View source for 22.2

{{DISPLAYTITLE:Multidimensional Theta Functions - 22.2 Definitions}}

<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/22.2#Ex1 22.2#Ex1] || <math qid="Q6922">k = \frac{\Jacobithetaq{2}^{2}@{0}{q}}{\Jacobithetaq{3}^{2}@{0}{q}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>k = \frac{\Jacobithetaq{2}^{2}@{0}{q}}{\Jacobithetaq{3}^{2}@{0}{q}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>k = ((JacobiTheta2(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))^(2))/((JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>k == Divide[(EllipticTheta[2, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]])^(2),(EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]])^(2)]</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 3]
|- 
| [https://dlmf.nist.gov/22.2#Ex2 22.2#Ex2] || <math qid="Q6923">k^{\prime} = \frac{\Jacobithetaq{4}^{2}@{0}{q}}{\Jacobithetaq{3}^{2}@{0}{q}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>k^{\prime} = \frac{\Jacobithetaq{4}^{2}@{0}{q}}{\Jacobithetaq{3}^{2}@{0}{q}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sqrt(1 - (k)^(2)) = ((JacobiTheta4(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))^(2))/((JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[1 - (k)^(2)] == Divide[(EllipticTheta[4, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]])^(2),(EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]])^(2)]</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 3]
|- 
| [https://dlmf.nist.gov/22.2#Ex3 22.2#Ex3] || <math qid="Q6924">\compellintKk@{k} = \frac{\pi}{2}\Jacobithetaq{3}^{2}@{0}{q}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\compellintKk@{k} = \frac{\pi}{2}\Jacobithetaq{3}^{2}@{0}{q}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>EllipticK(k) = (Pi)/(2)*(JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticK[(k)^2] == Divide[Pi,2]*(EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]])^(2)</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: DirectedInfinity[]
Test Values: {Rule[k, 1]}</syntaxhighlight><br></div></div>
|- 
| [https://dlmf.nist.gov/22.2.E3 22.2.E3] || <math qid="Q6925">\zeta = \frac{\pi z}{2\compellintKk@{k}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\zeta = \frac{\pi z}{2\compellintKk@{k}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>zeta = (Pi*z)/(2*EllipticK(k))</syntaxhighlight> || <syntaxhighlight lang=mathematica>\[Zeta] == Divide[Pi*z,2*EllipticK[(k)^2]]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [210 / 210]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.8660254037844387, 0.49999999999999994]
Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.7059984047169785, -0.6365247818792681]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.2.E4 22.2.E4] || <math qid="Q6926">\Jacobiellsnk@{z}{k} = \frac{\Jacobithetaq{3}@{0}{q}}{\Jacobithetaq{2}@{0}{q}}\frac{\Jacobithetaq{1}@{\zeta}{q}}{\Jacobithetaq{4}@{\zeta}{q}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobiellsnk@{z}{k} = \frac{\Jacobithetaq{3}@{0}{q}}{\Jacobithetaq{2}@{0}{q}}\frac{\Jacobithetaq{1}@{\zeta}{q}}{\Jacobithetaq{4}@{\zeta}{q}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiSN(z, k) = (JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta2(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta1(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta4(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiSN[z, (k)^2] == Divide[EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[2, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[1, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[4, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [140 / 210]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.1017958925630662, 9.78035129055685*^-4]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.08293092681074243, -0.5359189266558633]
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.2.E4 22.2.E4] || <math qid="Q6926">\frac{\Jacobithetaq{3}@{0}{q}}{\Jacobithetaq{2}@{0}{q}}\frac{\Jacobithetaq{1}@{\zeta}{q}}{\Jacobithetaq{4}@{\zeta}{q}} = \frac{1}{\Jacobiellnsk@{z}{k}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\Jacobithetaq{3}@{0}{q}}{\Jacobithetaq{2}@{0}{q}}\frac{\Jacobithetaq{1}@{\zeta}{q}}{\Jacobithetaq{4}@{\zeta}{q}} = \frac{1}{\Jacobiellnsk@{z}{k}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta2(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta1(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta4(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k)))) = (1)/(JacobiNS(z, k))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[2, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[1, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[4, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]] == Divide[1,JacobiNS[z, (k)^2]]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [140 / 210]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.1017958925630662, -9.780351290556814*^-4]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.08293092681074243, 0.5359189266558634]
