="position: sticky; top: 0;" | Mathematica
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
|- 
| [https://dlmf.nist.gov/19.9#Ex1 19.9#Ex1] || [[Item:Q6242|<math>\ln@@{4} \leq \compellintKk@{k}+\ln@@{k^{\prime}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ln@@{4} \leq \compellintKk@{k}+\ln@@{k^{\prime}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>ln(4) <= EllipticK(k)+ ln(sqrt(1 - (k)^(2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Log[4] <= EllipticK[(k)^2]+ Log[Sqrt[1 - (k)^(2)]]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[1.3862943611198906, Indeterminate]
Test Values: {Rule[k, 1]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[1.3862943611198906, Complex[1.392181321740353, 0.49253850304507485]]
Test Values: {Rule[k, 2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.9#Ex1 19.9#Ex1] || [[Item:Q6242|<math>\compellintKk@{k}+\ln@@{k^{\prime}} \leq \pi/2</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\compellintKk@{k}+\ln@@{k^{\prime}} \leq \pi/2</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>EllipticK(k)+ ln(sqrt(1 - (k)^(2))) <= Pi/2</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticK[(k)^2]+ Log[Sqrt[1 - (k)^(2)]] <= Pi/2</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[Indeterminate, 1.5707963267948966]
Test Values: {Rule[k, 1]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[1.392181321740353, 0.49253850304507485], 1.5707963267948966]
Test Values: {Rule[k, 2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.9#Ex2 19.9#Ex2] || [[Item:Q6243|<math>1 \leq \compellintEk@{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>1 \leq \compellintEk@{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>1 <= EllipticE(k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>1 <= EllipticE[(k)^2]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[1.0, Complex[0.40629888645996043, 1.343854231387098]]
Test Values: {Rule[k, 2]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[1.0, Complex[0.2655964076372759, 2.498348127732516]]
Test Values: {Rule[k, 3]}</syntaxhighlight><br></div></div>
|- 
| [https://dlmf.nist.gov/19.9#Ex2 19.9#Ex2] || [[Item:Q6243|<math>\compellintEk@{k} \leq \pi/2</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\compellintEk@{k} \leq \pi/2</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>EllipticE(k) <= Pi/2</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticE[(k)^2] <= Pi/2</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[0.40629888645996043, 1.343854231387098], 1.5707963267948966]
Test Values: {Rule[k, 2]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[0.2655964076372759, 2.498348127732516], 1.5707963267948966]
Test Values: {Rule[k, 3]}</syntaxhighlight><br></div></div>
|- 
| [https://dlmf.nist.gov/19.9#Ex3 19.9#Ex3] || [[Item:Q6244|<math>1 \leq (2/\pi)\sqrt{1-\alpha^{2}}\compellintPik@{\alpha^{2}}{k}\leq 1/k^{\prime}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>1 \leq (2/\pi)\sqrt{1-\alpha^{2}}\compellintPik@{\alpha^{2}}{k}\leq 1/k^{\prime}</syntaxhighlight> || <math>\alpha^{2} < 1</math> || <syntaxhighlight lang=mathematica>1 <= (2/Pi)*sqrt(1 - (alpha)^(2))*EllipticPi((alpha)^(2), k) <= 1/(sqrt(1 - (k)^(2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>1 <= (2/Pi)*Sqrt[1 - \[Alpha]^(2)]*EllipticPi[\[Alpha]^(2), (k)^2] <= 1/(Sqrt[1 - (k)^(2)])</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[1.0, DirectedInfinity[], DirectedInfinity[]]
Test Values: {Rule[k, 1], Rule[α, 0.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[1.0, Complex[0.4804983499812288, -0.6957733039705274], Complex[0.0, -0.5773502691896258]]
