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{{DISPLAYTITLE:Theta Functions - 20.7 Identities}}

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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/20.7.E1 20.7.E1] || [[Item:Q6796|<math>\Jacobithetaq{3}^{2}@{0}{q}\Jacobithetaq{3}^{2}@{z}{q} = \Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{4}^{2}@{z}{q}+\Jacobithetaq{2}^{2}@{0}{q}\Jacobithetaq{2}^{2}@{z}{q}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobithetaq{3}^{2}@{0}{q}\Jacobithetaq{3}^{2}@{z}{q} = \Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{4}^{2}@{z}{q}+\Jacobithetaq{2}^{2}@{0}{q}\Jacobithetaq{2}^{2}@{z}{q}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(JacobiTheta3(0, q))^(2)* (JacobiTheta3(z, q))^(2) = (JacobiTheta4(0, q))^(2)* (JacobiTheta4(z, q))^(2)+ (JacobiTheta2(0, q))^(2)* (JacobiTheta2(z, q))^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(EllipticTheta[3, 0, q])^(2)* (EllipticTheta[3, z, q])^(2) == (EllipticTheta[4, 0, q])^(2)* (EllipticTheta[4, z, q])^(2)+ (EllipticTheta[2, 0, q])^(2)* (EllipticTheta[2, z, q])^(2)</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 70]
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| [https://dlmf.nist.gov/20.7.E2 20.7.E2] || [[Item:Q6797|<math>\Jacobithetaq{3}^{2}@{0}{q}\Jacobithetaq{4}^{2}@{z}{q} = \Jacobithetaq{2}^{2}@{0}{q}\Jacobithetaq{1}^{2}@{z}{q}+\Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{3}^{2}@{z}{q}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobithetaq{3}^{2}@{0}{q}\Jacobithetaq{4}^{2}@{z}{q} = \Jacobithetaq{2}^{2}@{0}{q}\Jacobithetaq{1}^{2}@{z}{q}+\Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{3}^{2}@{z}{q}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(JacobiTheta3(0, q))^(2)* (JacobiTheta4(z, q))^(2) = (JacobiTheta2(0, q))^(2)* (JacobiTheta1(z, q))^(2)+ (JacobiTheta4(0, q))^(2)* (JacobiTheta3(z, q))^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(EllipticTheta[3, 0, q])^(2)* (EllipticTheta[4, z, q])^(2) == (EllipticTheta[2, 0, q])^(2)* (EllipticTheta[1, z, q])^(2)+ (EllipticTheta[4, 0, q])^(2)* (EllipticTheta[3, z, q])^(2)</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 70]
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| [https://dlmf.nist.gov/20.7.E3 20.7.E3] || [[Item:Q6798|<math>\Jacobithetaq{2}^{2}@{0}{q}\Jacobithetaq{4}^{2}@{z}{q} = \Jacobithetaq{3}^{2}@{0}{q}\Jacobithetaq{1}^{2}@{z}{q}+\Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{2}^{2}@{z}{q}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobithetaq{2}^{2}@{0}{q}\Jacobithetaq{4}^{2}@{z}{q} = \Jacobithetaq{3}^{2}@{0}{q}\Jacobithetaq{1}^{2}@{z}{q}+\Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{2}^{2}@{z}{q}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(JacobiTheta2(0, q))^(2)* (JacobiTheta4(z, q))^(2) = (JacobiTheta3(0, q))^(2)* (JacobiTheta1(z, q))^(2)+ (JacobiTheta4(0, q))^(2)* (JacobiTheta2(z, q))^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(EllipticTheta[2, 0, q])^(2)* (EllipticTheta[4, z, q])^(2) == (EllipticTheta[3, 0, q])^(2)* (EllipticTheta[1, z, q])^(2)+ (EllipticTheta[4, 0, q])^(2)* (EllipticTheta[2, z, q])^(2)</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 70]
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| [https://dlmf.nist.gov/20.7.E4 20.7.E4] || [[Item:Q6799|<math>\Jacobithetaq{2}^{2}@{0}{q}\Jacobithetaq{3}^{2}@{z}{q} = \Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{1}^{2}@{z}{q}+\Jacobithetaq{3}^{2}@{0}{q}\Jacobithetaq{2}^{2}@{z}{q}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobithetaq{2}^{2}@{0}{q}\Jacobithetaq{3}^{2}@{z}{q} = \Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{1}^{2}@{z}{q}+\Jacobithetaq{3}^{2}@{0}{q}\Jacobithetaq{2}^{2}@{z}{q}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(JacobiTheta2(0, q))^(2)* (JacobiTheta3(z, q))^(2) = (JacobiTheta4(0, q))^(2)* (JacobiTheta1(z, q))^(2)+ (JacobiTheta3(0, q))^(2)* (JacobiTheta2(z, q))^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(EllipticTheta[2, 0, q])^(2)* (EllipticTheta[3, z, q])^(2) == (EllipticTheta[4, 0, q])^(2)* (EllipticTheta[1, z, q])^(2)+ (EllipticTheta[3, 0, q])^(2)* (EllipticTheta[2, z, q])^(2)</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 70]
