Results of Theta Functions: Difference between revisions

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; Notation : [[20.1|20.1 Special Notation]]<br>
! DLMF !! Formula !! Constraints !! Maple !! Mathematica !! Symbolic<br>Maple !! Symbolic<br>Mathematica !! Numeric<br>Maple !! Numeric<br>Mathematica
; Properties : [[20.2|20.2 Definitions and Periodic Properties]]<br>[[20.3|20.3 Graphics]]<br>[[20.4|20.4 Values at <math>z</math> = 0]]<br>[[20.5|20.5 Infinite Products and Related Results]]<br>[[20.6|20.6 Power Series]]<br>[[20.7|20.7 Identities]]<br>[[20.8|20.8 Watson’s Expansions]]<br>[[20.9|20.9 Relations to Other Functions]]<br>[[20.10|20.10 Integrals]]<br>[[20.11|20.11 Generalizations and Analogs]]<br>
|-
; Applications : [[20.12|20.12 Mathematical Applications]]<br>[[20.13|20.13 Physical Applications]]<br>
| [https://dlmf.nist.gov/20.2.E1 20.2.E1] || [[Item:Q6741|<math>\Jacobithetatau{1}@{z}{\tau} = \Jacobithetaq{1}@{z}{q}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta1(z,exp(I*Pi*tau)) = JacobiTheta1(z, q)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[1, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[1, z, q]</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 300]
; Computation : [[20.14|20.14 Methods of Computation]]<br>[[20.15|20.15 Tables]]<br>[[20.16|20.16 Software]]<br>
|-
</div>
| [https://dlmf.nist.gov/20.2.E2 20.2.E2] || [[Item:Q6742|<math>\Jacobithetatau{2}@{z}{\tau} = \Jacobithetaq{2}@{z}{q}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta2(z,exp(I*Pi*tau)) = JacobiTheta2(z, q)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[2, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[2, z, q]</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 300]
|-
| [https://dlmf.nist.gov/20.2.E3 20.2.E3] || [[Item:Q6743|<math>\Jacobithetatau{3}@{z}{\tau} = \Jacobithetaq{3}@{z}{q}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta3(z,exp(I*Pi*tau)) = JacobiTheta3(z, q)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[3, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[3, z, q]</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 300]
|-
| [https://dlmf.nist.gov/20.2.E4 20.2.E4] || [[Item:Q6744|<math>\Jacobithetatau{4}@{z}{\tau} = \Jacobithetaq{4}@{z}{q}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta4(z,exp(I*Pi*tau)) = JacobiTheta4(z, q)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[4, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[4, z, q]</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 300]
|-
| [https://dlmf.nist.gov/20.2.E5 20.2.E5] || [[Item:Q6745|<math>z_{m,n} = (m+n\tau)\pi</math>]] || <math></math> || <syntaxhighlight lang=mathematica>z[m , n] = (m + n*tau)*Pi</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[z, m , n] == (m + n*\[Tau])*Pi</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/20.2.E6 20.2.E6] || [[Item:Q6746|<math>\Jacobithetatau{1}@{z+(m+n\tau)\pi}{\tau} = (-1)^{m+n}q^{-n^{2}}e^{-2inz}\Jacobithetatau{1}@{z}{\tau}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta1(z +(m + n*tau)*Pi,exp(I*Pi*tau)) = (- 1)^(m + n)* (q)^(- (n)^(2))* exp(- 2*I*n*z)*JacobiTheta1(z,exp(I*Pi*tau))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[1, z +(m + n*\[Tau])*Pi, Exp[I*Pi*(\[Tau])]] == (- 1)^(m + n)* (q)^(- (n)^(2))* Exp[- 2*I*n*z]*EllipticTheta[1, 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: -18.62843952+6.320473139*I
Test Values: {q = 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, m = 1, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -4187.991134+3174.249087*I
Test Values: {q = 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, m = 1, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [72 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-18.628439525286133, 6.320473094431787]
Test Values: {Rule[m, 1], Rule[n, 1], Rule[q, 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[-4187.991166649552, 3174.249038247393]
Test Values: {Rule[m, 1], Rule[n, 2], Rule[q, 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/20.2.E7 20.2.E7] || [[Item:Q6747|<math>\Jacobithetatau{2}@{z+(m+n\tau)\pi}{\tau} = (-1)^{m}q^{-n^{2}}e^{-2inz}\Jacobithetatau{2}@{z}{\tau}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta2(z +(m + n*tau)*Pi,exp(I*Pi*tau)) = (- 1)^(m)* (q)^(- (n)^(2))* exp(- 2*I*n*z)*JacobiTheta2(z,exp(I*Pi*tau))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[2, z +(m + n*\[Tau])*Pi, Exp[I*Pi*(\[Tau])]] == (- 1)^(m)* (q)^(- (n)^(2))* Exp[- 2*I*n*z]*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: 3.950576529-14.16574159*I
Test Values: {q = 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, m = 1, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 194.3416227+3923.809342*I
Test Values: {q = 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, m = 1, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [72 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[3.9505765593957305, -14.165741580817551]
Test Values: {Rule[m, 1], Rule[n, 1], Rule[q, 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[194.34158297403354, 3923.809350793304]
Test Values: {Rule[m, 1], Rule[n, 2], Rule[q, 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/20.2.E8 20.2.E8] || [[Item:Q6748|<math>\Jacobithetatau{3}@{z+(m+n\tau)\pi}{\tau} = q^{-n^{2}}e^{-2inz}\Jacobithetatau{3}@{z}{\tau}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta3(z +(m + n*tau)*Pi,exp(I*Pi*tau)) = (q)^(- (n)^(2))* exp(- 2*I*n*z)*JacobiTheta3(z,exp(I*Pi*tau))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[3, z +(m + n*\[Tau])*Pi, Exp[I*Pi*(\[Tau])]] == (q)^(- (n)^(2))* Exp[- 2*I*n*z]*EllipticTheta[3, 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: -8.181021151+18.44680448*I
Test Values: {q = 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, m = 1, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 516.5479372-5365.925849*I
Test Values: {q = 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, m = 1, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [72 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-8.181021187984683, 18.446804447343553]
Test Values: {Rule[m, 1], Rule[n, 1], Rule[q, 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[516.547995329447, -5365.925840115722]