Test Values: {Rule[k, 2], Rule[α, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.9.E2 19.9.E2] || [[Item:Q6245|<math>1+\frac{{k^{\prime}}^{2}}{8} < \frac{\compellintKk@{k}}{\ln@{4/k^{\prime}}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>1+\frac{{k^{\prime}}^{2}}{8} < \frac{\compellintKk@{k}}{\ln@{4/k^{\prime}}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>1 +(1 - (k)^(2))/(8) < (EllipticK(k))/(ln(4/(sqrt(1 - (k)^(2)))))</syntaxhighlight> || <syntaxhighlight lang=mathematica>1 +Divide[1 - (k)^(2),8] < Divide[EllipticK[(k)^2],Log[4/(Sqrt[1 - (k)^(2)])]]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Less[1.0, Indeterminate]
Test Values: {Rule[k, 1]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Less[0.625, Complex[0.7573351019929213, 0.13305010797062605]]
Test Values: {Rule[k, 2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.9.E2 19.9.E2] || [[Item:Q6245|<math>\frac{\compellintKk@{k}}{\ln@{4/k^{\prime}}} < 1+\frac{{k^{\prime}}^{2}}{4}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\compellintKk@{k}}{\ln@{4/k^{\prime}}} < 1+\frac{{k^{\prime}}^{2}}{4}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(EllipticK(k))/(ln(4/(sqrt(1 - (k)^(2))))) < 1 +(1 - (k)^(2))/(4)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[EllipticK[(k)^2],Log[4/(Sqrt[1 - (k)^(2)])]] < 1 +Divide[1 - (k)^(2),4]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Less[Indeterminate, 1.0]
Test Values: {Rule[k, 1]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Less[Complex[0.7573351019929213, 0.13305010797062605], 0.25]
Test Values: {Rule[k, 2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.9.E3 19.9.E3] || [[Item:Q6246|<math>9+\frac{k^{2}{k^{\prime}}^{2}}{8} < \frac{(8+k^{2})\compellintKk@{k}}{\ln@{4/k^{\prime}}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>9+\frac{k^{2}{k^{\prime}}^{2}}{8} < \frac{(8+k^{2})\compellintKk@{k}}{\ln@{4/k^{\prime}}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>9 +((k)^(2)*1 - (k)^(2))/(8) < ((8 + (k)^(2))*EllipticK(k))/(ln(4/(sqrt(1 - (k)^(2)))))</syntaxhighlight> || <syntaxhighlight lang=mathematica>9 +Divide[(k)^(2)*1 - (k)^(2),8] < Divide[(8 + (k)^(2))*EllipticK[(k)^2],Log[4/(Sqrt[1 - (k)^(2)])]]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Less[9.0, Indeterminate]
Test Values: {Rule[k, 1]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Less[9.0, Complex[9.088021223915057, 1.5966012956475137]]
Test Values: {Rule[k, 2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.9.E3 19.9.E3] || [[Item:Q6246|<math>\frac{(8+k^{2})\compellintKk@{k}}{\ln@{4/k^{\prime}}} < 9.096</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{(8+k^{2})\compellintKk@{k}}{\ln@{4/k^{\prime}}} < 9.096</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>((8 + (k)^(2))*EllipticK(k))/(ln(4/(sqrt(1 - (k)^(2))))) < 9.096</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[(8 + (k)^(2))*EllipticK[(k)^2],Log[4/(Sqrt[1 - (k)^(2)])]] < 9.096</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Less[Indeterminate, 9.096]
Test Values: {Rule[k, 1]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Less[Complex[9.088021223915057, 1.5966012956475137], 9.096]
Test Values: {Rule[k, 2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.9.E4 19.9.E4] || [[Item:Q6247|<math>\left(\frac{1+{k^{\prime}}^{3/2}}{2}\right)^{2/3} \leq \frac{2}{\pi}\compellintEk@{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(\frac{1+{k^{\prime}}^{3/2}}{2}\right)^{2/3} \leq \frac{2}{\pi}\compellintEk@{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>((1 +(sqrt(1 - (k)^(2)))^(3/2))/(2))^(2/3) <= (2)/(Pi)*EllipticE(k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(Divide[1 +(Sqrt[1 - (k)^(2)])^(3/2),2])^(2/3) <= Divide[2,Pi]*EllipticE[(k)^2]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[0.2518251425072316, 0.8700591952646104], Complex[0.2586579046113418, 0.8555241748808654]]