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| [https://dlmf.nist.gov/20.7.E5 20.7.E5] || [[Item:Q6800|<math>\Jacobithetaq{3}^{4}@{0}{q} = \Jacobithetaq{2}^{4}@{0}{q}+\Jacobithetaq{4}^{4}@{0}{q}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobithetaq{3}^{4}@{0}{q} = \Jacobithetaq{2}^{4}@{0}{q}+\Jacobithetaq{4}^{4}@{0}{q}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(JacobiTheta3(0, q))^(4) = (JacobiTheta2(0, q))^(4)+ (JacobiTheta4(0, q))^(4)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(EllipticTheta[3, 0, q])^(4) == (EllipticTheta[2, 0, q])^(4)+ (EllipticTheta[4, 0, q])^(4)</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 10]
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| [https://dlmf.nist.gov/20.7.E6 20.7.E6] || [[Item:Q6801|<math>\Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{1}@{w+z}{q}\Jacobithetaq{1}@{w-z}{q} = \Jacobithetaq{3}^{2}@{w}{q}\Jacobithetaq{2}^{2}@{z}{q}-\Jacobithetaq{2}^{2}@{w}{q}\Jacobithetaq{3}^{2}@{z}{q}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{1}@{w+z}{q}\Jacobithetaq{1}@{w-z}{q} = \Jacobithetaq{3}^{2}@{w}{q}\Jacobithetaq{2}^{2}@{z}{q}-\Jacobithetaq{2}^{2}@{w}{q}\Jacobithetaq{3}^{2}@{z}{q}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(JacobiTheta4(0, q))^(2)* JacobiTheta1(w + z, q)*JacobiTheta1(w - z, q) = (JacobiTheta3(w, q))^(2)* (JacobiTheta2(z, q))^(2)- (JacobiTheta2(w, q))^(2)* (JacobiTheta3(z, q))^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(EllipticTheta[4, 0, q])^(2)* EllipticTheta[1, w + z, q]*EllipticTheta[1, w - z, q] == (EllipticTheta[3, w, q])^(2)* (EllipticTheta[2, z, q])^(2)- (EllipticTheta[2, w, q])^(2)* (EllipticTheta[3, z, q])^(2)</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 300]
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| [https://dlmf.nist.gov/20.7.E7 20.7.E7] || [[Item:Q6802|<math>\Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{2}@{w+z}{q}\Jacobithetaq{2}@{w-z}{q} = \Jacobithetaq{4}^{2}@{w}{q}\Jacobithetaq{2}^{2}@{z}{q}-\Jacobithetaq{1}^{2}@{w}{q}\Jacobithetaq{3}^{2}@{z}{q}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{2}@{w+z}{q}\Jacobithetaq{2}@{w-z}{q} = \Jacobithetaq{4}^{2}@{w}{q}\Jacobithetaq{2}^{2}@{z}{q}-\Jacobithetaq{1}^{2}@{w}{q}\Jacobithetaq{3}^{2}@{z}{q}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(JacobiTheta4(0, q))^(2)* JacobiTheta2(w + z, q)*JacobiTheta2(w - z, q) = (JacobiTheta4(w, q))^(2)* (JacobiTheta2(z, q))^(2)- (JacobiTheta1(w, q))^(2)* (JacobiTheta3(z, q))^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(EllipticTheta[4, 0, q])^(2)* EllipticTheta[2, w + z, q]*EllipticTheta[2, w - z, q] == (EllipticTheta[4, w, q])^(2)* (EllipticTheta[2, z, q])^(2)- (EllipticTheta[1, w, q])^(2)* (EllipticTheta[3, z, q])^(2)</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 300]
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| [https://dlmf.nist.gov/20.7.E8 20.7.E8] || [[Item:Q6803|<math>\Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{3}@{w+z}{q}\Jacobithetaq{3}@{w-z}{q} = \Jacobithetaq{4}^{2}@{w}{q}\Jacobithetaq{3}^{2}@{z}{q}-\Jacobithetaq{1}^{2}@{w}{q}\Jacobithetaq{2}^{2}@{z}{q}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{3}@{w+z}{q}\Jacobithetaq{3}@{w-z}{q} = \Jacobithetaq{4}^{2}@{w}{q}\Jacobithetaq{3}^{2}@{z}{q}-\Jacobithetaq{1}^{2}@{w}{q}\Jacobithetaq{2}^{2}@{z}{q}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(JacobiTheta4(0, q))^(2)* JacobiTheta3(w + z, q)*JacobiTheta3(w - z, q) = (JacobiTheta4(w, q))^(2)* (JacobiTheta3(z, q))^(2)- (JacobiTheta1(w, q))^(2)* (JacobiTheta2(z, q))^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(EllipticTheta[4, 0, q])^(2)* EllipticTheta[3, w + z, q]*EllipticTheta[3, w - z, q] == (EllipticTheta[4, w, q])^(2)* (EllipticTheta[3, z, q])^(2)- (EllipticTheta[1, w, q])^(2)* (EllipticTheta[2, z, q])^(2)</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 300]