Test Values: {Rule[m, 1], Rule[n, 2], Rule[q, 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/20.2.E9 20.2.E9] || [[Item:Q6749|<math>\Jacobithetatau{4}@{z+(m+n\tau)\pi}{\tau} = (-1)^{n}q^{-n^{2}}e^{-2inz}\Jacobithetatau{4}@{z}{\tau}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta4(z +(m + n*tau)*Pi,exp(I*Pi*tau)) = (- 1)^(n)* (q)^(- (n)^(2))* exp(- 2*I*n*z)*JacobiTheta4(z,exp(I*Pi*tau))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[4, z +(m + n*\[Tau])*Pi, Exp[I*Pi*(\[Tau])]] == (- 1)^(n)* (q)^(- (n)^(2))* Exp[- 2*I*n*z]*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: -4.510694228-11.16801166*I
Test Values: {q = 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, m = 1, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -2085.869632-2449.864344*I
Test Values: {q = 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, m = 1, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [72 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-4.5106942149502025, -11.168011665083736]
Test Values: {Rule[m, 1], Rule[n, 1], Rule[q, 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[-2085.8696157878926, -2449.864367431773]
Test Values: {Rule[m, 1], Rule[n, 2], Rule[q, 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/20.2.E11 20.2.E11] || [[Item:Q6751|<math>\Jacobithetatau{1}@{z}{\tau} = -\Jacobithetatau{2}@{z+\tfrac{1}{2}\pi}{\tau}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta1(z,exp(I*Pi*tau)) = - JacobiTheta2(z +(1)/(2)*Pi,exp(I*Pi*tau))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[1, z, Exp[I*Pi*(\[Tau])]] == - EllipticTheta[2, z +Divide[1,2]*Pi, Exp[I*Pi*(\[Tau])]]</syntaxhighlight> || Successful || Failure || Skip - symbolical successful subtest || Successful [Tested: 70]
|-
| [https://dlmf.nist.gov/20.2.E11 20.2.E11] || [[Item:Q6751|<math>-\Jacobithetatau{2}@{z+\tfrac{1}{2}\pi}{\tau} = -iM\Jacobithetatau{4}@{z+\tfrac{1}{2}\pi\tau}{\tau}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>- JacobiTheta2(z +(1)/(2)*Pi,exp(I*Pi*tau)) = - I*M*JacobiTheta4(z +(1)/(2)*Pi*tau,exp(I*Pi*tau))</syntaxhighlight> || <syntaxhighlight lang=mathematica>- EllipticTheta[2, z +Divide[1,2]*Pi, Exp[I*Pi*(\[Tau])]] == - I*M*EllipticTheta[4, z +Divide[1,2]*Pi*\[Tau], 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: -2.656130280+.8441101403*I
Test Values: {M = 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: 2.726401771+.6812031274*I
Test Values: {M = 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[-2.65613027348202, 0.8441101301235214]
Test Values: {Rule[M, 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[-2.1127027753782777, -0.09362434622808774]
Test Values: {Rule[M, 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/20.2.E11 20.2.E11] || [[Item:Q6751|<math>-iM\Jacobithetatau{4}@{z+\tfrac{1}{2}\pi\tau}{\tau} = -iM\Jacobithetatau{3}@{z+\tfrac{1}{2}\pi+\tfrac{1}{2}\pi\tau}{\tau}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>- I*M*JacobiTheta4(z +(1)/(2)*Pi*tau,exp(I*Pi*tau)) = - I*M*JacobiTheta3(z +(1)/(2)*Pi +(1)/(2)*Pi*tau,exp(I*Pi*tau))</syntaxhighlight> || <syntaxhighlight lang=mathematica>- I*M*EllipticTheta[4, z +Divide[1,2]*Pi*\[Tau], Exp[I*Pi*(\[Tau])]] == - I*M*EllipticTheta[3, z +Divide[1,2]*Pi +Divide[1,2]*Pi*\[Tau], Exp[I*Pi*(\[Tau])]]</syntaxhighlight> || Successful || Failure || Skip - symbolical successful subtest || Successful [Tested: 300]
|-
| [https://dlmf.nist.gov/20.2.E12 20.2.E12] || [[Item:Q6752|<math>\Jacobithetatau{2}@{z}{\tau} = \Jacobithetatau{1}@{z+\tfrac{1}{2}\pi}{\tau}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta2(z,exp(I*Pi*tau)) = JacobiTheta1(z +(1)/(2)*Pi,exp(I*Pi*tau))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[2, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[1, z +Divide[1,2]*Pi, Exp[I*Pi*(\[Tau])]]</syntaxhighlight> || Successful || Failure || Skip - symbolical successful subtest || Successful [Tested: 70]
|-
| [https://dlmf.nist.gov/20.2.E12 20.2.E12] || [[Item:Q6752|<math>\Jacobithetatau{1}@{z+\tfrac{1}{2}\pi}{\tau} = M\Jacobithetatau{3}@{z+\tfrac{1}{2}\pi\tau}{\tau}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta1(z +(1)/(2)*Pi,exp(I*Pi*tau)) = M*JacobiTheta3(z +(1)/(2)*Pi*tau,exp(I*Pi*tau))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[1, z +Divide[1,2]*Pi, Exp[I*Pi*(\[Tau])]] == M*EllipticTheta[3, z +Divide[1,2]*Pi*\[Tau], 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: -.5985410657+1.995750316*I
Test Values: {M = 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: -.54909285e-2-5.605651596*I
Test Values: {M = 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[-0.5985410729973577, 1.9957503125524838]
Test Values: {Rule[M, 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[-1.3220776692839347, 1.382964384147599]
Test Values: {Rule[M, 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/20.2.E12 20.2.E12] || [[Item:Q6752|<math>M\Jacobithetatau{3}@{z+\tfrac{1}{2}\pi\tau}{\tau} = M\Jacobithetatau{4}@{z+\tfrac{1}{2}\pi+\tfrac{1}{2}\pi\tau}{\tau}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>M*JacobiTheta3(z +(1)/(2)*Pi*tau,exp(I*Pi*tau)) = M*JacobiTheta4(z +(1)/(2)*Pi +(1)/(2)*Pi*tau,exp(I*Pi*tau))</syntaxhighlight> || <syntaxhighlight lang=mathematica>M*EllipticTheta[3, z +Divide[1,2]*Pi*\[Tau], Exp[I*Pi*(\[Tau])]] == M*EllipticTheta[4, z +Divide[1,2]*Pi +Divide[1,2]*Pi*\[Tau], Exp[I*Pi*(\[Tau])]]</syntaxhighlight> || Successful || Failure || Skip - symbolical successful subtest || Successful [Tested: 300]
|-
| [https://dlmf.nist.gov/20.2.E13 20.2.E13] || [[Item:Q6753|<math>\Jacobithetatau{3}@{z}{\tau} = \Jacobithetatau{4}@{z+\tfrac{1}{2}\pi}{\tau}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta3(z,exp(I*Pi*tau)) = JacobiTheta4(z +(1)/(2)*Pi,exp(I*Pi*tau))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[3, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[4, z +Divide[1,2]*Pi, Exp[I*Pi*(\[Tau])]]</syntaxhighlight> || Successful || Failure || Skip - symbolical successful subtest || Successful [Tested: 70]
|-
| [https://dlmf.nist.gov/20.2.E13 20.2.E13] || [[Item:Q6753|<math>\Jacobithetatau{4}@{z+\tfrac{1}{2}\pi}{\tau} = M\Jacobithetatau{2}@{z+\tfrac{1}{2}\pi\tau}{\tau}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta4(z +(1)/(2)*Pi,exp(I*Pi*tau)) = M*JacobiTheta2(z +(1)/(2)*Pi*tau,exp(I*Pi*tau))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[4, z +Divide[1,2]*Pi, Exp[I*Pi*(\[Tau])]] == M*EllipticTheta[2, z +Divide[1,2]*Pi*\[Tau], 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: -1.209558888+2.590545189*I
Test Values: {M = 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: -.8537358258+1.281173247*I
Test Values: {M = 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[-1.2095588901959111, 2.5905451776573183]
Test Values: {Rule[M, 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[-1.901447370098885, -0.21207958455265288]