Test Values: {Rule[k, 2]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[0.1858923839966674, 1.6059081831429025], Complex[0.16908392457168991, 1.5904978163720476]]
Test Values: {Rule[k, 3]}</syntaxhighlight><br></div></div>
|- 
| [https://dlmf.nist.gov/19.9.E4 19.9.E4] || [[Item:Q6247|<math>\frac{2}{\pi}\compellintEk@{k} \leq \left(\frac{1+{k^{\prime}}^{2}}{2}\right)^{1/2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{2}{\pi}\compellintEk@{k} \leq \left(\frac{1+{k^{\prime}}^{2}}{2}\right)^{1/2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(2)/(Pi)*EllipticE(k) <= ((1 +1 - (k)^(2))/(2))^(1/2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[2,Pi]*EllipticE[(k)^2] <= (Divide[1 +1 - (k)^(2),2])^(1/2)</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[0.2586579046113418, 0.8555241748808654], Complex[0.0, 1.0]]
Test Values: {Rule[k, 2]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[0.16908392457168991, 1.5904978163720476], Complex[0.0, 1.8708286933869707]]
Test Values: {Rule[k, 3]}</syntaxhighlight><br></div></div>
|- 
| [https://dlmf.nist.gov/19.9.E5 19.9.E5] || [[Item:Q6248|<math>\ln@@{\frac{(1+\sqrt{k^{\prime}})^{2}}{k}} < \frac{\pi\ccompellintKk@{k}}{2\compellintKk@{k}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ln@@{\frac{(1+\sqrt{k^{\prime}})^{2}}{k}} < \frac{\pi\ccompellintKk@{k}}{2\compellintKk@{k}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>ln(((1 +sqrt(sqrt(1 - (k)^(2))))^(2))/(k)) < (Pi*EllipticCK(k))/(2*EllipticK(k))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Log[Divide[(1 +Sqrt[Sqrt[1 - (k)^(2)]])^(2),k]] < Divide[Pi*EllipticK[1-(k)^2],2*EllipticK[(k)^2]]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: False
Test Values: {Rule[k, 1]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Less[Complex[0.8314429455293103, 0.8983332083070389], Complex[0.762166367418117, 0.9750101446769989]]
Test Values: {Rule[k, 2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.9.E5 19.9.E5] || [[Item:Q6248|<math>\frac{\pi\ccompellintKk@{k}}{2\compellintKk@{k}} < \ln@@{\frac{2(1+k^{\prime})}{k}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\pi\ccompellintKk@{k}}{2\compellintKk@{k}} < \ln@@{\frac{2(1+k^{\prime})}{k}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(Pi*EllipticCK(k))/(2*EllipticK(k)) < ln((2*(1 +sqrt(1 - (k)^(2))))/(k))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Pi*EllipticK[1-(k)^2],2*EllipticK[(k)^2]] < Log[Divide[2*(1 +Sqrt[1 - (k)^(2)]),k]]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Less[Complex[0.762166367418117, 0.9750101446769989], Complex[0.6931471805599452, 1.0471975511965976]]
Test Values: {Rule[k, 2]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Less[Complex[0.7130154358988758, 1.1147297033963086], Complex[0.6931471805599453, 1.2309594173407747]]
Test Values: {Rule[k, 3]}</syntaxhighlight><br></div></div>
|- 
| [https://dlmf.nist.gov/19.9.E6 19.9.E6] || [[Item:Q6249|<math>(1-\tfrac{3}{4}k^{2})^{-1/2} < \frac{4}{\pi k^{2}}(\compellintKk@{k}-\compellintEk@{k})</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(1-\tfrac{3}{4}k^{2})^{-1/2} < \frac{4}{\pi k^{2}}(\compellintKk@{k}-\compellintEk@{k})</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(1 -(3)/(4)*(k)^(2))^(- 1/2) < (4)/(Pi*(k)^(2))*(EllipticK(k)- EllipticE(k))</syntaxhighlight> || <syntaxhighlight lang=mathematica>(1 -Divide[3,4]*(k)^(2))^(- 1/2) < Divide[4,Pi*(k)^(2)]*(EllipticK[(k)^2]- EllipticE[(k)^2])</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Less[2.0, DirectedInfinity[]]
Test Values: {Rule[k, 1]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Less[Complex[0.0, -0.7071067811865475], Complex[0.13896654948167025, -0.7709822125950203]]