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.2.E5 22.2.E5] || <math qid="Q6927">\Jacobiellcnk@{z}{k} = \frac{\Jacobithetaq{4}@{0}{q}}{\Jacobithetaq{2}@{0}{q}}\frac{\Jacobithetaq{2}@{\zeta}{q}}{\Jacobithetaq{4}@{\zeta}{q}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobiellcnk@{z}{k} = \frac{\Jacobithetaq{4}@{0}{q}}{\Jacobithetaq{2}@{0}{q}}\frac{\Jacobithetaq{2}@{\zeta}{q}}{\Jacobithetaq{4}@{\zeta}{q}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiCN(z, k) = (JacobiTheta4(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta2(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta2(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta4(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiCN[z, (k)^2] == Divide[EllipticTheta[4, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[2, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[2, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[4, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [140 / 210]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.08257811120249814, 0.0027270134984790223]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.13231049687767538, 0.2777560839806882]
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.2.E5 22.2.E5] || <math qid="Q6927">\frac{\Jacobithetaq{4}@{0}{q}}{\Jacobithetaq{2}@{0}{q}}\frac{\Jacobithetaq{2}@{\zeta}{q}}{\Jacobithetaq{4}@{\zeta}{q}} = \frac{1}{\Jacobiellnck@{z}{k}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\Jacobithetaq{4}@{0}{q}}{\Jacobithetaq{2}@{0}{q}}\frac{\Jacobithetaq{2}@{\zeta}{q}}{\Jacobithetaq{4}@{\zeta}{q}} = \frac{1}{\Jacobiellnck@{z}{k}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(JacobiTheta4(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta2(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta2(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta4(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k)))) = (1)/(JacobiNC(z, k))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[EllipticTheta[4, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[2, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[2, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[4, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]] == Divide[1,JacobiNC[z, (k)^2]]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [140 / 210]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.08257811120249814, -0.002727013498479031]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.13231049687767538, -0.2777560839806882]
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.2.E6 22.2.E6] || <math qid="Q6928">\Jacobielldnk@{z}{k} = \frac{\Jacobithetaq{4}@{0}{q}}{\Jacobithetaq{3}@{0}{q}}\frac{\Jacobithetaq{3}@{\zeta}{q}}{\Jacobithetaq{4}@{\zeta}{q}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobielldnk@{z}{k} = \frac{\Jacobithetaq{4}@{0}{q}}{\Jacobithetaq{3}@{0}{q}}\frac{\Jacobithetaq{3}@{\zeta}{q}}{\Jacobithetaq{4}@{\zeta}{q}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiDN(z, k) = (JacobiTheta4(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta3(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta4(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiDN[z, (k)^2] == Divide[EllipticTheta[4, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[3, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[4, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [140 / 210]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.11217526698173597, -1.5044574583405517]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-1.6119897435833945, -2.3508894631681736]
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.2.E6 22.2.E6] || <math qid="Q6928">\frac{\Jacobithetaq{4}@{0}{q}}{\Jacobithetaq{3}@{0}{q}}\frac{\Jacobithetaq{3}@{\zeta}{q}}{\Jacobithetaq{4}@{\zeta}{q}} = \frac{1}{\Jacobiellndk@{z}{k}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\Jacobithetaq{4}@{0}{q}}{\Jacobithetaq{3}@{0}{q}}\frac{\Jacobithetaq{3}@{\zeta}{q}}{\Jacobithetaq{4}@{\zeta}{q}} = \frac{1}{\Jacobiellndk@{z}{k}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(JacobiTheta4(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta3(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta4(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k)))) = (1)/(JacobiND(z, k))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[EllipticTheta[4, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[3, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[4, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]] == Divide[1,JacobiND[z, (k)^2]]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [140 / 210]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.112175266981736, 1.5044574583405517]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.6119897435833943, 2.350889463168173]
