View source for 19.24

{{DISPLAYTITLE:Elliptic Integrals - 19.24 Inequalities}}

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! 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
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| [https://dlmf.nist.gov/19.24.E1 19.24.E1] || <math qid="Q6482">\ln@@{4} \leq \sqrt{z}\CarlsonsymellintRF@{0}{y}{z}+\ln@@{\sqrt{y/z}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ln@@{4} \leq \sqrt{z}\CarlsonsymellintRF@{0}{y}{z}+\ln@@{\sqrt{y/z}}</syntaxhighlight> || <math>0 < y, y \leq z</math> || <syntaxhighlight lang=mathematica>ln(4) <= sqrt(x + y*I)*0.5*int(1/(sqrt(t+0)*sqrt(t+y)*sqrt(t+x + y*I)), t = 0..infinity)+ ln(sqrt(y/(x + y*I)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Log[4] <= Sqrt[x + y*I]*EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]/Sqrt[x + y*I-0]+ Log[Sqrt[y/(x + y*I)]]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 9]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[1.3862943611198906, Complex[0.5672499697282593, -1.7874177081206242]]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[1.3862943611198906, Complex[0.6277320470267476, -0.9602476282953896]]
Test Values: {Rule[x, 1.5], Rule[y, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/19.24.E1 19.24.E1] || <math qid="Q6482">\sqrt{z}\CarlsonsymellintRF@{0}{y}{z}+\ln@@{\sqrt{y/z}} \leq \tfrac{1}{2}\pi</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sqrt{z}\CarlsonsymellintRF@{0}{y}{z}+\ln@@{\sqrt{y/z}} \leq \tfrac{1}{2}\pi</syntaxhighlight> || <math>0 < y, y \leq z</math> || <syntaxhighlight lang=mathematica>sqrt(x + y*I)*0.5*int(1/(sqrt(t+0)*sqrt(t+y)*sqrt(t+x + y*I)), t = 0..infinity)+ ln(sqrt(y/(x + y*I))) <= (1)/(2)*Pi</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[x + y*I]*EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]/Sqrt[x + y*I-0]+ Log[Sqrt[y/(x + y*I)]] <= Divide[1,2]*Pi</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 9]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[0.5672499697282593, -1.7874177081206242], 1.5707963267948966]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[0.6277320470267476, -0.9602476282953896], 1.5707963267948966]
Test Values: {Rule[x, 1.5], Rule[y, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/19.24.E2 19.24.E2] || <math qid="Q6483">\tfrac{1}{2} \leq z^{-1/2}\CarlsonsymellintRG@{0}{y}{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\tfrac{1}{2} \leq z^{-1/2}\CarlsonsymellintRG@{0}{y}{z}</syntaxhighlight> || <math>0 \leq y, y \leq z</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,2] <= (x + y*I)^(- 1/2)* Sqrt[x + y*I-0]*(EllipticE[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]+(Cot[ArcCos[Sqrt[0/x + y*I]]])^2*EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]+Cot[ArcCos[Sqrt[0/x + y*I]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/x + y*I]]]^2])</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 9]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[0.5, Plus[Complex[1.0084590214609772, 0.7147093671486319], Times[Complex[0.2643318009908678, 0.8730286325904596], Power[Plus[1.0, Times[Complex[-1.0, 1.5], Power[k, 2]]], Rational[1, 2]]]]]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[0.5, Plus[Complex[1.0897585107701309, 0.2919625251300463], Times[Complex[0.3515775842541431, 0.5688644810057831], Power[Plus[1.0, Times[Complex[-1.0, 0.5], Power[k, 2]]], Rational[1, 2]]]]]
Test Values: {Rule[x, 1.5], Rule[y, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/19.24.E2 19.24.E2] || <math qid="Q6483">z^{-1/2}\CarlsonsymellintRG@{0}{y}{z} \leq \tfrac{1}{4}\pi</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z^{-1/2}\CarlsonsymellintRG@{0}{y}{z} \leq \tfrac{1}{4}\pi</syntaxhighlight> || <math>0 \leq y, y \leq z</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>(x + y*I)^(- 1/2)* Sqrt[x + y*I-0]*(EllipticE[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]+(Cot[ArcCos[Sqrt[0/x + y*I]]])^2*EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]+Cot[ArcCos[Sqrt[0/x + y*I]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/x + y*I]]]^2]) <= Divide[1,4]*Pi</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 9]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[Plus[Complex[1.0084590214609772, 0.7147093671486319], Times[Complex[0.2643318009908678, 0.8730286325904596], Power[Plus[1.0, Times[Complex[-1.0, 1.5], Power[k, 2]]], Rational[1, 2]]]], 0.7853981633974483]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[Plus[Complex[1.0897585107701309, 0.2919625251300463], Times[Complex[0.3515775842541431, 0.5688644810057831], Power[Plus[1.0, Times[Complex[-1.0, 0.5], Power[k, 2]]], Rational[1, 2]]]], 0.7853981633974483]
Test Values: {Rule[x, 1.5], Rule[y, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.24.E3 19.24.E3] || <math qid="Q6484">\left(\frac{y^{3/2}+z^{3/2}}{2}\right)^{2/3} \leq \frac{4}{\pi}\CarlsonsymellintRG@{0}{y^{2}}{z^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(\frac{y^{3/2}+z^{3/2}}{2}\right)^{2/3} \leq \frac{4}{\pi}\CarlsonsymellintRG@{0}{y^{2}}{z^{2}}</syntaxhighlight> || <math>y > 0, z > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>(Divide[(y)^(3/2)+(x + y*I)^(3/2),2])^(2/3) <= Divide[4,Pi]*Sqrt[(x + y*I)^(2)-0]*(EllipticE[ArcCos[Sqrt[0/(x + y*I)^(2)]],((x + y*I)^(2)-(y)^(2))/((x + y*I)^(2)-0)]+(Cot[ArcCos[Sqrt[0/(x + y*I)^(2)]]])^2*EllipticF[ArcCos[Sqrt[0/(x + y*I)^(2)]],((x + y*I)^(2)-(y)^(2))/((x + y*I)^(2)-0)]+Cot[ArcCos[Sqrt[0/(x + y*I)^(2)]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/(x + y*I)^(2)]]]^2])</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 9]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[1.4250443092558214, 0.7875512141675095], Complex[2.850438542245679, 1.5730146161508307]]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[1.0588191704631045, 0.29794136993360365], Complex[2.118851869395612, 0.5983245902184247]]
Test Values: {Rule[x, 1.5], Rule[y, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.24.E3 19.24.E3] || <math qid="Q6484">\frac{4}{\pi}\CarlsonsymellintRG@{0}{y^{2}}{z^{2}} \leq \left(\frac{y^{2}+z^{2}}{2}\right)^{1/2}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{4}{\pi}\CarlsonsymellintRG@{0}{y^{2}}{z^{2}} \leq \left(\frac{y^{2}+z^{2}}{2}\right)^{1/2}</syntaxhighlight> || <math>y > 0, z > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[4,Pi]*Sqrt[(x + y*I)^(2)-0]*(EllipticE[ArcCos[Sqrt[0/(x + y*I)^(2)]],((x + y*I)^(2)-(y)^(2))/((x + y*I)^(2)-0)]+(Cot[ArcCos[Sqrt[0/(x + y*I)^(2)]]])^2*EllipticF[ArcCos[Sqrt[0/(x + y*I)^(2)]],((x + y*I)^(2)-(y)^(2))/((x + y*I)^(2)-0)]+Cot[ArcCos[Sqrt[0/(x + y*I)^(2)]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/(x + y*I)^(2)]]]^2]) <= (Divide[(y)^(2)+(x + y*I)^(2),2])^(1/2)</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 9]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[2.850438542245679, 1.5730146161508307], Complex[1.3491805799609005, 0.8338394553771318]]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[2.118851869395612, 0.5983245902184247], Complex[1.112897508375995, 0.3369582528288897]]
Test Values: {Rule[x, 1.5], Rule[y, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.24.E4 19.24.E4] || <math qid="Q6485">\frac{2}{\sqrt{p}}(2yz+yp+zp)^{-1/2} \leq \frac{4}{3\pi}\CarlsonsymellintRJ@{0}{y}{z}{p}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{2}{\sqrt{p}}(2yz+yp+zp)^{-1/2} \leq \frac{4}{3\pi}\CarlsonsymellintRJ@{0}{y}{z}{p}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[2,Sqrt[p]]*(2*y*(x + y*I)+ y*p +(x + y*I)*p)^(- 1/2) <= Divide[4,3*Pi]*3*(x + y*I-0)/(x + y*I-p)*(EllipticPi[(x + y*I-p)/(x + y*I-0),ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/Sqrt[x + y*I-0]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [180 / 180]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[0.13508456755677706, -1.1829936015765863], Complex[-0.3213270063391195, -0.3051912044731223]]
Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[0.7797231369520263, -0.6247258696161743], Complex[-0.6706782382611747, 0.54526856836685]]
Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.24.E4 19.24.E4] || <math qid="Q6485">\frac{4}{3\pi}\CarlsonsymellintRJ@{0}{y}{z}{p} \leq (yzp^{2})^{-3/8}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{4}{3\pi}\CarlsonsymellintRJ@{0}{y}{z}{p} \leq (yzp^{2})^{-3/8}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[4,3*Pi]*3*(x + y*I-0)/(x + y*I-p)*(EllipticPi[(x + y*I-p)/(x + y*I-0),ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/Sqrt[x + y*I-0] <= (y*(x + y*I)*(p)^(2))^(- 3/8)</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [180 / 180]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[-0.3213270063391195, -0.3051912044731223], Complex[0.5136265917030035, 0.9609277658721954]]
Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[-0.6706782382611747, 0.54526856836685], Complex[0.8422602311268256, -0.6912251080442312]]
Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.24.E5 19.24.E5] || <math qid="Q6486">\frac{1}{a_{n}} \leq \frac{2}{\pi}\CarlsonsymellintRF@{0}{a_{0}^{2}}{g_{0}^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{a_{n}} \leq \frac{2}{\pi}\CarlsonsymellintRF@{0}{a_{0}^{2}}{g_{0}^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(1)/(a[n]) <= (2)/(Pi)*0.5*int(1/(sqrt(t+0)*sqrt(t+(a[0])^(2))*sqrt(t+(g[0])^(2))), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,Subscript[a, n]] <= Divide[2,Pi]*EllipticF[ArcCos[Sqrt[0/(Subscript[g, 0])^(2)]],((Subscript[g, 0])^(2)-(Subscript[a, 0])^(2))/((Subscript[g, 0])^(2)-0)]/Sqrt[(Subscript[g, 0])^(2)-0]</syntaxhighlight> || Aborted || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[1.7320508075688774, -0.9999999999999999], Times[2.0, Power[Times[Complex[0.5000000000000001, 0.8660254037844386], g], Rational[-1, 2]], EllipticK[Times[Complex[2.0000000000000004, -3.4641016151377544], Plus[Times[Complex[-0.12500000000000003, -0.21650635094610965], a], Times[Complex[0.12500000000000003, 0.21650635094610965], g]], Power[g, -1]]]]]
Test Values: {Rule[n, 3], Rule[Subscript[a, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[a, n], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[g, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[1.7320508075688774, -0.9999999999999999], Times[2.0, Power[Times[Complex[-0.4999999999999998, -0.8660254037844387], g], Rational[-1, 2]], EllipticK[Times[Complex[-1.9999999999999991, 3.464101615137755], Plus[Times[Complex[-0.12500000000000003, -0.21650635094610965], a], Times[Complex[-0.12499999999999994, -0.21650635094610968], g]], Power[g, -1]]]]]
Test Values: {Rule[n, 3], Rule[Subscript[a, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[a, n], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[g, 0], 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/19.24.E5 19.24.E5] || <math qid="Q6486">\frac{2}{\pi}\CarlsonsymellintRF@{0}{a_{0}^{2}}{g_{0}^{2}} \leq \frac{1}{g_{n}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{2}{\pi}\CarlsonsymellintRF@{0}{a_{0}^{2}}{g_{0}^{2}} \leq \frac{1}{g_{n}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(2)/(Pi)*0.5*int(1/(sqrt(t+0)*sqrt(t+(a[0])^(2))*sqrt(t+(g[0])^(2))), t = 0..infinity) <= (1)/(g[n])</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[2,Pi]*EllipticF[ArcCos[Sqrt[0/(Subscript[g, 0])^(2)]],((Subscript[g, 0])^(2)-(Subscript[a, 0])^(2))/((Subscript[g, 0])^(2)-0)]/Sqrt[(Subscript[g, 0])^(2)-0] <= Divide[1,Subscript[g, n]]</syntaxhighlight> || Aborted || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[Times[2.0, Power[Times[Complex[0.5000000000000001, 0.8660254037844386], g], Rational[-1, 2]], EllipticK[Times[Complex[2.0000000000000004, -3.4641016151377544], Plus[Times[Complex[-0.12500000000000003, -0.21650635094610965], a], Times[Complex[0.12500000000000003, 0.21650635094610965], g]], Power[g, -1]]]], Complex[1.7320508075688774, -0.9999999999999999]]