Test Values: {Rule[M, 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/20.2.E13 20.2.E13] || [[Item:Q6753|<math>M\Jacobithetatau{2}@{z+\tfrac{1}{2}\pi\tau}{\tau} = M\Jacobithetatau{1}@{z+\tfrac{1}{2}\pi+\tfrac{1}{2}\pi\tau}{\tau}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>M*JacobiTheta2(z +(1)/(2)*Pi*tau,exp(I*Pi*tau)) = M*JacobiTheta1(z +(1)/(2)*Pi +(1)/(2)*Pi*tau,exp(I*Pi*tau))</syntaxhighlight> || <syntaxhighlight lang=mathematica>M*EllipticTheta[2, z +Divide[1,2]*Pi*\[Tau], Exp[I*Pi*(\[Tau])]] == M*EllipticTheta[1, z +Divide[1,2]*Pi +Divide[1,2]*Pi*\[Tau], Exp[I*Pi*(\[Tau])]]</syntaxhighlight> || Successful || Failure || Skip - symbolical successful subtest || Successful [Tested: 300]
|-
| [https://dlmf.nist.gov/20.2.E14 20.2.E14] || [[Item:Q6754|<math>\Jacobithetatau{4}@{z}{\tau} = \Jacobithetatau{3}@{z+\tfrac{1}{2}\pi}{\tau}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta4(z,exp(I*Pi*tau)) = JacobiTheta3(z +(1)/(2)*Pi,exp(I*Pi*tau))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[4, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[3, z +Divide[1,2]*Pi, Exp[I*Pi*(\[Tau])]]</syntaxhighlight> || Successful || Failure || Skip - symbolical successful subtest || Successful [Tested: 70]
|-
| [https://dlmf.nist.gov/20.2.E14 20.2.E14] || [[Item:Q6754|<math>\Jacobithetatau{3}@{z+\tfrac{1}{2}\pi}{\tau} = -iM\Jacobithetatau{1}@{z+\tfrac{1}{2}\pi\tau}{\tau}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta3(z +(1)/(2)*Pi,exp(I*Pi*tau)) = - I*M*JacobiTheta1(z +(1)/(2)*Pi*tau,exp(I*Pi*tau))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[3, z +Divide[1,2]*Pi, Exp[I*Pi*(\[Tau])]] == - I*M*EllipticTheta[1, z +Divide[1,2]*Pi*\[Tau], 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: .6082553523+1.594370406*I
Test Values: {M = 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: -3.995156823-3.872683361*I
Test Values: {M = 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[0.6082553477594059, 1.594370409676146]
Test Values: {Rule[M, 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[-2.5995966648178563, -0.1152010311326023]
Test Values: {Rule[M, 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/20.2.E14 20.2.E14] || [[Item:Q6754|<math>-iM\Jacobithetatau{1}@{z+\tfrac{1}{2}\pi\tau}{\tau} = iM\Jacobithetatau{2}@{z+\tfrac{1}{2}\pi+\tfrac{1}{2}\pi\tau}{\tau}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>- I*M*JacobiTheta1(z +(1)/(2)*Pi*tau,exp(I*Pi*tau)) = I*M*JacobiTheta2(z +(1)/(2)*Pi +(1)/(2)*Pi*tau,exp(I*Pi*tau))</syntaxhighlight> || <syntaxhighlight lang=mathematica>- I*M*EllipticTheta[1, z +Divide[1,2]*Pi*\[Tau], Exp[I*Pi*(\[Tau])]] == I*M*EllipticTheta[2, z +Divide[1,2]*Pi +Divide[1,2]*Pi*\[Tau], Exp[I*Pi*(\[Tau])]]</syntaxhighlight> || Successful || Failure || Skip - symbolical successful subtest || Successful [Tested: 300]
|-
| [https://dlmf.nist.gov/20.4.E1 20.4.E1] || [[Item:Q6755|<math>\Jacobithetaq{1}@{0}{q} = \Jacobithetaq{2}'@{0}{q}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta1(0, q) = diff( JacobiTheta2(0, q), 0$(1) )</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[1, 0, q] == D[EllipticTheta[2, 0, q], {0, 1}]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Times[-1.0, D[EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]]
Test Values: {0.0, 1.0}]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Times[-1.0, D[EllipticTheta[2, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]]
Test Values: {0.0, 1.0}]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/20.4.E1 20.4.E1] || [[Item:Q6755|<math>\Jacobithetaq{2}'@{0}{q} = \Jacobithetaq{3}'@{0}{q}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>diff( JacobiTheta2(0, q), 0$(1) ) = diff( JacobiTheta3(0, q), 0$(1) )</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[EllipticTheta[2, 0, q], {0, 1}] == D[EllipticTheta[3, 0, q], {0, 1}]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[D[EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]]
Test Values: {0.0, 1.0}], Times[-1.0, D[EllipticTheta[3, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], {0.0, 1.0}]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[D[EllipticTheta[2, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]]
Test Values: {0.0, 1.0}], Times[-1.0, D[EllipticTheta[3, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]], {0.0, 1.0}]]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/20.4.E1 20.4.E1] || [[Item:Q6755|<math>\Jacobithetaq{3}'@{0}{q} = \Jacobithetaq{4}'@{0}{q}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>diff( JacobiTheta3(0, q), 0$(1) ) = diff( JacobiTheta4(0, q), 0$(1) )</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[EllipticTheta[3, 0, q], {0, 1}] == D[EllipticTheta[4, 0, q], {0, 1}]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[D[EllipticTheta[3, 0.0, Complex[0.8660254037844387, 0.49999999999999994]]
Test Values: {0.0, 1.0}], Times[-1.0, D[EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], {0.0, 1.0}]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[D[EllipticTheta[3, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]]
Test Values: {0.0, 1.0}], Times[-1.0, D[EllipticTheta[4, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]], {0.0, 1.0}]]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/20.4.E1 20.4.E1] || [[Item:Q6755|<math>\Jacobithetaq{4}'@{0}{q} = 0</math>]] || <math></math> || <syntaxhighlight lang=mathematica>diff( JacobiTheta4(0, q), 0$(1) ) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[EllipticTheta[4, 0, q], {0, 1}] == 0</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: D[EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]]
Test Values: {0.0, 1.0}], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: D[EllipticTheta[4, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]]
Test Values: {0.0, 1.0}], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/20.4.E6 20.4.E6] || [[Item:Q6761|<math>\Jacobithetaq{1}'@{0}{q} = \Jacobithetaq{2}@{0}{q}\Jacobithetaq{3}@{0}{q}\Jacobithetaq{4}@{0}{q}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>diff( JacobiTheta1(0, q), 0$(1) ) = JacobiTheta2(0, q)*JacobiTheta3(0, q)*JacobiTheta4(0, q)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[EllipticTheta[1, 0, q], {0, 1}] == EllipticTheta[2, 0, q]*EllipticTheta[3, 0, q]*EllipticTheta[4, 0, q]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[D[0.0
Test Values: {0.0, 1.0}], Times[-1.0, EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], EllipticTheta[3, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[D[0.0