Test Values: {Rule[k, 2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.9.E6 19.9.E6] || [[Item:Q6249|<math>\frac{4}{\pi k^{2}}(\compellintKk@{k}-\compellintEk@{k}) < (k^{\prime})^{-3/4}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{4}{\pi k^{2}}(\compellintKk@{k}-\compellintEk@{k}) < (k^{\prime})^{-3/4}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(4)/(Pi*(k)^(2))*(EllipticK(k)- EllipticE(k)) < (sqrt(1 - (k)^(2)))^(- 3/4)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[4,Pi*(k)^(2)]*(EllipticK[(k)^2]- EllipticE[(k)^2]) < (Sqrt[1 - (k)^(2)])^(- 3/4)</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Less[DirectedInfinity[], DirectedInfinity[]]
Test Values: {Rule[k, 1]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Less[Complex[0.13896654948167025, -0.7709822125950203], Complex[0.2534656958546175, -0.6119203205285516]]
Test Values: {Rule[k, 2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.9.E7 19.9.E7] || [[Item:Q6250|<math>(1-\tfrac{1}{4}k^{2})^{-1/2} < \frac{4}{\pi k^{2}}(\compellintEk@{k}-{k^{\prime}}^{2}\compellintKk@{k})</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(1-\tfrac{1}{4}k^{2})^{-1/2} < \frac{4}{\pi k^{2}}(\compellintEk@{k}-{k^{\prime}}^{2}\compellintKk@{k})</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(1 -(1)/(4)*(k)^(2))^(- 1/2) < (4)/(Pi*(k)^(2))*(EllipticE(k)-1 - (k)^(2)*EllipticK(k))</syntaxhighlight> || <syntaxhighlight lang=mathematica>(1 -Divide[1,4]*(k)^(2))^(- 1/2) < Divide[4,Pi*(k)^(2)]*(EllipticE[(k)^2]-1 - (k)^(2)*EllipticK[(k)^2])</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Less[1.1547005383792517, DirectedInfinity[]]
Test Values: {Rule[k, 1]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Less[DirectedInfinity[], Complex[-1.2621629410274844, 1.800642588058783]]
Test Values: {Rule[k, 2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.9.E8 19.9.E8] || [[Item:Q6251|<math>k^{\prime} < \frac{\compellintEk@{k}}{\compellintKk@{k}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>k^{\prime} < \frac{\compellintEk@{k}}{\compellintKk@{k}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sqrt(1 - (k)^(2)) < (EllipticE(k))/(EllipticK(k))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[1 - (k)^(2)] < Divide[EllipticE[(k)^2],EllipticK[(k)^2]]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: False
Test Values: {Rule[k, 1]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Less[Complex[0.0, 1.7320508075688772], Complex[-0.5907718728609501, 0.8386174564999851]]
Test Values: {Rule[k, 2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.9.E8 19.9.E8] || [[Item:Q6251|<math>\frac{\compellintEk@{k}}{\compellintKk@{k}} < \left(\frac{1+k^{\prime}}{2}\right)^{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\compellintEk@{k}}{\compellintKk@{k}} < \left(\frac{1+k^{\prime}}{2}\right)^{2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(EllipticE(k))/(EllipticK(k)) < ((1 +sqrt(1 - (k)^(2)))/(2))^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[EllipticE[(k)^2],EllipticK[(k)^2]] < (Divide[1 +Sqrt[1 - (k)^(2)],2])^(2)</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Less[Complex[-0.5907718728609501, 0.8386174564999851], Complex[-0.4999999999999999, 0.8660254037844386]]
Test Values: {Rule[k, 2]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Less[Complex[-1.9604512687154212, 1.5690726247192568], Complex[-1.7500000000000004, 1.4142135623730951]]
Test Values: {Rule[k, 3]}</syntaxhighlight><br></div></div>
|- 