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| [https://dlmf.nist.gov/20.7.E9 20.7.E9] || [[Item:Q6804|<math>\Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{4}@{w+z}{q}\Jacobithetaq{4}@{w-z}{q} = \Jacobithetaq{3}^{2}@{w}{q}\Jacobithetaq{3}^{2}@{z}{q}-\Jacobithetaq{2}^{2}@{w}{q}\Jacobithetaq{2}^{2}@{z}{q}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{4}@{w+z}{q}\Jacobithetaq{4}@{w-z}{q} = \Jacobithetaq{3}^{2}@{w}{q}\Jacobithetaq{3}^{2}@{z}{q}-\Jacobithetaq{2}^{2}@{w}{q}\Jacobithetaq{2}^{2}@{z}{q}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(JacobiTheta4(0, q))^(2)* JacobiTheta4(w + z, q)*JacobiTheta4(w - z, q) = (JacobiTheta3(w, q))^(2)* (JacobiTheta3(z, q))^(2)- (JacobiTheta2(w, q))^(2)* (JacobiTheta2(z, q))^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(EllipticTheta[4, 0, q])^(2)* EllipticTheta[4, w + z, q]*EllipticTheta[4, w - z, q] == (EllipticTheta[3, w, q])^(2)* (EllipticTheta[3, z, q])^(2)- (EllipticTheta[2, w, q])^(2)* (EllipticTheta[2, z, q])^(2)</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 300]
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| [https://dlmf.nist.gov/20.7.E10 20.7.E10] || [[Item:Q6805|<math>\Jacobithetaq{1}@{2z}{q} = 2\frac{\Jacobithetaq{1}@{z}{q}\Jacobithetaq{2}@{z}{q}\Jacobithetaq{3}@{z}{q}\Jacobithetaq{4}@{z}{q}}{\Jacobithetaq{2}@{0}{q}\Jacobithetaq{3}@{0}{q}\Jacobithetaq{4}@{0}{q}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobithetaq{1}@{2z}{q} = 2\frac{\Jacobithetaq{1}@{z}{q}\Jacobithetaq{2}@{z}{q}\Jacobithetaq{3}@{z}{q}\Jacobithetaq{4}@{z}{q}}{\Jacobithetaq{2}@{0}{q}\Jacobithetaq{3}@{0}{q}\Jacobithetaq{4}@{0}{q}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta1(2*z, q) = 2*(JacobiTheta1(z, q)*JacobiTheta2(z, q)*JacobiTheta3(z, q)*JacobiTheta4(z, q))/(JacobiTheta2(0, q)*JacobiTheta3(0, q)*JacobiTheta4(0, q))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[1, 2*z, q] == 2*Divide[EllipticTheta[1, z, q]*EllipticTheta[2, z, q]*EllipticTheta[3, z, q]*EllipticTheta[4, z, q],EllipticTheta[2, 0, q]*EllipticTheta[3, 0, q]*EllipticTheta[4, 0, q]]</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 70]
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| [https://dlmf.nist.gov/20.7.E11 20.7.E11] || [[Item:Q6806|<math>\frac{\Jacobithetaq{1}@{z}{q}\Jacobithetaq{2}@{z}{q}}{\Jacobithetaq{1}@{2z}{q^{2}}} = \frac{\Jacobithetaq{3}@{z}{q}\Jacobithetaq{4}@{z}{q}}{\Jacobithetaq{4}@{2z}{q^{2}}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\Jacobithetaq{1}@{z}{q}\Jacobithetaq{2}@{z}{q}}{\Jacobithetaq{1}@{2z}{q^{2}}} = \frac{\Jacobithetaq{3}@{z}{q}\Jacobithetaq{4}@{z}{q}}{\Jacobithetaq{4}@{2z}{q^{2}}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(JacobiTheta1(z, q)*JacobiTheta2(z, q))/(JacobiTheta1(2*z, (q)^(2))) = (JacobiTheta3(z, q)*JacobiTheta4(z, q))/(JacobiTheta4(2*z, (q)^(2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[EllipticTheta[1, z, q]*EllipticTheta[2, z, q],EllipticTheta[1, 2*z, (q)^(2)]] == Divide[EllipticTheta[3, z, q]*EllipticTheta[4, z, q],EllipticTheta[4, 2*z, (q)^(2)]]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.5078048710711283, 0.5078048710711279]
Test Values: {Rule[q, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.5078048710711284, 0.5078048710711281]
Test Values: {Rule[q, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/20.7.E11 20.7.E11] || [[Item:Q6806|<math>\frac{\Jacobithetaq{3}@{z}{q}\Jacobithetaq{4}@{z}{q}}{\Jacobithetaq{4}@{2z}{q^{2}}} = \Jacobithetaq{4}@{0}{q^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\Jacobithetaq{3}@{z}{q}\Jacobithetaq{4}@{z}{q}}{\Jacobithetaq{4}@{2z}{q^{2}}} = \Jacobithetaq{4}@{0}{q^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(JacobiTheta3(z, q)*JacobiTheta4(z, q))/(JacobiTheta4(2*z, (q)^(2))) = JacobiTheta4(0, (q)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[EllipticTheta[3, z, q]*EllipticTheta[4, z, q],EllipticTheta[4, 2*z, (q)^(2)]] == EllipticTheta[4, 0, (q)^(2)]</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 70]
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| [https://dlmf.nist.gov/20.7.E12 20.7.E12] || [[Item:Q6807|<math>\frac{\Jacobithetaq{1}@{z}{q^{2}}\Jacobithetaq{4}@{z}{q^{2}}}{\Jacobithetaq{1}@{z}{q}} = \frac{\Jacobithetaq{2}@{z}{q^{2}}\Jacobithetaq{3}@{z}{q^{2}}}{\Jacobithetaq{2}@{z}{q}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\Jacobithetaq{1}@{z}{q^{2}}\Jacobithetaq{4}@{z}{q^{2}}}{\Jacobithetaq{1}@{z}{q}} = \frac{\Jacobithetaq{2}@{z}{q^{2}}\Jacobithetaq{3}@{z}{q^{2}}}{\Jacobithetaq{2}@{z}{q}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(JacobiTheta1(z, (q)^(2))*JacobiTheta4(z, (q)^(2)))/(JacobiTheta1(z, q)) = (JacobiTheta2(z, (q)^(2))*JacobiTheta3(z, (q)^(2)))/(JacobiTheta2(z, q))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[EllipticTheta[1, z, (q)^(2)]*EllipticTheta[4, z, (q)^(2)],EllipticTheta[1, z, q]] == Divide[EllipticTheta[2, z, (q)^(2)]*EllipticTheta[3, z, (q)^(2)],EllipticTheta[2, z, q]]</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 70]