Test Values: {0.0, 1.0}], Times[-1.0, EllipticTheta[2, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]], EllipticTheta[3, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]], EllipticTheta[4, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]]]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/20.4.E7 20.4.E7] || [[Item:Q6762|<math>\Jacobithetaq{1}''(0,q) = \Jacobithetaq{2}'''@{0}{q}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>subs( temp=(0 , q) , diff( JacobiTheta1(temp, =), temp$(2) ) )*diff( JacobiTheta2(0, q), 0$(3) )</syntaxhighlight> || <syntaxhighlight lang=mathematica>(D[EllipticTheta[1, temp, ==], {temp, 2}]/.temp-> (0 , q) )*D[EllipticTheta[2, 0, q], {0, 3}]</syntaxhighlight> || Translation Error || Translation Error || - || Skip - symbolical successful subtest
|-
| [https://dlmf.nist.gov/20.4.E7 20.4.E7] || [[Item:Q6762|<math>\Jacobithetaq{2}'''@{0}{q} = \Jacobithetaq{3}'''@{0}{q}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>diff( JacobiTheta2(0, q), 0$(3) ) = diff( JacobiTheta3(0, q), 0$(3) )</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[EllipticTheta[2, 0, q], {0, 3}] == D[EllipticTheta[3, 0, q], {0, 3}]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[D[EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]]
Test Values: {0.0, 3.0}], Times[-1.0, D[EllipticTheta[3, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], {0.0, 3.0}]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[D[EllipticTheta[2, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]]
Test Values: {0.0, 3.0}], Times[-1.0, D[EllipticTheta[3, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]], {0.0, 3.0}]]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/20.4.E7 20.4.E7] || [[Item:Q6762|<math>\Jacobithetaq{3}'''@{0}{q} = \Jacobithetaq{4}'''@{0}{q}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>diff( JacobiTheta3(0, q), 0$(3) ) = diff( JacobiTheta4(0, q), 0$(3) )</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[EllipticTheta[3, 0, q], {0, 3}] == D[EllipticTheta[4, 0, q], {0, 3}]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[D[EllipticTheta[3, 0.0, Complex[0.8660254037844387, 0.49999999999999994]]
Test Values: {0.0, 3.0}], Times[-1.0, D[EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], {0.0, 3.0}]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[D[EllipticTheta[3, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]]
Test Values: {0.0, 3.0}], Times[-1.0, D[EllipticTheta[4, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]], {0.0, 3.0}]]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/20.4.E7 20.4.E7] || [[Item:Q6762|<math>\Jacobithetaq{4}'''@{0}{q} = 0</math>]] || <math></math> || <syntaxhighlight lang=mathematica>diff( JacobiTheta4(0, q), 0$(3) ) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[EllipticTheta[4, 0, q], {0, 3}] == 0</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: D[EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]]
Test Values: {0.0, 3.0}], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: D[EllipticTheta[4, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]]
Test Values: {0.0, 3.0}], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/20.4.E8 20.4.E8] || [[Item:Q6763|<math>\frac{\Jacobithetaq{1}'''@{0}{q}}{\Jacobithetaq{1}'@{0}{q}} = -1+24\sum_{n=1}^{\infty}\frac{q^{2n}}{(1-q^{2n})^{2}}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>(diff( JacobiTheta1(0, q), 0$(3) ))/(diff( JacobiTheta1(0, q), 0$(1) )) = - 1 + 24*sum(((q)^(2*n))/((1 - (q)^(2*n))^(2)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[D[EllipticTheta[1, 0, q], {0, 3}],D[EllipticTheta[1, 0, q], {0, 1}]] == - 1 + 24*Sum[Divide[(q)^(2*n),(1 - (q)^(2*n))^(2)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[1.0, Times[Power[D[0.0
Test Values: {0.0, 1.0}], -1], D[0.0, {0.0, 3.0}]], Times[-24.0, NSum[Times[Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Times[2, n]], Power[Plus[1, Times[-1, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Times[2, n]]]], -2]], {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[1.0, Times[Power[D[0.0
Test Values: {0.0, 1.0}], -1], D[0.0, {0.0, 3.0}]], Times[-24.0, NSum[Times[Power[Power[E, Times[Complex[0, Rational[2, 3]], Pi]], Times[2, n]], Power[Plus[1, Times[-1, Power[Power[E, Times[Complex[0, Rational[2, 3]], Pi]], Times[2, n]]]], -2]], {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/20.4.E9 20.4.E9] || [[Item:Q6764|<math>\frac{\Jacobithetaq{2}''@{0}{q}}{\Jacobithetaq{2}@{0}{q}} = -1-8\sum_{n=1}^{\infty}\frac{q^{2n}}{(1+q^{2n})^{2}}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>(diff( JacobiTheta2(0, q), 0$(2) ))/(JacobiTheta2(0, q)) = - 1 - 8*sum(((q)^(2*n))/((1 + (q)^(2*n))^(2)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[D[EllipticTheta[2, 0, q], {0, 2}],EllipticTheta[2, 0, q]] == - 1 - 8*Sum[Divide[(q)^(2*n),(1 + (q)^(2*n))^(2)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/20.4.E10 20.4.E10] || [[Item:Q6765|<math>\frac{\Jacobithetaq{3}''@{0}{q}}{\Jacobithetaq{3}@{0}{q}} = -8\sum_{n=1}^{\infty}\frac{q^{2n-1}}{(1+q^{2n-1})^{2}}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>(diff( JacobiTheta3(0, q), 0$(2) ))/(JacobiTheta3(0, q)) = - 8*sum(((q)^(2*n - 1))/((1 + (q)^(2*n - 1))^(2)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[D[EllipticTheta[3, 0, q], {0, 2}],EllipticTheta[3, 0, q]] == - 8*Sum[Divide[(q)^(2*n - 1),(1 + (q)^(2*n - 1))^(2)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Error || Aborted || - || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/20.4.E11 20.4.E11] || [[Item:Q6766|<math>\frac{\Jacobithetaq{4}''@{0}{q}}{\Jacobithetaq{4}@{0}{q}} = 8\sum_{n=1}^{\infty}\frac{q^{2n-1}}{(1-q^{2n-1})^{2}}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>(diff( JacobiTheta4(0, q), 0$(2) ))/(JacobiTheta4(0, q)) = 8*sum(((q)^(2*n - 1))/((1 - (q)^(2*n - 1))^(2)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[D[EllipticTheta[4, 0, q], {0, 2}],EllipticTheta[4, 0, q]] == 8*Sum[Divide[(q)^(2*n - 1),(1 - (q)^(2*n - 1))^(2)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Error || Aborted || - || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/20.4.E12 20.4.E12] || [[Item:Q6767|<math>\frac{\Jacobithetaq{1}'''@{0}{q}}{\Jacobithetaq{1}'@{0}{q}} = \frac{\Jacobithetaq{2}''@{0}{q}}{\Jacobithetaq{2}@{0}{q}}+\frac{\Jacobithetaq{3}''@{0}{q}}{\Jacobithetaq{3}@{0}{q}}+\frac{\Jacobithetaq{4}''@{0}{q}}{\Jacobithetaq{4}@{0}{q}}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>(diff( JacobiTheta1(0, q), 0$(3) ))/(diff( JacobiTheta1(0, q), 0$(1) )) = (diff( JacobiTheta2(0, q), 0$(2) ))/(JacobiTheta2(0, q))+(diff( JacobiTheta3(0, q), 0$(2) ))/(JacobiTheta3(0, q))+(diff( JacobiTheta4(0, q), 0$(2) ))/(JacobiTheta4(0, q))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[D[EllipticTheta[1, 0, q], {0, 3}],D[EllipticTheta[1, 0, q], {0, 1}]] == Divide[D[EllipticTheta[2, 0, q], {0, 2}],EllipticTheta[2, 0, q]]+Divide[D[EllipticTheta[3, 0, q], {0, 2}],EllipticTheta[3, 0, q]]+Divide[D[EllipticTheta[4, 0, q], {0, 2}],EllipticTheta[4, 0, q]]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Times[Power[D[0.0