Test Values: {Rule[n, 3], Rule[Subscript[a, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[g, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[g, n], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[Times[2.0, Power[Times[Complex[0.5000000000000001, 0.8660254037844386], g], Rational[-1, 2]], EllipticK[Times[Complex[2.0000000000000004, -3.4641016151377544], Plus[Times[Complex[-0.12500000000000003, -0.21650635094610965], a], Times[Complex[0.12500000000000003, 0.21650635094610965], g]], Power[g, -1]]]], Complex[-0.9999999999999996, -1.7320508075688774]]
Test Values: {Rule[n, 3], Rule[Subscript[a, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[g, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[g, n], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/19.24#Ex1 19.24#Ex1] || <math qid="Q6487">a_{n+1} = (a_{n}+g_{n})/2</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>a_{n+1} = (a_{n}+g_{n})/2</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a[n + 1] = (a[n]+ g[n])/2</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[a, n + 1] == (Subscript[a, n]+ Subscript[g, n])/2</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/19.24#Ex2 19.24#Ex2] || <math qid="Q6488">g_{n+1} = \sqrt{a_{n}g_{n}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>g_{n+1} = \sqrt{a_{n}g_{n}}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">g[n + 1] = sqrt(a[n]*g[n])</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[g, n + 1] == Sqrt[Subscript[a, n]*Subscript[g, n]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- 
| [https://dlmf.nist.gov/19.24.E7 19.24.E7] || <math qid="Q6489">L(a,b) = 8\CarlsonsymellintRG@{0}{a^{2}}{b^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>L(a,b) = 8\CarlsonsymellintRG@{0}{a^{2}}{b^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>L[a , b] == 8*Sqrt[(b)^(2)-0]*(EllipticE[ArcCos[Sqrt[0/(b)^(2)]],((b)^(2)-(a)^(2))/((b)^(2)-0)]+(Cot[ArcCos[Sqrt[0/(b)^(2)]]])^2*EllipticF[ArcCos[Sqrt[0/(b)^(2)]],((b)^(2)-(a)^(2))/((b)^(2)-0)]+Cot[ArcCos[Sqrt[0/(b)^(2)]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/(b)^(2)]]]^2])</syntaxhighlight> || Missing Macro Error || Failure || - || Error
|- 
| [https://dlmf.nist.gov/19.24#Ex3 19.24#Ex3] || <math qid="Q6490">\CarlsonsymellintRF@{x}{y}{0}\CarlsonsymellintRG@{x}{y}{0} > \tfrac{1}{8}\pi^{2}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\CarlsonsymellintRF@{x}{y}{0}\CarlsonsymellintRG@{x}{y}{0} > \tfrac{1}{8}\pi^{2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticF[ArcCos[Sqrt[x/0]],(0-y)/(0-x)]/Sqrt[0-x]*Sqrt[0-x]*(EllipticE[ArcCos[Sqrt[x/0]],(0-y)/(0-x)]+(Cot[ArcCos[Sqrt[x/0]]])^2*EllipticF[ArcCos[Sqrt[x/0]],(0-y)/(0-x)]+Cot[ArcCos[Sqrt[x/0]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[x/0]]]^2]) > Divide[1,8]*(Pi)^(2)</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 18]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Greater[Indeterminate, 1.2337005501361697]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Greater[Indeterminate, 1.2337005501361697]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.24#Ex4 19.24#Ex4] || <math qid="Q6491">\CarlsonsymellintRF@{x}{y}{0}+2\CarlsonsymellintRG@{x}{y}{0} > \pi</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\CarlsonsymellintRF@{x}{y}{0}+2\CarlsonsymellintRG@{x}{y}{0} > \pi</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticF[ArcCos[Sqrt[x/0]],(0-y)/(0-x)]/Sqrt[0-x]+ 2*Sqrt[0-x]*(EllipticE[ArcCos[Sqrt[x/0]],(0-y)/(0-x)]+(Cot[ArcCos[Sqrt[x/0]]])^2*EllipticF[ArcCos[Sqrt[x/0]],(0-y)/(0-x)]+Cot[ArcCos[Sqrt[x/0]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[x/0]]]^2]) > Pi</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 18]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Greater[Indeterminate, 3.141592653589793]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Greater[Indeterminate, 3.141592653589793]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.24.E9 19.24.E9] || <math qid="Q6492">\frac{1}{2}\,g_{1}^{2} \leq \frac{\CarlsonsymellintRG@{a_{0}^{2}}{g_{0}^{2}}{0}}{\CarlsonsymellintRF@{a_{0}^{2}}{g_{0}^{2}}{0}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{2}\,g_{1}^{2} \leq \frac{\CarlsonsymellintRG@{a_{0}^{2}}{g_{0}^{2}}{0}}{\CarlsonsymellintRF@{a_{0}^{2}}{g_{0}^{2}}{0}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,2]*(Subscript[g, 