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| [https://dlmf.nist.gov/20.7.E12 20.7.E12] || [[Item:Q6807|<math>\frac{\Jacobithetaq{2}@{z}{q^{2}}\Jacobithetaq{3}@{z}{q^{2}}}{\Jacobithetaq{2}@{z}{q}} = \tfrac{1}{2}\Jacobithetaq{2}@{0}{q}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\Jacobithetaq{2}@{z}{q^{2}}\Jacobithetaq{3}@{z}{q^{2}}}{\Jacobithetaq{2}@{z}{q}} = \tfrac{1}{2}\Jacobithetaq{2}@{0}{q}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(JacobiTheta2(z, (q)^(2))*JacobiTheta3(z, (q)^(2)))/(JacobiTheta2(z, q)) = (1)/(2)*JacobiTheta2(0, q)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[EllipticTheta[2, z, (q)^(2)]*EllipticTheta[3, z, (q)^(2)],EllipticTheta[2, z, q]] == Divide[1,2]*EllipticTheta[2, 0, q]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.1102230246251565*^-16, -1.5053817239177183]
Test Values: {Rule[q, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[3.3306690738754696*^-16, -1.5053817239177185]
Test Values: {Rule[q, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/20.7.E13 20.7.E13] || [[Item:Q6808|<math>\Jacobithetaq{1}@{z}{q}\Jacobithetaq{1}@{w}{q} = \Jacobithetaq{3}@{z+w}{q^{2}}\Jacobithetaq{2}@{z-w}{q^{2}}-\Jacobithetaq{2}@{z+w}{q^{2}}\Jacobithetaq{3}@{z-w}{q^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobithetaq{1}@{z}{q}\Jacobithetaq{1}@{w}{q} = \Jacobithetaq{3}@{z+w}{q^{2}}\Jacobithetaq{2}@{z-w}{q^{2}}-\Jacobithetaq{2}@{z+w}{q^{2}}\Jacobithetaq{3}@{z-w}{q^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta1(z, q)*JacobiTheta1(w, q) = JacobiTheta3(z + w, (q)^(2))*JacobiTheta2(z - w, (q)^(2))- JacobiTheta2(z + w, (q)^(2))*JacobiTheta3(z - w, (q)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[1, z, q]*EllipticTheta[1, w, q] == EllipticTheta[3, z + w, (q)^(2)]*EllipticTheta[2, z - w, (q)^(2)]- EllipticTheta[2, z + w, (q)^(2)]*EllipticTheta[3, z - w, (q)^(2)]</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 300]
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| [https://dlmf.nist.gov/20.7.E14 20.7.E14] || [[Item:Q6809|<math>\Jacobithetaq{3}@{z}{q}\Jacobithetaq{3}@{w}{q} = \Jacobithetaq{3}@{z+w}{q^{2}}\Jacobithetaq{3}@{z-w}{q^{2}}+\Jacobithetaq{2}@{z+w}{q^{2}}\Jacobithetaq{2}@{z-w}{q^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobithetaq{3}@{z}{q}\Jacobithetaq{3}@{w}{q} = \Jacobithetaq{3}@{z+w}{q^{2}}\Jacobithetaq{3}@{z-w}{q^{2}}+\Jacobithetaq{2}@{z+w}{q^{2}}\Jacobithetaq{2}@{z-w}{q^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta3(z, q)*JacobiTheta3(w, q) = JacobiTheta3(z + w, (q)^(2))*JacobiTheta3(z - w, (q)^(2))+ JacobiTheta2(z + w, (q)^(2))*JacobiTheta2(z - w, (q)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[3, z, q]*EllipticTheta[3, w, q] == EllipticTheta[3, z + w, (q)^(2)]*EllipticTheta[3, z - w, (q)^(2)]+ EllipticTheta[2, z + w, (q)^(2)]*EllipticTheta[2, z - w, (q)^(2)]</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 300]
|- 
| [https://dlmf.nist.gov/20.7.E16 20.7.E16] || [[Item:Q6811|<math>\Jacobithetatau{1}@{2z}{2\tau} = A\Jacobithetatau{1}@{z}{\tau}\Jacobithetatau{2}@{z}{\tau}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobithetatau{1}@{2z}{2\tau} = A\Jacobithetatau{1}@{z}{\tau}\Jacobithetatau{2}@{z}{\tau}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta1(2*z,exp(I*Pi*2*tau)) = A*JacobiTheta1(z,exp(I*Pi*tau))*JacobiTheta2(z,exp(I*Pi*tau))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[1, 2*z, Exp[I*Pi*(2*\[Tau])]] == A*EllipticTheta[1, z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[2, z, Exp[I*Pi*(\[Tau])]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.631641333-1.744983248*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, tau = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.353330373+4.008308689*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, tau = 1/2*3^(1/2)+1/2*I, z = -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 [60 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.6316413333035786, -1.7449832486391479]