Test Values: {0.0, 1.0}], -1], D[0.0, {0.0, 3.0}]], Times[-1.0, D[EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], {0.0, 2.0}], Power[EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], -1]], Times[-1.0, D[EllipticTheta[3, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], {0.0, 2.0}], Power[EllipticTheta[3, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], -1]], Times[-1.0, D[EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], {0.0, 2.0}], Power[EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], -1]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Times[Power[D[0.0
Test Values: {0.0, 1.0}], -1], D[0.0, {0.0, 3.0}]], Times[-1.0, D[EllipticTheta[2, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]], {0.0, 2.0}], Power[EllipticTheta[2, 0.0, Co</div></div>
|-
| [https://dlmf.nist.gov/20.5.E5 20.5.E5] || [[Item:Q6772|<math>\Jacobithetatau{1}@{z}{\tau} = \Jacobithetatau{1}'@{0}{\tau}\sin@@{z}\prod_{n=1}^{\infty}\frac{\sin@{n\pi\tau+z}\sin@{n\pi\tau-z}}{\sin^{2}@{n\pi\tau}}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta1(z,exp(I*Pi*tau)) = diff( JacobiTheta1(0,exp(I*Pi*tau)), 0$(1) )*sin(z)*product((sin(n*Pi*tau + z)*sin(n*Pi*tau - z))/((sin(n*Pi*tau))^(2)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[1, z, Exp[I*Pi*(\[Tau])]] == D[EllipticTheta[1, 0, Exp[I*Pi*(\[Tau])]], {0, 1}]*Sin[z]*Product[Divide[Sin[n*Pi*\[Tau]+ z]*Sin[n*Pi*\[Tau]- z],(Sin[n*Pi*\[Tau]])^(2)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Error || Aborted || - || Skipped - Because timed out
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| [https://dlmf.nist.gov/20.5.E6 20.5.E6] || [[Item:Q6773|<math>\Jacobithetatau{2}@{z}{\tau} = \Jacobithetatau{2}@{0}{\tau}\cos@@{z}\prod_{n=1}^{\infty}\frac{\cos@{n\pi\tau+z}\cos@{n\pi\tau-z}}{\cos^{2}@{n\pi\tau}}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta2(z,exp(I*Pi*tau)) = JacobiTheta2(0,exp(I*Pi*tau))*cos(z)*product((cos(n*Pi*tau + z)*cos(n*Pi*tau - z))/((cos(n*Pi*tau))^(2)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[2, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[2, 0, Exp[I*Pi*(\[Tau])]]*Cos[z]*Product[Divide[Cos[n*Pi*\[Tau]+ z]*Cos[n*Pi*\[Tau]- z],(Cos[n*Pi*\[Tau]])^(2)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Error || Skipped - Because timed out
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| [https://dlmf.nist.gov/20.5.E7 20.5.E7] || [[Item:Q6774|<math>\Jacobithetatau{3}@{z}{\tau} = \Jacobithetatau{3}@{0}{\tau}\prod_{n=1}^{\infty}\frac{\cos@{(n-\tfrac{1}{2})\pi\tau+z}\cos@{(n-\tfrac{1}{2})\pi\tau-z}}{\cos^{2}@{(n-\tfrac{1}{2})\pi\tau}}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta3(z,exp(I*Pi*tau)) = JacobiTheta3(0,exp(I*Pi*tau))*product((cos((n -(1)/(2))*Pi*tau + z)*cos((n -(1)/(2))*Pi*tau - z))/((cos((n -(1)/(2))*Pi*tau))^(2)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[3, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[3, 0, Exp[I*Pi*(\[Tau])]]*Product[Divide[Cos[(n -Divide[1,2])*Pi*\[Tau]+ z]*Cos[(n -Divide[1,2])*Pi*\[Tau]- z],(Cos[(n -Divide[1,2])*Pi*\[Tau]])^(2)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Error || Skipped - Because timed out
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| [https://dlmf.nist.gov/20.5.E8 20.5.E8] || [[Item:Q6775|<math>\Jacobithetatau{4}@{z}{\tau} = \Jacobithetatau{4}@{0}{\tau}\prod_{n=1}^{\infty}\frac{\sin@{(n-\tfrac{1}{2})\pi\tau+z}\sin@{(n-\tfrac{1}{2})\pi\tau-z}}{\sin^{2}@{(n-\tfrac{1}{2})\pi\tau}}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta4(z,exp(I*Pi*tau)) = JacobiTheta4(0,exp(I*Pi*tau))*product((sin((n -(1)/(2))*Pi*tau + z)*sin((n -(1)/(2))*Pi*tau - z))/((sin((n -(1)/(2))*Pi*tau))^(2)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[4, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[4, 0, Exp[I*Pi*(\[Tau])]]*Product[Divide[Sin[(n -Divide[1,2])*Pi*\[Tau]+ z]*Sin[(n -Divide[1,2])*Pi*\[Tau]- z],(Sin[(n -Divide[1,2])*Pi*\[Tau]])^(2)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Error || Skipped - Because timed out
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| [https://dlmf.nist.gov/20.5.E9 20.5.E9] || [[Item:Q6776|<math>\Jacobithetatau{3}@{\pi z}{\tau} = \sum_{n=-\infty}^{\infty}p^{2n}q^{n^{2}}\\</math>]] || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta3(Pi*z,exp(I*Pi*tau)) = sum((p)^(2*n)* (q)^((n)^(2)), n = - infinity..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[3, Pi*z, Exp[I*Pi*(\[Tau])]] == Sum[(p)^(2*n)* (q)^((n)^(2)), {n, - Infinity, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [140 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], 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: Indeterminate
Test Values: {Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], 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>
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| [https://dlmf.nist.gov/20.5.E9 20.5.E9] || [[Item:Q6776|<math>\sum_{n=-\infty}^{\infty}p^{2n}q^{n^{2}}\\ = \prod_{n=1}^{\infty}\left(1-q^{2n}\right)\left(1+q^{2n-1}p^{2}\right)\left(1+q^{2n-1}p^{-2}\right)</math>]] || <math></math> || <syntaxhighlight lang=mathematica>sum((p)^(2*n)* (q)^((n)^(2)), n = - infinity..infinity) = product((1 - (q)^(2*n))*(1 + (q)^(2*n - 1)* (p)^(2))*(1 + (q)^(2*n - 1)* (p)^(- 2)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(p)^(2*n)* (q)^((n)^(2)), {n, - Infinity, Infinity}, GenerateConditions->None] == Product[(1 - (q)^(2*n))*(1 + (q)^(2*n - 1)* (p)^(2))*(1 + (q)^(2*n - 1)* (p)^(- 2)), {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [20 / 100]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: DirectedInfinity[]