| [https://dlmf.nist.gov/19.9.E9 19.9.E9] || [[Item:Q6252|<math>L(a,b) = 4a\compellintEk@{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>L(a,b) = 4a\compellintEk@{k}</syntaxhighlight> || <math>k^{2} = 1-(b^{2}/a^{2}), a > b</math> || <syntaxhighlight lang=mathematica>L(a , b) = 4*a*EllipticE(k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>L[a , b] == 4*a*EllipticE[(k)^2]</syntaxhighlight> || Error || Failure || - || Error
|- 
| [https://dlmf.nist.gov/19.9.E11 19.9.E11] || [[Item:Q6254|<math>\phi \leq \incellintFk@{\phi}{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\phi \leq \incellintFk@{\phi}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>phi <= EllipticF(sin(phi), k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>\[Phi] <= EllipticF[\[Phi], (k)^2]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [4 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.500000000 <= -3.340677542
Test Values: {phi = -3/2, k = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.5000000000 <= -.5222381033
Test Values: {phi = -1/2, k = 1}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [28 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.9.E12 19.9.E12] || [[Item:Q6255|<math>\incellintEk@{\phi}{k} \leq \phi</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incellintEk@{\phi}{k} \leq \phi</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>EllipticE(sin(phi), k) <= phi</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticE[\[Phi], (k)^2] <= \[Phi]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [4 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.9974949866 <= -1.500000000
Test Values: {phi = -3/2, k = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.4794255386 <= -.5000000000
Test Values: {phi = -1/2, k = 1}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [27 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.9.E13 19.9.E13] || [[Item:Q6256|<math>\incellintPik@{\phi}{\alpha^{2}}{0} \leq \incellintPik@{\phi}{\alpha^{2}}{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incellintPik@{\phi}{\alpha^{2}}{0} \leq \incellintPik@{\phi}{\alpha^{2}}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>EllipticPi(sin(phi), (alpha)^(2), 0) <= EllipticPi(sin(phi), (alpha)^(2), k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticPi[\[Alpha]^(2), \[Phi],(0)^2] <= EllipticPi[\[Alpha]^(2), \[Phi],(k)^2]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [8 / 90]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.6351972518 <= -.6692391842
Test Values: {alpha = 3/2, phi = -1/2, k = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.6351972518 <= -.9273807742
Test Values: {alpha = 3/2, phi = -1/2, k = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [84 / 90]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.9.E14 19.9.E14] || [[Item:Q6257|<math>\frac{3}{1+\Delta+\cos@@{\phi}} < \frac{\incellintFk@{\phi}{k}}{\sin@@{\phi}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{3}{1+\Delta+\cos@@{\phi}} < \frac{\incellintFk@{\phi}{k}}{\sin@@{\phi}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(3)/(1 + Delta + cos(phi)) < (EllipticF(sin(phi), k))/(sin(phi))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[3,1 + \[CapitalDelta]+ Cos[\[Phi]]] < Divide[EllipticF[\[Phi], (k)^2],Sin[\[Phi]]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [16 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 7.945282179 < 1.089299717
Test Values: {Delta = -3/2, phi = -1/2, k = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 7.945282179 < 1.412977582