Test Values: {Rule[A, 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: Complex[-0.25205232655780907, -0.3227610482702816]
Test Values: {Rule[A, 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[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/20.7.E17 20.7.E17] || [[Item:Q6812|<math>\Jacobithetatau{2}@{2z}{2\tau} = A\Jacobithetatau{1}@{\tfrac{1}{4}\pi-z}{\tau}\Jacobithetatau{1}@{\tfrac{1}{4}\pi+z}{\tau}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobithetatau{2}@{2z}{2\tau} = A\Jacobithetatau{1}@{\tfrac{1}{4}\pi-z}{\tau}\Jacobithetatau{1}@{\tfrac{1}{4}\pi+z}{\tau}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta2(2*z,exp(I*Pi*2*tau)) = A*JacobiTheta1((1)/(4)*Pi - z,exp(I*Pi*tau))*JacobiTheta1((1)/(4)*Pi + z,exp(I*Pi*tau))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[2, 2*z, Exp[I*Pi*(2*\[Tau])]] == A*EllipticTheta[1, Divide[1,4]*Pi - z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[1, Divide[1,4]*Pi + z, Exp[I*Pi*(\[Tau])]]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [60 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.4403734484961686, -1.1891981543571708]
Test Values: {Rule[A, 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: Complex[-0.23150096143650367, 0.21570115304796234]
Test Values: {Rule[A, 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[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/20.7.E18 20.7.E18] || [[Item:Q6813|<math>\Jacobithetatau{3}@{2z}{2\tau} = A\Jacobithetatau{3}@{\tfrac{1}{4}\pi-z}{\tau}\Jacobithetatau{3}@{\tfrac{1}{4}\pi+z}{\tau}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobithetatau{3}@{2z}{2\tau} = A\Jacobithetatau{3}@{\tfrac{1}{4}\pi-z}{\tau}\Jacobithetatau{3}@{\tfrac{1}{4}\pi+z}{\tau}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta3(2*z,exp(I*Pi*2*tau)) = A*JacobiTheta3((1)/(4)*Pi - z,exp(I*Pi*tau))*JacobiTheta3((1)/(4)*Pi + z,exp(I*Pi*tau))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[3, 2*z, Exp[I*Pi*(2*\[Tau])]] == A*EllipticTheta[3, Divide[1,4]*Pi - z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[3, Divide[1,4]*Pi + z, Exp[I*Pi*(\[Tau])]]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [60 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.3438479503598899, -0.39372543999621956]
Test Values: {Rule[A, 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: Complex[0.12535543238516544, -0.5211900545642698]
Test Values: {Rule[A, 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[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/20.7.E19 20.7.E19] || [[Item:Q6814|<math>\Jacobithetatau{4}@{2z}{2\tau} = A\Jacobithetatau{3}@{z}{\tau}\Jacobithetatau{4}@{z}{\tau}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobithetatau{4}@{2z}{2\tau} = A\Jacobithetatau{3}@{z}{\tau}\Jacobithetatau{4}@{z}{\tau}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta4(2*z,exp(I*Pi*2*tau)) = A*JacobiTheta3(z,exp(I*Pi*tau))*JacobiTheta4(z,exp(I*Pi*tau))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[4, 2*z, Exp[I*Pi*(2*\[Tau])]] == A*EllipticTheta[3, z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[4, z, Exp[I*Pi*(\[Tau])]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .88393938e-1-.6601554491*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, tau = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.5678871113-.5102031247*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, tau = 1/2*3^(1/2)+1/2*I, z = -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 [60 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.08839393747885427, -0.6601554493410663]
Test Values: {Rule[A, 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: Complex[0.12758234205780994, -0.4874768056112989]