Test Values: {Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[q, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/20.5.E10 20.5.E10] || [[Item:Q6777|<math>\frac{\Jacobithetaq{1}'@{z}{q}}{\Jacobithetaq{1}@{z}{q}}-\cot@@{z} = 4\sin@{2z}\sum_{n=1}^{\infty}\frac{q^{2n}}{1-2q^{2n}\cos@{2z}+q^{4n}}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>(diff( JacobiTheta1(z, q), z$(1) ))/(JacobiTheta1(z, q))- cot(z) = 4*sin(2*z)*sum(((q)^(2*n))/(1 - 2*(q)^(2*n)* cos(2*z)+ (q)^(4*n)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[D[EllipticTheta[1, z, q], {z, 1}],EllipticTheta[1, z, q]]- Cot[z] == 4*Sin[2*z]*Sum[Divide[(q)^(2*n),1 - 2*(q)^(2*n)* Cos[2*z]+ (q)^(4*n)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/20.5.E10 20.5.E10] || [[Item:Q6777|<math>4\sin@{2z}\sum_{n=1}^{\infty}\frac{q^{2n}}{1-2q^{2n}\cos@{2z}+q^{4n}} = 4\sum_{n=1}^{\infty}\frac{q^{2n}}{1-q^{2n}}\sin@{2nz}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>4*sin(2*z)*sum(((q)^(2*n))/(1 - 2*(q)^(2*n)* cos(2*z)+ (q)^(4*n)), n = 1..infinity) = 4*sum(((q)^(2*n))/(1 - (q)^(2*n))*sin(2*n*z), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>4*Sin[2*z]*Sum[Divide[(q)^(2*n),1 - 2*(q)^(2*n)* Cos[2*z]+ (q)^(4*n)], {n, 1, Infinity}, GenerateConditions->None] == 4*Sum[Divide[(q)^(2*n),1 - (q)^(2*n)]*Sin[2*n*z], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/20.5.E11 20.5.E11] || [[Item:Q6778|<math>\frac{\Jacobithetaq{2}'@{z}{q}}{\Jacobithetaq{2}@{z}{q}}+\tan@@{z} = -4\sin@{2z}\sum_{n=1}^{\infty}\frac{q^{2n}}{1+2q^{2n}\cos@{2z}+q^{4n}}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>(diff( JacobiTheta2(z, q), z$(1) ))/(JacobiTheta2(z, q))+ tan(z) = - 4*sin(2*z)*sum(((q)^(2*n))/(1 + 2*(q)^(2*n)* cos(2*z)+ (q)^(4*n)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[D[EllipticTheta[2, z, q], {z, 1}],EllipticTheta[2, z, q]]+ Tan[z] == - 4*Sin[2*z]*Sum[Divide[(q)^(2*n),1 + 2*(q)^(2*n)* Cos[2*z]+ (q)^(4*n)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/20.5.E11 20.5.E11] || [[Item:Q6778|<math>-4\sin@{2z}\sum_{n=1}^{\infty}\frac{q^{2n}}{1+2q^{2n}\cos@{2z}+q^{4n}} = 4\sum_{n=1}^{\infty}(-1)^{n}\frac{q^{2n}}{1-q^{2n}}\sin@{2nz}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>- 4*sin(2*z)*sum(((q)^(2*n))/(1 + 2*(q)^(2*n)* cos(2*z)+ (q)^(4*n)), n = 1..infinity) = 4*sum((- 1)^(n)*((q)^(2*n))/(1 - (q)^(2*n))*sin(2*n*z), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>- 4*Sin[2*z]*Sum[Divide[(q)^(2*n),1 + 2*(q)^(2*n)* Cos[2*z]+ (q)^(4*n)], {n, 1, Infinity}, GenerateConditions->None] == 4*Sum[(- 1)^(n)*Divide[(q)^(2*n),1 - (q)^(2*n)]*Sin[2*n*z], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/20.5.E12 20.5.E12] || [[Item:Q6779|<math>\frac{\Jacobithetaq{3}'@{z}{q}}{\Jacobithetaq{3}@{z}{q}} = -4\sin@{2z}\sum_{n=1}^{\infty}\frac{q^{2n-1}}{1+2q^{2n-1}\cos@{2z}+q^{4n-2}}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>(diff( JacobiTheta3(z, q), z$(1) ))/(JacobiTheta3(z, q)) = - 4*sin(2*z)*sum(((q)^(2*n - 1))/(1 + 2*(q)^(2*n - 1)* cos(2*z)+ (q)^(4*n - 2)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[D[EllipticTheta[3, z, q], {z, 1}],EllipticTheta[3, z, q]] == - 4*Sin[2*z]*Sum[Divide[(q)^(2*n - 1),1 + 2*(q)^(2*n - 1)* Cos[2*z]+ (q)^(4*n - 2)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/20.5.E12 20.5.E12] || [[Item:Q6779|<math>-4\sin@{2z}\sum_{n=1}^{\infty}\frac{q^{2n-1}}{1+2q^{2n-1}\cos@{2z}+q^{4n-2}} = 4\sum_{n=1}^{\infty}(-1)^{n}\frac{q^{n}}{1-q^{2n}}\sin@{2nz}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>- 4*sin(2*z)*sum(((q)^(2*n - 1))/(1 + 2*(q)^(2*n - 1)* cos(2*z)+ (q)^(4*n - 2)), n = 1..infinity) = 4*sum((- 1)^(n)*((q)^(n))/(1 - (q)^(2*n))*sin(2*n*z), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>- 4*Sin[2*z]*Sum[Divide[(q)^(2*n - 1),1 + 2*(q)^(2*n - 1)* Cos[2*z]+ (q)^(4*n - 2)], {n, 1, Infinity}, GenerateConditions->None] == 4*Sum[(- 1)^(n)*Divide[(q)^(n),1 - (q)^(2*n)]*Sin[2*n*z], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/20.5.E13 20.5.E13] || [[Item:Q6780|<math>\frac{\Jacobithetaq{4}'@{z}{q}}{\Jacobithetaq{4}@{z}{q}} = 4\sin@{2z}\sum_{n=1}^{\infty}\frac{q^{2n-1}}{1-2q^{2n-1}\cos@{2z}+q^{4n-2}}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>(diff( JacobiTheta4(z, q), z$(1) ))/(JacobiTheta4(z, q)) = 4*sin(2*z)*sum(((q)^(2*n - 1))/(1 - 2*(q)^(2*n - 1)* cos(2*z)+ (q)^(4*n - 2)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[D[EllipticTheta[4, z, q], {z, 1}],EllipticTheta[4, z, q]] == 4*Sin[2*z]*Sum[Divide[(q)^(2*n - 1),1 - 2*(q)^(2*n - 1)* Cos[2*z]+ (q)^(4*n - 2)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/20.5.E13 20.5.E13] || [[Item:Q6780|<math>4\sin@{2z}\sum_{n=1}^{\infty}\frac{q^{2n-1}}{1-2q^{2n-1}\cos@{2z}+q^{4n-2}} = 4\sum_{n=1}^{\infty}\frac{q^{n}}{1-q^{2n}}\sin@{2nz}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>4*sin(2*z)*sum(((q)^(2*n - 1))/(1 - 2*(q)^(2*n - 1)* cos(2*z)+ (q)^(4*n - 2)), n = 1..infinity) = 4*sum(((q)^(n))/(1 - (q)^(2*n))*sin(2*n*z), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>4*Sin[2*z]*Sum[Divide[(q)^(2*n - 1),1 - 2*(q)^(2*n - 1)* Cos[2*z]+ (q)^(4*n - 2)], {n, 1, Infinity}, GenerateConditions->None] == 4*Sum[Divide[(q)^(n),1 - (q)^(2*n)]*Sin[2*n*z], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/20.5.E15 20.5.E15] || [[Item:Q6782|<math>\Jacobithetatau{2}@{z}{\tau} = \Jacobithetatau{2}@{0}{\tau}\*\lim_{N\to\infty}\prod_{n=-N}^{N}\lim_{M\to\infty}\prod_{m=1-M}^{M}\left(1+\frac{z}{(m-\tfrac{1}{2}+n\tau)\pi}\right)</math>]] || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta2(z,exp(I*Pi*tau)) = JacobiTheta2(0,exp(I*Pi*tau))* limit(product(limit(product(1 +(z)/((m -(1)/(2)+ n*tau)*Pi), m = 1 - M..M), M = infinity), n = - N..N), N = infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[2, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[2, 0, Exp[I*Pi*(\[Tau])]]* Limit[Product[Limit[Product[1 +Divide[z,(m -Divide[1,2]+ n*\[Tau])*Pi], {m, 1 - M, M}, GenerateConditions->None], M -> Infinity, GenerateConditions->None], {n, - N, N}, GenerateConditions->None], N -> Infinity, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/20.5.E16 20.5.E16] || [[Item:Q6783|<math>\Jacobithetatau{3}@{z}{\tau} = \Jacobithetatau{3}@{0}{\tau}\*\lim_{N\to\infty}\prod_{n=1-N}^{N}\lim_{M\to\infty}\prod_{m=1-M}^{M}\left(1+\frac{z}{(m-\tfrac{1}{2}+(n-\tfrac{1}{2})\tau)\pi}\right)</math>]] || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta3(z,exp(I*Pi*tau)) = JacobiTheta3(0,exp(I*Pi*tau))* limit(product(limit(product(1 +(z)/((m -(1)/(2)+(n -(1)/(2))*tau)*Pi), m = 1 - M..M), M = infinity), n = 1 - N..N), N = infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[3, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[3, 0, Exp[I*Pi*(\[Tau])]]* Limit[Product[Limit[Product[1 +Divide[z,(m -Divide[1,2]+(n -Divide[1,2])*\[Tau])*Pi], {m, 1 - M, M}, GenerateConditions->None], M -> Infinity, GenerateConditions->None], {n, 1 - N, N}, GenerateConditions->None], N -> Infinity, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/20.5.E17 20.5.E17] || [[Item:Q6784|<math>\Jacobithetatau{4}@{z}{\tau} = \Jacobithetatau{4}@{0}{\tau}\*\lim_{N\to\infty}\prod_{n=1-N}^{N}\lim_{M\to\infty}\prod_{m=-M}^{M}\left(1+\frac{z}{(m+(n-\tfrac{1}{2})\tau)\pi}\right)</math>]] || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta4(z,exp(I*Pi*tau)) = JacobiTheta4(0,exp(I*Pi*tau))* limit(product(limit(product(1 +(z)/((m +(n -(1)/(2))*tau)*Pi), m = - M..M), M = infinity), n = 1 - N..N), N = infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[4, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[4, 0, Exp[I*Pi*(\[Tau])]]* Limit[Product[Limit[Product[1 +Divide[z,(m +(n -Divide[1,2])*\[Tau])*Pi], {m, - M, M}, GenerateConditions->None], M -> Infinity, GenerateConditions->None], {n, 1 - N, N}, GenerateConditions->None], N -> Infinity, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/20.6.E3 20.6.E3] || [[Item:Q6787|<math>\Jacobithetatau{2}@{\pi z}{\tau} = \Jacobithetatau{2}@{0}{\tau}\exp@{-\sum_{j=1}^{\infty}\frac{1}{2j}\alpha_{2j}(\tau)z^{2j}}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta2(Pi*z,exp(I*Pi*tau)) = JacobiTheta2(0,exp(I*Pi*tau))*exp(- sum((1)/(2*j)*(sum(sum((m -(1)/(2)+ n*tau)^(- 2*j), m = - infinity..infinity), n = - infinity..infinity))*(z)^(2*j), j = 1..infinity))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[2, Pi*z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[2, 0, Exp[I*Pi*(\[Tau])]]*Exp[- Sum[Divide[1,2*j]*(Sum[Sum[(m -Divide[1,2]+ n*\[Tau])^(- 2*j), {m, - Infinity, Infinity}, GenerateConditions->None], {n, - Infinity, Infinity}, GenerateConditions->None])*(z)^(2*j), {j, 1, Infinity}, GenerateConditions->None]]</syntaxhighlight> || Aborted || Aborted || Skipped - Because timed out || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/20.6.E4 20.6.E4] || [[Item:Q6788|<math>\Jacobithetatau{3}@{\pi z}{\tau} = \Jacobithetatau{3}@{0}{\tau}\exp@{-\sum_{j=1}^{\infty}\frac{1}{2j}\beta_{2j}(\tau)z^{2j}}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta3(Pi*z,exp(I*Pi*tau)) = JacobiTheta3(0,exp(I*Pi*tau))*exp(- sum((1)/(2*j)*(sum(sum((m -(1)/(2)+(n -(1)/(2))*tau)^(- 2*j), m = - infinity..infinity), n = - infinity..infinity))*(z)^(2*j), j = 1..infinity))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[3, Pi*z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[3, 0, Exp[I*Pi*(\[Tau])]]*Exp[- Sum[Divide[1,2*j]*(Sum[Sum[(m -Divide[1,2]+(n -Divide[1,2])*\[Tau])^(- 2*j), {m, - Infinity, Infinity}, GenerateConditions->None], {n, - Infinity, Infinity}, GenerateConditions->None])*(z)^(2*j), {j, 1, Infinity}, GenerateConditions->None]]</syntaxhighlight> || Aborted || Aborted || Skipped - Because timed out || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/20.6.E5 20.6.E5] || [[Item:Q6789|<math>\Jacobithetatau{4}@{\pi z}{\tau} = \Jacobithetatau{4}@{0}{\tau}\exp@{-\sum_{j=1}^{\infty}\frac{1}{2j}\gamma_{2j}(\tau)z^{2j}}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>JacobiTheta4(Pi*z,exp(I*Pi*tau)) = JacobiTheta4(0,exp(I*Pi*tau))*exp(- sum((1)/(2*j)*(sum(sum((m +(n -(1)/(2))*tau)^(- 2*j), m = - infinity..infinity), n = - infinity..infinity))*(z)^(2*j), j = 1..infinity))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticTheta[4, Pi*z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[4, 0, Exp[I*Pi*(\[Tau])]]*Exp[- Sum[Divide[1,2*j]*(Sum[Sum[(m +(n -Divide[1,2])*\[Tau])^(- 2*j), {m, - Infinity, Infinity}, GenerateConditions->None], {n, - Infinity, Infinity}, GenerateConditions->None])*(z)^(2*j), {j, 1, Infinity}, GenerateConditions->None]]</syntaxhighlight> || Aborted || Aborted || Skipped - Because timed out || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/20.6#Ex2 20.6#Ex2] || [[Item:Q6795|<math>\beta_{2j}(\tau) = 2^{2j}\gamma_{2j}(2\tau)-\gamma_{2j}(\tau)</math>]] || <math></math> || <syntaxhighlight lang=mathematica>(sum(sum((m -(1)/(2)+(n -(1)/(2))*tau)^(- 2*j), m = - infinity..infinity), n = - infinity..infinity)) = (2)^(2*j)* gamma[2*j](2*tau)-(sum(sum((m +(n -(1)/(2))*tau)^(- 2*j), m = - infinity..infinity), n = - infinity..infinity))</syntaxhighlight> || <syntaxhighlight lang=mathematica>(Sum[Sum[(m -Divide[1,2]+(n -Divide[1,2])*\[Tau])^(- 2*j), {m, - Infinity, Infinity}, GenerateConditions->None], {n, - Infinity, Infinity}, GenerateConditions->None]) == (2)^(2*j)* Subscript[\[Gamma], 2*j][2*\[Tau]]-(Sum[Sum[(m +(n -Divide[1,2])*\[Tau])^(- 2*j), {m, - Infinity, Infinity}, GenerateConditions->None], {n, - Infinity, Infinity}, GenerateConditions->None])</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [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>]] || <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]
|-
| [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>]] || <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]
|-
| [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>]] || <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]
|-
| [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>]] || <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]
|-
| [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>]] || <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]
|-
| [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>]] || <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]
|-
| [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>]] || <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]
|-
| [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>]] || <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]
|-
| [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>]] || <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]
|-
| [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>]] || <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]
|-
| [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>]] || <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>
|-
| [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>]] || <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]
|-
| [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>]] || <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]
|-
| [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>]] || <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>
|-
| [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>]] || <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]
|-