Test Values: {Delta = -3/2, phi = -1/2, k = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [284 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.9.E14 19.9.E14] || [[Item:Q6257|<math>\frac{\incellintFk@{\phi}{k}}{\sin@@{\phi}} < \frac{1}{(\Delta\cos@@{\phi})^{1/3}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\incellintFk@{\phi}{k}}{\sin@@{\phi}} < \frac{1}{(\Delta\cos@@{\phi})^{1/3}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(EllipticF(sin(phi), k))/(sin(phi)) < (1)/((Delta*cos(phi))^(1/3))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[EllipticF[\[Phi], (k)^2],Sin[\[Phi]]] < Divide[1,(\[CapitalDelta]*Cos[\[Phi]])^(1/3)]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [20 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.675417084 < 1.170093898
Test Values: {Delta = -3/2, phi = -2, k = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.675417084 < 1.170093898
Test Values: {Delta = -3/2, phi = 2, k = 1}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [298 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.9.E15 19.9.E15] || [[Item:Q6258|<math>1 < \incellintFk@{\phi}{k}\bigg{/}\left((\sin@@{\phi})\ln@{\frac{4}{\Delta+\cos@@{\phi}}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>1 < \incellintFk@{\phi}{k}\bigg{/}\left((\sin@@{\phi})\ln@{\frac{4}{\Delta+\cos@@{\phi}}}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>1 < EllipticF(sin(phi), k)/((sin(phi))*ln((4)/(Delta + cos(phi))))</syntaxhighlight> || <syntaxhighlight lang=mathematica>1 < EllipticF[\[Phi], (k)^2]/((Sin[\[Phi]])*Log[Divide[4,\[CapitalDelta]+ Cos[\[Phi]]]])</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1. < .4615167558
Test Values: {Delta = -1/2, phi = -1/2, k = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1. < .5986532627
Test Values: {Delta = -1/2, phi = -1/2, k = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [288 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.9.E15 19.9.E15] || [[Item:Q6258|<math>\incellintFk@{\phi}{k}\bigg{/}\left((\sin@@{\phi})\ln@{\frac{4}{\Delta+\cos@@{\phi}}}\right) < \frac{4}{2+(1+k^{2})\sin^{2}@@{\phi}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incellintFk@{\phi}{k}\bigg{/}\left((\sin@@{\phi})\ln@{\frac{4}{\Delta+\cos@@{\phi}}}\right) < \frac{4}{2+(1+k^{2})\sin^{2}@@{\phi}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>EllipticF(sin(phi), k)/((sin(phi))*ln((4)/(Delta + cos(phi)))) < (4)/(2 +(1 + (k)^(2))*(sin(phi))^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticF[\[Phi], (k)^2]/((Sin[\[Phi]])*Log[Divide[4,\[CapitalDelta]+ Cos[\[Phi]]]]) < Divide[4,2 +(1 + (k)^(2))*(Sin[\[Phi]])^(2)]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [20 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 3.582850518 < 1.002508151
Test Values: {Delta = 3/2, phi = -3/2, k = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 3.582850518 < 1.002508151
Test Values: {Delta = 3/2, phi = 3/2, k = 1}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [296 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.9.E16 19.9.E16] || [[Item:Q6259|<math>\incellintFk@{\phi}{k} = \frac{2}{\pi}\compellintKk@{k^{\prime}}\ln@{\frac{4}{\Delta+\cos@@{\phi}}}-\theta\Delta^{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incellintFk@{\phi}{k} = \frac{2}{\pi}\compellintKk@{k^{\prime}}\ln@{\frac{4}{\Delta+\cos@@{\phi}}}-\theta\Delta^{2}</syntaxhighlight> || <math>(\sin@@{\phi})/8 < \theta, \theta < (\ln@@{2})/(k^{2}\sin@@{\phi})</math> || <syntaxhighlight lang=mathematica>EllipticF(sin(phi), k) = (2)/(Pi)*EllipticK(sqrt(1 - (k)^(2)))*ln((4)/(Delta + cos(phi)))- theta*(Delta)^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticF[\[Phi], (k)^2] == Divide[2,Pi]*EllipticK[(Sqrt[1 - (k)^(2)])^2]*Log[Divide[4,\[CapitalDelta]+ Cos[\[Phi]]]]- \[Theta]*\[CapitalDelta]^(2)</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 2.264395299+.9232968251*I
Test Values: {Delta = 1/2*3^(1/2)+1/2*I, phi = 3/2, theta = 1/2, k = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.185868314e-1+.7122804653*I