Test Values: {Rule[A, 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[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/20.7.E21 20.7.E21] || [[Item:Q6816|<math>\Jacobithetatau{1}@{4z}{4\tau} = B\Jacobithetatau{1}@{z}{\tau}\Jacobithetatau{1}@{\tfrac{1}{4}\pi-z}{\tau}\*\Jacobithetatau{1}@{\tfrac{1}{4}\pi+z}{\tau}\Jacobithetatau{2}@{z}{\tau}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobithetatau{1}@{4z}{4\tau} = B\Jacobithetatau{1}@{z}{\tau}\Jacobithetatau{1}@{\tfrac{1}{4}\pi-z}{\tau}\*\Jacobithetatau{1}@{\tfrac{1}{4}\pi+z}{\tau}\Jacobithetatau{2}@{z}{\tau}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta1(4*z,exp(I*Pi*4*tau)) = B*JacobiTheta1(z,exp(I*Pi*tau))*JacobiTheta1((1)/(4)*Pi - z,exp(I*Pi*tau))* JacobiTheta1((1)/(4)*Pi + z,exp(I*Pi*tau))*JacobiTheta2(z,exp(I*Pi*tau))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[1, 4*z, Exp[I*Pi*(4*\[Tau])]] == B*EllipticTheta[1, z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[1, Divide[1,4]*Pi - z, Exp[I*Pi*(\[Tau])]]* EllipticTheta[1, Divide[1,4]*Pi + z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[2, z, Exp[I*Pi*(\[Tau])]]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [60 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.1596846442931608, -2.448595776474227]
Test Values: {Rule[B, 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: Complex[0.3218907084595235, -0.36082838804303224]
Test Values: {Rule[B, 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[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/20.7.E22 20.7.E22] || [[Item:Q6817|<math>\Jacobithetatau{2}@{4z}{4\tau} = B\Jacobithetatau{2}@{\tfrac{1}{8}\pi-z}{\tau}\Jacobithetatau{2}@{\tfrac{1}{8}\pi+z}{\tau}\*\Jacobithetatau{2}@{\tfrac{3}{8}\pi-z}{\tau}\Jacobithetatau{2}@{\tfrac{3}{8}\pi+z}{\tau}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobithetatau{2}@{4z}{4\tau} = B\Jacobithetatau{2}@{\tfrac{1}{8}\pi-z}{\tau}\Jacobithetatau{2}@{\tfrac{1}{8}\pi+z}{\tau}\*\Jacobithetatau{2}@{\tfrac{3}{8}\pi-z}{\tau}\Jacobithetatau{2}@{\tfrac{3}{8}\pi+z}{\tau}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta2(4*z,exp(I*Pi*4*tau)) = B*JacobiTheta2((1)/(8)*Pi - z,exp(I*Pi*tau))*JacobiTheta2((1)/(8)*Pi + z,exp(I*Pi*tau))* JacobiTheta2((3)/(8)*Pi - z,exp(I*Pi*tau))*JacobiTheta2((3)/(8)*Pi + z,exp(I*Pi*tau))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[2, 4*z, Exp[I*Pi*(4*\[Tau])]] == B*EllipticTheta[2, Divide[1,8]*Pi - z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[2, Divide[1,8]*Pi + z, Exp[I*Pi*(\[Tau])]]* EllipticTheta[2, Divide[3,8]*Pi - z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[2, Divide[3,8]*Pi + z, Exp[I*Pi*(\[Tau])]]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [60 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-2.54672123948714, 1.1372871673366372]
Test Values: {Rule[B, 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: Complex[-0.36415557562453404, -0.3395547407401721]
Test Values: {Rule[B, 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[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/20.7.E23 20.7.E23] || [[Item:Q6818|<math>\Jacobithetatau{3}@{4z}{4\tau} = B\Jacobithetatau{3}@{\tfrac{1}{8}\pi-z}{\tau}\Jacobithetatau{3}@{\tfrac{1}{8}\pi+z}{\tau}\*\Jacobithetatau{3}@{\tfrac{3}{8}\pi-z}{\tau}\Jacobithetatau{3}@{\tfrac{3}{8}\pi+z}{\tau}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobithetatau{3}@{4z}{4\tau} = B\Jacobithetatau{3}@{\tfrac{1}{8}\pi-z}{\tau}\Jacobithetatau{3}@{\tfrac{1}{8}\pi+z}{\tau}\*\Jacobithetatau{3}@{\tfrac{3}{8}\pi-z}{\tau}\Jacobithetatau{3}@{\tfrac{3}{8}\pi+z}{\tau}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta3(4*z,exp(I*Pi*4*tau)) = B*JacobiTheta3((1)/(8)*Pi - z,exp(I*Pi*tau))*JacobiTheta3((1)/(8)*Pi + z,exp(I*Pi*tau))* JacobiTheta3((3)/(8)*Pi - z,exp(I*Pi*tau))*JacobiTheta3((3)/(8)*Pi + z,exp(I*Pi*tau))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[3, 4*z, Exp[I*Pi*(4*\[Tau])]] == B*EllipticTheta[3, Divide[1,8]*Pi - z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[3, Divide[1,8]*Pi + z, Exp[I*Pi*(\[Tau])]]* EllipticTheta[3, Divide[3,8]*Pi - z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[3, Divide[3,8]*Pi + z, Exp[I*Pi*(\[Tau])]]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [60 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.2353615104715142, -0.5335293147703523]
Test Values: {Rule[B, 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: Complex[0.11871524589758675, -0.5091754766273449]