1])^(2) <= Divide[Sqrt[0-(Subscript[a, 0])^(2)]*(EllipticE[ArcCos[Sqrt[(Subscript[a, 0])^(2)/0]],(0-(Subscript[g, 0])^(2))/(0-(Subscript[a, 0])^(2))]+(Cot[ArcCos[Sqrt[(Subscript[a, 0])^(2)/0]]])^2*EllipticF[ArcCos[Sqrt[(Subscript[a, 0])^(2)/0]],(0-(Subscript[g, 0])^(2))/(0-(Subscript[a, 0])^(2))]+Cot[ArcCos[Sqrt[(Subscript[a, 0])^(2)/0]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[(Subscript[a, 0])^(2)/0]]]^2]),EllipticF[ArcCos[Sqrt[(Subscript[a, 0])^(2)/0]],(0-(Subscript[g, 0])^(2))/(0-(Subscript[a, 0])^(2))]/Sqrt[0-(Subscript[a, 0])^(2)]]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[0.06250000000000001, 0.10825317547305482], Indeterminate]
Test Values: {Rule[Subscript[a, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[g, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[g, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[-0.06249999999999997, -0.10825317547305484], Indeterminate]
Test Values: {Rule[Subscript[a, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[g, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[g, 1], 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/19.24.E9 19.24.E9] || <math qid="Q6492">\frac{\CarlsonsymellintRG@{a_{0}^{2}}{g_{0}^{2}}{0}}{\CarlsonsymellintRF@{a_{0}^{2}}{g_{0}^{2}}{0}} \leq \frac{1}{2}\,a_{1}^{2}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\CarlsonsymellintRG@{a_{0}^{2}}{g_{0}^{2}}{0}}{\CarlsonsymellintRF@{a_{0}^{2}}{g_{0}^{2}}{0}} \leq \frac{1}{2}\,a_{1}^{2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Sqrt[0-(Subscript[a, 0])^(2)]*(EllipticE[ArcCos[Sqrt[(Subscript[a, 0])^(2)/0]],(0-(Subscript[g, 0])^(2))/(0-(Subscript[a, 0])^(2))]+(Cot[ArcCos[Sqrt[(Subscript[a, 0])^(2)/0]]])^2*EllipticF[ArcCos[Sqrt[(Subscript[a, 0])^(2)/0]],(0-(Subscript[g, 0])^(2))/(0-(Subscript[a, 0])^(2))]+Cot[ArcCos[Sqrt[(Subscript[a, 0])^(2)/0]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[(Subscript[a, 0])^(2)/0]]]^2]),EllipticF[ArcCos[Sqrt[(Subscript[a, 0])^(2)/0]],(0-(Subscript[g, 0])^(2))/(0-(Subscript[a, 0])^(2))]/Sqrt[0-(Subscript[a, 0])^(2)]] <= Divide[1,2]*(Subscript[a, 1])^(2)</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[Indeterminate, Complex[0.06250000000000001, 0.10825317547305482]]
Test Values: {Rule[Subscript[a, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[a, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[g, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[Indeterminate, Complex[0.06250000000000001, 0.10825317547305482]]
Test Values: {Rule[Subscript[a, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[a, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[g, 0], 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/19.24.E10 19.24.E10] || <math qid="Q6493">\frac{3}{\sqrt{x}+\sqrt{y}+\sqrt{z}} \leq \CarlsonsymellintRF@{x}{y}{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{3}{\sqrt{x}+\sqrt{y}+\sqrt{z}} \leq \CarlsonsymellintRF@{x}{y}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(3)/(sqrt(x)+sqrt(y)+sqrt(x + y*I)) <= 0.5*int(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+x + y*I)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[3,Sqrt[x]+Sqrt[y]+Sqrt[x + y*I]] <= EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]</syntaxhighlight> || Aborted || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 18]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[1.0934408788539995, -0.2839050517129825], Complex[-0.16214470973156064, 0.6784437678906974]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[0.7738030002696183, -0.11364498174072818], Complex[-0.28823404661462, -0.7809212115368181]]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.24.E10 19.24.E10] || <math qid="Q6493">\CarlsonsymellintRF@{x}{y}{z} \leq \frac{1}{(xyz)^{1/6}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\CarlsonsymellintRF@{x}{y}{z} \leq \frac{1}{(xyz)^{1/6}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>0.5*int(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+x + y*I)), t = 0..infinity) <= (1)/((x*y*(x + y*I))^(1/6))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x] <= Divide[1,(x*y*(x + y*I))^(1/6)]</syntaxhighlight> || Aborted || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 18]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[-0.16214470973156064, 0.6784437678906974], Complex[0.7120063770987297, -0.29492269789042613]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[-0.28823404661462, -0.7809212115368181], Complex[0.7640769591692358, -0.10059264002361257]]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.24.E11 19.24.E11] || <math qid="Q6494">\left(\frac{5}{\sqrt{x}+\sqrt{y}+\sqrt{z}+2\sqrt{p}}\right)^{3} \leq \CarlsonsymellintRJ@{x}{y}{z}{p}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(\frac{5}{\sqrt{x}+\sqrt{y}+\sqrt{z}+2\sqrt{p}}\right)^{3} \leq \CarlsonsymellintRJ@{x}{y}{z}{p}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>(Divide[5,Sqrt[x]+Sqrt[y]+Sqrt[x + y*I]+ 2*Sqrt[p]])^(3) <= 3*(x + y*I-x)/(x + y*I-p)*(EllipticPi[(x + y*I-p)/(x + y*I-x),ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/Sqrt[x + y*I-x]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [180 / 180]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[1.3310335634294785, -1.2911719373315522], Complex[-0.2876927312707393, -0.327259429717868]]
Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[0.7477899794343462, -0.4392695700678081], Complex[-0.36602768453446033, 0.5058947820270108]]
Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.24.E11 19.24.E11] || <math qid="Q6494">\CarlsonsymellintRJ@{x}{y}{z}{p} \leq (xyzp^{2})^{-3/10}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\CarlsonsymellintRJ@{x}{y}{z}{p} \leq (xyzp^{2})^{-3/10}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>3*(x + y*I-x)/(x + y*I-p)*(EllipticPi[(x + y*I-p)/(x + y*I-x),ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/Sqrt[x + y*I-x] <= (x*y*(x + y*I)*(p)^(2))^(- 3/10)</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [180 / 180]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[-0.2876927312707393, -0.327259429717868], Complex[0.6159220908806466, 0.7211521128667333]]
Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[-0.36602768453446033, 0.5058947820270108], Complex[0.8086249764673956, -0.49552602288885395]]
Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.24.E12 19.24.E12] || <math qid="Q6495">\tfrac{1}{3}(\sqrt{x}+\sqrt{y}+\sqrt{z}) \leq \CarlsonsymellintRG@{x}{y}{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\tfrac{1}{3}(\sqrt{x}+\sqrt{y}+\sqrt{z}) \leq \CarlsonsymellintRG@{x}{y}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,3]*(Sqrt[x]+Sqrt[y]+Sqrt[x + y*I]) <= Sqrt[x + y*I-x]*(EllipticE[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]+(Cot[ArcCos[Sqrt[x/x + y*I]]])^2*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]+Cot[ArcCos[Sqrt[x/x + y*I]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[x/x + y*I]]]^2])</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 18]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[0.8567842015469013, 0.22245863288189585], Times[Complex[0.8660254037844386, -0.8660254037844385], Plus[Complex[0.9985512968581824, 0.2012315241723115], Times[Complex[0.3176872874027722, -1.049249833251038], Power[Plus[1.0, Times[Complex[0.0, -1.5], Power[k, 2]]], Rational[1, 2]]]]]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[1.2650324920107643, 0.1857896575819671], Times[Complex[0.8660254037844386, 0.8660254037844385], Plus[Complex[1.0566228789425183, 0.3443432776585209], Times[Complex[0.3176872874027722, 1.049249833251038], Power[Plus[1.0, Times[Complex[0.0, 1.5], Power[k, 2]]], Rational[1, 2]]]]]]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.24#Ex7 19.24#Ex7] || <math qid="Q6498">\CarlsonsymellintRF@{x}{y}{z}\CarlsonsymellintRG@{x}{y}{z} > 1</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\CarlsonsymellintRF@{x}{y}{z}\CarlsonsymellintRG@{x}{y}{z} > 1</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]*Sqrt[x + y*I-x]*(EllipticE[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]+(Cot[ArcCos[Sqrt[x/x + y*I]]])^2*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]+Cot[ArcCos[Sqrt[x/x + y*I]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[x/x + y*I]]]^2]) > 1</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 18]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Greater[Times[Complex[0.44712810031579164, 0.7279709757493625], Plus[Complex[0.9985512968581824, 0.2012315241723115], Times[Complex[0.3176872874027722, -1.049249833251038], Power[Plus[1.0, Times[Complex[0.0, -1.5], Power[k, 2]]], Rational[1, 2]]]]], 1.0]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Greater[Times[Complex[0.42667960094115687, -0.925915614148855], Plus[Complex[1.0566228789425183, 0.3443432776585209], Times[Complex[0.3176872874027722, 1.049249833251038], Power[Plus[1.0, Times[Complex[0.0, 1.5], Power[k, 2]]], Rational[1, 2]]]]], 1.0]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.24#Ex8 19.24#Ex8] || <math qid="Q6499">\CarlsonsymellintRF@{x}{y}{z}+\CarlsonsymellintRG@{x}{y}{z} > 2</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\CarlsonsymellintRF@{x}{y}{z}+\CarlsonsymellintRG@{x}{y}{z} > 2</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]+ Sqrt[x + y*I-x]*(EllipticE[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]+(Cot[ArcCos[Sqrt[x/x + y*I]]])^2*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]+Cot[ArcCos[Sqrt[x/x + y*I]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[x/x + y*I]]]^2]) > 2</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 18]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Greater[Plus[Complex[-0.16214470973156064, 0.6784437678906974], Times[Complex[0.8660254037844386, -0.8660254037844385], Plus[Complex[0.9985512968581824, 0.2012315241723115], Times[Complex[0.3176872874027722, -1.049249833251038], Power[Plus[1.0, Times[Complex[0.0, -1.5], Power[k, 2]]], Rational[1, 2]]]]]], 2.0]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Greater[Plus[Complex[-0.28823404661462, -0.7809212115368181], Times[Complex[0.8660254037844386, 0.8660254037844385], Plus[Complex[1.0566228789425183, 0.3443432776585209], Times[Complex[0.3176872874027722, 1.049249833251038], Power[Plus[1.0, Times[Complex[0.0, 1.5], Power[k, 2]]], Rational[1, 2]]]]]], 2.0]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.24.E15 19.24.E15] || <math qid="Q6500">\CarlsonellintRC@{x}{\tfrac{1}{2}(y+z)} \leq \CarlsonsymellintRF@{x}{y}{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\CarlsonellintRC@{x}{\tfrac{1}{2}(y+z)} \leq \CarlsonsymellintRF@{x}{y}{z}</syntaxhighlight> || <math>x \geq 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>1/Sqrt[Divide[1,2]*(y +(x + y*I))]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(Divide[1,2]*(y +(x + y*I)))] <= EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 18]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[0.9580693887321644, 0.49152363500125495], Complex[-0.16214470973156064, 0.6784437678906974]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[0.7805167095081702, -0.12346643314922054], Complex[-0.28823404661462, -0.7809212115368181]]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/19.24.E15 19.24.E15] || <math qid="Q6500">\CarlsonsymellintRF@{x}{y}{z} \leq \CarlsonellintRC@{x}{\sqrt{yz}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\CarlsonsymellintRF@{x}{y}{z} \leq \CarlsonellintRC@{x}{\sqrt{yz}}</syntaxhighlight> || <math>x \geq 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x] <= 1/Sqrt[Sqrt[y*(x + y*I)]]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(Sqrt[y*(x + y*I)])]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 18]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[-0.16214470973156064, 0.6784437678906974], Complex[0.7308447207533646, -0.31118718328917466]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[-0.28823404661462, -0.7809212115368181], Complex[0.765857524311696, -0.1031964554328576]]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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