Test Values: {Delta = 1/2*3^(1/2)+1/2*I, phi = 1/2, theta = 1/2, k = 1}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.9.E17 19.9.E17] || [[Item:Q6260|<math>L \leq \incellintFk@{\phi}{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>L \leq \incellintFk@{\phi}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>L <= EllipticF(sin(phi), k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>L <= EllipticF[\[Phi], (k)^2]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [24 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.500000000 <= -3.340677542
Test Values: {L = -3/2, phi = -3/2, k = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.500000000 <= -1.523452443
Test Values: {L = -3/2, phi = -2, k = 1}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [288 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.9.E17 19.9.E17] || [[Item:Q6260|<math>\incellintFk@{\phi}{k} \leq \sqrt{UL}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incellintFk@{\phi}{k} \leq \sqrt{UL}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>EllipticF(sin(phi), k) <= sqrt(U*L)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticF[\[Phi], (k)^2] <= Sqrt[U*L]</syntaxhighlight> || Failure || Failure || Successful [Tested: 300] || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.9.E17 19.9.E17] || [[Item:Q6260|<math>\sqrt{UL} \leq \tfrac{1}{2}(U+L)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sqrt{UL} \leq \tfrac{1}{2}(U+L)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sqrt(U*L) <= (1)/(2)*(U + L)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[U*L] <= Divide[1,2]*(U + L)</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 100]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.500000000 <= -1.500000000
Test Values: {L = -3/2, U = -3/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .8660254040 <= -1.
Test Values: {L = -3/2, U = -1/2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [91 / 100]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.9.E17 19.9.E17] || [[Item:Q6260|<math>\tfrac{1}{2}(U+L) \leq U</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\tfrac{1}{2}(U+L) \leq U</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(1)/(2)*(U + L) <= U</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,2]*(U + L) <= U</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [15 / 100]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.750000000 <= -2.
Test Values: {L = -3/2, U = -2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 0. <= -1.500000000
Test Values: {L = 3/2, U = -3/2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [79 / 100]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.9#Ex4 19.9#Ex4] || [[Item:Q6261|<math>L = (1/\sigma)\atanh@{\sigma\sin@@{\phi}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>L = (1/\sigma)\atanh@{\sigma\sin@@{\phi}}</syntaxhighlight> || <math>\sigma = \sqrt{(1+k^{2})/2}</math> || <syntaxhighlight lang=mathematica>L = (1/sigma)*arctanh(sigma*sin(phi))</syntaxhighlight> || <syntaxhighlight lang=mathematica>L == (1/\[Sigma])*ArcTanh[\[Sigma]*Sin[\[Phi]]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.9#Ex5 19.9#Ex5] || [[Item:Q6262|<math>U = \tfrac{1}{2}\atanh@{\sin@@{\phi}}+\tfrac{1}{2}k^{-1}\atanh@{k\sin@@{\phi}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>U = \tfrac{1}{2}\atanh@{\sin@@{\phi}}+\tfrac{1}{2}k^{-1}\atanh@{k\sin@@{\phi}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>U = (1)/(2)*arctanh(sin(phi))+(1)/(2)*(k)^(- 1)* arctanh(k*sin(phi))</syntaxhighlight> || <syntaxhighlight lang=mathematica>U == Divide[1,2]*ArcTanh[Sin[\[Phi]]]+Divide[1,2]*(k)^(- 1)* ArcTanh[k*Sin[\[Phi]]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|}
</div>