Test Values: {Rule[B, 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[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/20.7.E24 20.7.E24] || [[Item:Q6819|<math>\Jacobithetatau{4}@{4z}{4\tau} = B\Jacobithetatau{4}@{z}{\tau}\Jacobithetatau{4}@{\tfrac{1}{4}\pi-z}{\tau}\*\Jacobithetatau{4}@{\tfrac{1}{4}\pi+z}{\tau}\Jacobithetatau{3}@{z}{\tau}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobithetatau{4}@{4z}{4\tau} = B\Jacobithetatau{4}@{z}{\tau}\Jacobithetatau{4}@{\tfrac{1}{4}\pi-z}{\tau}\*\Jacobithetatau{4}@{\tfrac{1}{4}\pi+z}{\tau}\Jacobithetatau{3}@{z}{\tau}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta4(4*z,exp(I*Pi*4*tau)) = B*JacobiTheta4(z,exp(I*Pi*tau))*JacobiTheta4((1)/(4)*Pi - z,exp(I*Pi*tau))* JacobiTheta4((1)/(4)*Pi + z,exp(I*Pi*tau))*JacobiTheta3(z,exp(I*Pi*tau))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[4, 4*z, Exp[I*Pi*(4*\[Tau])]] == B*EllipticTheta[4, z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[4, Divide[1,4]*Pi - z, Exp[I*Pi*(\[Tau])]]* EllipticTheta[4, Divide[1,4]*Pi + z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[3, z, Exp[I*Pi*(\[Tau])]]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [60 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.3584730563399423, -0.5666107505620169]
Test Values: {Rule[B, 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: Complex[0.11914720780154586, -0.5081951100786072]
Test Values: {Rule[B, 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[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/20.7.E25 20.7.E25] || [[Item:Q6820|<math>\deriv{}{z}\left(\frac{\Jacobithetatau{2}@{z}{\tau}}{\Jacobithetatau{4}@{z}{\tau}}\right) = -\frac{\Jacobithetatau{3}^{2}@{0}{\tau}\Jacobithetatau{1}@{z}{\tau}\Jacobithetatau{3}@{z}{\tau}}{\Jacobithetatau{4}^{2}@{z}{\tau}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv{}{z}\left(\frac{\Jacobithetatau{2}@{z}{\tau}}{\Jacobithetatau{4}@{z}{\tau}}\right) = -\frac{\Jacobithetatau{3}^{2}@{0}{\tau}\Jacobithetatau{1}@{z}{\tau}\Jacobithetatau{3}@{z}{\tau}}{\Jacobithetatau{4}^{2}@{z}{\tau}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff((JacobiTheta2(z,exp(I*Pi*tau)))/(JacobiTheta4(z,exp(I*Pi*tau))), z) = -((JacobiTheta3(0,exp(I*Pi*tau)))^(2)* JacobiTheta1(z,exp(I*Pi*tau))*JacobiTheta3(z,exp(I*Pi*tau)))/((JacobiTheta4(z,exp(I*Pi*tau)))^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Divide[EllipticTheta[2, z, Exp[I*Pi*(\[Tau])]],EllipticTheta[4, z, Exp[I*Pi*(\[Tau])]]], z] == -Divide[(EllipticTheta[3, 0, Exp[I*Pi*(\[Tau])]])^(2)* EllipticTheta[1, z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[3, z, Exp[I*Pi*(\[Tau])]],(EllipticTheta[4, z, Exp[I*Pi*(\[Tau])]])^(2)]</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 70]
|- 
| [https://dlmf.nist.gov/20.7.E26 20.7.E26] || [[Item:Q6821|<math>\Jacobithetatau{1}@{z}{\tau+1} = e^{i\pi/4}\Jacobithetatau{1}@{z}{\tau}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobithetatau{1}@{z}{\tau+1} = e^{i\pi/4}\Jacobithetatau{1}@{z}{\tau}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta1(z,exp(I*Pi*tau + 1)) = exp(I*Pi/4)*JacobiTheta1(z,exp(I*Pi*tau))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[1, z, Exp[I*Pi*(\[Tau]+ 1)]] == Exp[I*Pi/4]*EllipticTheta[1, z, Exp[I*Pi*(\[Tau])]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .7294764132+1.608567858*I
Test Values: {tau = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.107791050+1.561378050*I
Test Values: {tau = 1/2*3^(1/2)+1/2*I, z = -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 [7 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.6985877827537141, -0.7949460182709149]
Test Values: {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: Complex[0.345921896794935, 1.4881712816971224]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], 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/20.7.E27 20.7.E27] || [[Item:Q6822|<math>\Jacobithetatau{2}@{z}{\tau+1} = e^{i\pi/4}\Jacobithetatau{2}@{z}{\tau}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobithetatau{2}@{z}{\tau+1} = e^{i\pi/4}\Jacobithetatau{2}@{z}{\tau}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta2(z,exp(I*Pi*tau + 1)) = exp(I*Pi/4)*JacobiTheta2(z,exp(I*Pi*tau))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[2, z, Exp[I*Pi*(\[Tau]+ 1)]] == Exp[I*Pi/4]*EllipticTheta[2, z, Exp[I*Pi*(\[Tau])]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.369621756e-1-.9012887423*I