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.2.E7 22.2.E7] || <math qid="Q6929">\Jacobiellsdk@{z}{k} = \frac{\Jacobithetaq{3}^{2}@{0}{q}}{\Jacobithetaq{2}@{0}{q}\Jacobithetaq{4}@{0}{q}}\frac{\Jacobithetaq{1}@{\zeta}{q}}{\Jacobithetaq{3}@{\zeta}{q}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobiellsdk@{z}{k} = \frac{\Jacobithetaq{3}^{2}@{0}{q}}{\Jacobithetaq{2}@{0}{q}\Jacobithetaq{4}@{0}{q}}\frac{\Jacobithetaq{1}@{\zeta}{q}}{\Jacobithetaq{3}@{\zeta}{q}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiSD(z, k) = ((JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))^(2))/(JacobiTheta2(0, exp(- Pi*EllipticCK(k)/EllipticK(k)))*JacobiTheta4(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta1(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta3(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiSD[z, (k)^2] == Divide[(EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]])^(2),EllipticTheta[2, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]*EllipticTheta[4, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[1, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[3, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [138 / 210]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.10264566281694597, 1.7190366283522571]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.005017214212665183, 0.8218706074973681]
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.2.E7 22.2.E7] || <math qid="Q6929">\frac{\Jacobithetaq{3}^{2}@{0}{q}}{\Jacobithetaq{2}@{0}{q}\Jacobithetaq{4}@{0}{q}}\frac{\Jacobithetaq{1}@{\zeta}{q}}{\Jacobithetaq{3}@{\zeta}{q}} = \frac{1}{\Jacobielldsk@{z}{k}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\Jacobithetaq{3}^{2}@{0}{q}}{\Jacobithetaq{2}@{0}{q}\Jacobithetaq{4}@{0}{q}}\frac{\Jacobithetaq{1}@{\zeta}{q}}{\Jacobithetaq{3}@{\zeta}{q}} = \frac{1}{\Jacobielldsk@{z}{k}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>((JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))^(2))/(JacobiTheta2(0, exp(- Pi*EllipticCK(k)/EllipticK(k)))*JacobiTheta4(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta1(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta3(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k)))) = (1)/(JacobiDS(z, k))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[(EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]])^(2),EllipticTheta[2, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]*EllipticTheta[4, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[1, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[3, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]] == Divide[1,JacobiDS[z, (k)^2]]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [138 / 210]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.1026456628169461, -1.7190366283522573]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.005017214212665148, -0.8218706074973681]
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.2.E8 22.2.E8] || <math qid="Q6930">\Jacobiellcdk@{z}{k} = \frac{\Jacobithetaq{3}@{0}{q}}{\Jacobithetaq{2}@{0}{q}}\frac{\Jacobithetaq{2}@{\zeta}{q}}{\Jacobithetaq{3}@{\zeta}{q}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobiellcdk@{z}{k} = \frac{\Jacobithetaq{3}@{0}{q}}{\Jacobithetaq{2}@{0}{q}}\frac{\Jacobithetaq{2}@{\zeta}{q}}{\Jacobithetaq{3}@{\zeta}{q}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiCD(z, k) = (JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta2(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta2(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta3(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiCD[z, (k)^2] == Divide[EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[2, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[2, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[3, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [140 / 210]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.23207264303523145, 2.174081147069575]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.3131092646447684, 1.178043032175558]
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.2.E8 22.2.E8] || <math qid="Q6930">\frac{\Jacobithetaq{3}@{0}{q}}{\Jacobithetaq{2}@{0}{q}}\frac{\Jacobithetaq{2}@{\zeta}{q}}{\Jacobithetaq{3}@{\zeta}{q}} = \frac{1}{\Jacobielldck@{z}{k}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\Jacobithetaq{3}@{0}{q}}{\Jacobithetaq{2}@{0}{q}}\frac{\Jacobithetaq{2}@{\zeta}{q}}{\Jacobithetaq{3}@{\zeta}{q}} = \frac{1}{\Jacobielldck@{z}{k}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta2(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta2(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta3(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k)))) = (1)/(JacobiDC(z, k))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[2, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[2, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[3, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]] == Divide[1,JacobiDC[z, (k)^2]]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [140 / 210]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.23207264303523142, -2.174081147069575]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.3131092646447683, -1.178043032175558]
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.2.E9 22.2.E9] || <math qid="Q6931">\Jacobiellsck@{z}{k} = \frac{\Jacobithetaq{3}@{0}{q}}{\Jacobithetaq{4}@{0}{q}}\frac{\Jacobithetaq{1}@{\zeta}{q}}{\Jacobithetaq{2}@{\zeta}{q}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobiellsck@{z}{k} = \frac{\Jacobithetaq{3}@{0}{q}}{\Jacobithetaq{4}@{0}{q}}\frac{\Jacobithetaq{1}@{\zeta}{q}}{\Jacobithetaq{2}@{\zeta}{q}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiSC(z, k) = (JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta4(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta1(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta2(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiSC[z, (k)^2] == Divide[EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[4, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[1, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[2, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [140 / 210]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.2180891710993932, -0.009050644683206828]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.13880139985550538, -0.6261898650931494]
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.2.E9 22.2.E9] || <math qid="Q6931">\frac{\Jacobithetaq{3}@{0}{q}}{\Jacobithetaq{4}@{0}{q}}\frac{\Jacobithetaq{1}@{\zeta}{q}}{\Jacobithetaq{2}@{\zeta}{q}} = \frac{1}{\Jacobiellcsk@{z}{k}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\Jacobithetaq{3}@{0}{q}}{\Jacobithetaq{4}@{0}{q}}\frac{\Jacobithetaq{1}@{\zeta}{q}}{\Jacobithetaq{2}@{\zeta}{q}} = \frac{1}{\Jacobiellcsk@{z}{k}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta4(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta1(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta2(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k)))) = (1)/(JacobiCS(z, k))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[4, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[1, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[2, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]] == Divide[1,JacobiCS[z, (k)^2]]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [140 / 210]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.2180891710993933, 0.009050644683206842]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.13880139985550533, 0.6261898650931494]
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.2.E11 22.2.E11] || <math qid="Q6933">\genJacobiellk{p}{q}@{z}{k} = \ifrac{\Jacobithetatau{p}@{z}{\tau}}{\Jacobithetatau{q}@{z}{\tau}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genJacobiellk{p}{q}@{z}{k} = \ifrac{\Jacobithetatau{p}@{z}{\tau}}{\Jacobithetatau{q}@{z}{\tau}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>genJacobiellk(p)*(exp(- Pi*EllipticCK(k)/EllipticK(k)))* z*k = (JacobiThetap(z,exp(I*Pi*tau)))/(JacobiThetaexp(- Pi*EllipticCK(k)/EllipticK(k))(z,exp(I*Pi*tau)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>genJacobiellk[p]*(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])* z*k == Divide[EllipticTheta[p, z, Exp[I*Pi*(\[Tau])]],EllipticTheta[Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]], z, Exp[I*Pi*(\[Tau])]]]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Times[Complex[0.5000000000000001, 0.8660254037844386], genJacobiellk], Times[Complex[-0.31964140165035193, 0.682988488811487], EllipticTheta[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Complex[0.8660254037844387, 0.49999999999999994], Complex[-0.1897367196265697, 0.08493465422971205]]]]
Test Values: {Rule[k, 1], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Times[Complex[0.26976733074627424, -0.3419272748333145], genJacobiellk], Times[-1.0, EllipticTheta[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Complex[0.8660254037844387, 0.49999999999999994], Complex[-0.1897367196265697, 0.08493465422971205]], Power[EllipticTheta[Power[E, Times[-1, Pi, EllipticK[-3], Power[EllipticK[4], -1]]], Complex[0.8660254037844387, 0.49999999999999994], Complex[-0.1897367196265697, 0.08493465422971205]], -1]]]
Test Values: {Rule[k, 2], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.2.E12 22.2.E12] || <math qid="Q6934">\tau = \ifrac{\iunit\ccompellintKk@{k}}{\compellintKk@{k}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\tau = \ifrac{\iunit\ccompellintKk@{k}}{\compellintKk@{k}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>tau = (I*EllipticCK(k))/(EllipticK(k))</syntaxhighlight> || <syntaxhighlight lang=mathematica>\[Tau] == Divide[I*EllipticK[1-(k)^2],EllipticK[(k)^2]]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.8660254037844387, 0.49999999999999994]
Test Values: {Rule[k, 1], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.4867361401447923, 0.0147898206680519]
Test Values: {Rule[k, 2], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|}
</div>