Test Values: {tau = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 4.590414642+4.526034042*I
Test Values: {tau = 1/2*3^(1/2)+1/2*I, z = -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 [7 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.22524015718924872, -1.3838317643459628]
Test Values: {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: Complex[2.711359141795916, -1.3916787489924032]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], 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/20.7.E28 20.7.E28] || [[Item:Q6823|<math>\Jacobithetatau{3}@{z}{\tau+1} = \Jacobithetatau{4}@{z}{\tau}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobithetatau{3}@{z}{\tau+1} = \Jacobithetatau{4}@{z}{\tau}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta3(z,exp(I*Pi*tau + 1)) = JacobiTheta4(z,exp(I*Pi*tau))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[3, z, Exp[I*Pi*(\[Tau]+ 1)]] == EllipticTheta[4, z, Exp[I*Pi*(\[Tau])]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.500564535+2.208881092*I
Test Values: {tau = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.492914692-.5532090072*I
Test Values: {tau = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 70]
|- 
| [https://dlmf.nist.gov/20.7.E29 20.7.E29] || [[Item:Q6824|<math>\Jacobithetatau{4}@{z}{\tau+1} = \Jacobithetatau{3}@{z}{\tau}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobithetatau{4}@{z}{\tau+1} = \Jacobithetatau{3}@{z}{\tau}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta4(z,exp(I*Pi*tau + 1)) = JacobiTheta3(z,exp(I*Pi*tau))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[4, z, Exp[I*Pi*(\[Tau]+ 1)]] == EllipticTheta[3, z, Exp[I*Pi*(\[Tau])]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.8770870366-.8516489897*I
Test Values: {tau = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 7.362801863+2.459098613*I
Test Values: {tau = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 70]
|- 
| [https://dlmf.nist.gov/20.7.E34 20.7.E34] || [[Item:Q6829|<math>\frac{\Jacobithetaq{1}@{z}{q^{2}}\Jacobithetaq{3}@{z}{q^{2}}}{\Jacobithetaq{1}@{z}{iq}} = \frac{\Jacobithetaq{2}@{z}{q^{2}}\Jacobithetaq{4}@{z}{q^{2}}}{\Jacobithetaq{2}@{z}{iq}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\Jacobithetaq{1}@{z}{q^{2}}\Jacobithetaq{3}@{z}{q^{2}}}{\Jacobithetaq{1}@{z}{iq}} = \frac{\Jacobithetaq{2}@{z}{q^{2}}\Jacobithetaq{4}@{z}{q^{2}}}{\Jacobithetaq{2}@{z}{iq}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(JacobiTheta1(z, (q)^(2))*JacobiTheta3(z, (q)^(2)))/(JacobiTheta1(z, I*q)) = (JacobiTheta2(z, (q)^(2))*JacobiTheta4(z, (q)^(2)))/(JacobiTheta2(z, I*q))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[EllipticTheta[1, z, (q)^(2)]*EllipticTheta[3, z, (q)^(2)],EllipticTheta[1, z, I*q]] == Divide[EllipticTheta[2, z, (q)^(2)]*EllipticTheta[4, z, (q)^(2)],EllipticTheta[2, z, I*q]]</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 70]
|- 
| [https://dlmf.nist.gov/20.7.E34 20.7.E34] || [[Item:Q6829|<math>\frac{\Jacobithetaq{2}@{z}{q^{2}}\Jacobithetaq{4}@{z}{q^{2}}}{\Jacobithetaq{2}@{z}{iq}} = i^{-1/4}\sqrt{\frac{\Jacobithetaq{2}@{0}{q^{2}}\Jacobithetaq{4}@{0}{q^{2}}}{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\Jacobithetaq{2}@{z}{q^{2}}\Jacobithetaq{4}@{z}{q^{2}}}{\Jacobithetaq{2}@{z}{iq}} = i^{-1/4}\sqrt{\frac{\Jacobithetaq{2}@{0}{q^{2}}\Jacobithetaq{4}@{0}{q^{2}}}{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(JacobiTheta2(z, (q)^(2))*JacobiTheta4(z, (q)^(2)))/(JacobiTheta2(z, I*q)) = (I)^(- 1/4)*sqrt((JacobiTheta2(0, (q)^(2))*JacobiTheta4(0, (q)^(2)))/(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[EllipticTheta[2, z, (q)^(2)]*EllipticTheta[4, z, (q)^(2)],EllipticTheta[2, z, I*q]] == (I)^(- 1/4)*Sqrt[Divide[EllipticTheta[2, 0, (q)^(2)]*EllipticTheta[4, 0, (q)^(2)],2]]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.1102230246251565*^-16, 0.47279727016045703]
Test Values: {Rule[q, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[4.440892098500626*^-16, 0.4727972701604571]
Test Values: {Rule[q, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|}
</div>