| [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>]] || <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>]] || <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>]] || <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>]] || <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>]] || <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>]] || <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>]] || <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>]] || <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>]] || <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>]] || <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>]] || <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>]] || <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>]] || <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>]] || <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>]] || <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>]] || <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>
|-
| [https://dlmf.nist.gov/20.8.E1 20.8.E1] || [[Item:Q6830|<math>\frac{\Jacobithetaq{2}@{0}{q}\Jacobithetaq{3}@{z}{q}\Jacobithetaq{4}@{z}{q}}{\Jacobithetaq{2}@{z}{q}} = 2\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}q^{n^{2}}e^{i2nz}}{q^{-n}e^{-iz}+q^{n}e^{iz}}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>(JacobiTheta2(0, q)*JacobiTheta3(z, q)*JacobiTheta4(z, q))/(JacobiTheta2(z, q)) = 2*sum(((- 1)^(n)* (q)^((n)^(2))* exp(I*2*n*z))/((q)^(- n)* exp(- I*z)+ (q)^(n)* exp(I*z)), n = - infinity..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[EllipticTheta[2, 0, q]*EllipticTheta[3, z, q]*EllipticTheta[4, z, q],EllipticTheta[2, z, q]] == 2*Sum[Divide[(- 1)^(n)* (q)^((n)^(2))* Exp[I*2*n*z],(q)^(- n)* Exp[- I*z]+ (q)^(n)* Exp[I*z]], {n, - Infinity, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/20.9#Ex1 20.9#Ex1] || [[Item:Q6832|<math>\compellintKk@{k} = \tfrac{1}{2}\pi\Jacobithetatau{3}^{2}@{0}{\tau}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>EllipticK((JacobiTheta2(0,exp(I*Pi*tau)))^(2)/(JacobiTheta3(0,exp(I*Pi*tau)))^(2)) = (1)/(2)*Pi*(JacobiTheta3(0,exp(I*Pi*tau)))^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticK[((EllipticTheta[2, 0, Exp[I*Pi*(\[Tau])]])^(2)/(EllipticTheta[3, 0, Exp[I*Pi*(\[Tau])]])^(2))^2] == Divide[1,2]*Pi*(EllipticTheta[3, 0, Exp[I*Pi*(\[Tau])]])^(2)</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 10]
|-
| [https://dlmf.nist.gov/20.9#Ex2 20.9#Ex2] || [[Item:Q6833|<math>\compellintKk'@{k} = -i\tau\compellintKk@{k}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>subs( temp=((JacobiTheta2(0,exp(I*Pi*tau)))^(2)/(JacobiTheta3(0,exp(I*Pi*tau)))^(2)), diff( EllipticK(temp), temp$(1) ) ) = - I*tau*EllipticK((JacobiTheta2(0,exp(I*Pi*tau)))^(2)/(JacobiTheta3(0,exp(I*Pi*tau)))^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>(D[EllipticK[(temp)^2], {temp, 1}]/.temp-> ((EllipticTheta[2, 0, Exp[I*Pi*(\[Tau])]])^(2)/(EllipticTheta[3, 0, Exp[I*Pi*(\[Tau])]])^(2))) == - I*\[Tau]*EllipticK[((EllipticTheta[2, 0, Exp[I*Pi*(\[Tau])]])^(2)/(EllipticTheta[3, 0, Exp[I*Pi*(\[Tau])]])^(2))^2]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.6481210221+.3604335389*I
Test Values: {tau = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.647990213-1.212940701*I
Test Values: {tau = -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 [2 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[2.220446049250313*^-16, 0.6473902356608235]
Test Values: {Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-2.220446049250313*^-16, -0.8272591738499964]
Test Values: {Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br></div></div>
|-
| [https://dlmf.nist.gov/20.9.E3 20.9.E3] || [[Item:Q6834|<math>\CarlsonsymellintRF@{\frac{\Jacobithetaq{2}^{2}@{z}{q}}{\Jacobithetaq{2}^{2}@{0}{q}}}{\frac{\Jacobithetaq{3}^{2}@{z}{q}}{\Jacobithetaq{3}^{2}@{0}{q}}}{\frac{\Jacobithetaq{4}^{2}@{z}{q}}{\Jacobithetaq{4}^{2}@{0}{q}}} = \frac{\Jacobithetaq{1}'@{0}{q}}{\Jacobithetaq{1}@{z}{q}}z</math>]] || <math></math> || <syntaxhighlight lang=mathematica>0.5*int(1/(sqrt(t+((JacobiTheta2(z, q))^(2))/((JacobiTheta2(0, q))^(2)))*sqrt(t+((JacobiTheta3(z, q))^(2))/((JacobiTheta3(0, q))^(2)))*sqrt(t+((JacobiTheta4(z, q))^(2))/((JacobiTheta4(0, q))^(2)))), t = 0..infinity) = (diff( JacobiTheta1(0, q), 0$(1) ))/(JacobiTheta1(z, q))*z</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticF[ArcCos[Sqrt[Divide[(EllipticTheta[2, z, q])^(2),(EllipticTheta[2, 0, q])^(2)]/Divide[(EllipticTheta[4, z, q])^(2),(EllipticTheta[4, 0, q])^(2)]]],(Divide[(EllipticTheta[4, z, q])^(2),(EllipticTheta[4, 0, q])^(2)]-Divide[(EllipticTheta[3, z, q])^(2),(EllipticTheta[3, 0, q])^(2)])/(Divide[(EllipticTheta[4, z, q])^(2),(EllipticTheta[4, 0, q])^(2)]-Divide[(EllipticTheta[2, z, q])^(2),(EllipticTheta[2, 0, q])^(2)])]/Sqrt[Divide[(EllipticTheta[4, z, q])^(2),(EllipticTheta[4, 0, q])^(2)]-Divide[(EllipticTheta[2, z, q])^(2),(EllipticTheta[2, 0, q])^(2)]] == Divide[D[EllipticTheta[1, 0, q], {0, 1}],EllipticTheta[1, z, q]]*z</syntaxhighlight> || Aborted || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Times[Complex[-0.8660254037844387, -0.49999999999999994], D[0.0
Test Values: {0.0, 1.0}], Power[EllipticTheta[1, Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]], -1]], Times[EllipticF[ArcCos[Power[Times[Power[EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], -2], Power[EllipticTheta[2, Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]], 2], Power[EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], 2], Power[EllipticTheta[4, Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]], -2]], Rational[1, 2]]], Times[Power[Plus[Times[-1.0, Power[EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], -2], Power[EllipticTheta[2, Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]], 2]], Times[Power[E</div></div>
|-
| [https://dlmf.nist.gov/20.9.E4 20.9.E4] || [[Item:Q6835|<math>\CarlsonsymellintRF@{0}{\Jacobithetaq{3}^{4}@{0}{q}}{\Jacobithetaq{4}^{4}@{0}{q}} = \tfrac{1}{2}\pi</math>]] || <math></math> || <syntaxhighlight lang=mathematica>0.5*int(1/(sqrt(t+0)*sqrt(t+(JacobiTheta3(0, q))^(4))*sqrt(t+(JacobiTheta4(0, q))^(4))), t = 0..infinity) = (1)/(2)*Pi</syntaxhighlight> || <syntaxhighlight lang=mathematica>EllipticF[ArcCos[Sqrt[0/(EllipticTheta[4, 0, q])^(4)]],((EllipticTheta[4, 0, q])^(4)-(EllipticTheta[3, 0, q])^(4))/((EllipticTheta[4, 0, q])^(4)-0)]/Sqrt[(EllipticTheta[4, 0, q])^(4)-0] == Divide[1,2]*Pi</syntaxhighlight> || Aborted || Failure || Skipped - Because timed out || Successful [Tested: 10]
|-
| [https://dlmf.nist.gov/20.10.E1 20.10.E1] || [[Item:Q6837|<math>\int_{0}^{\infty}x^{s-1}\Jacobithetatau{2}@{0}{ix^{2}}\diff{x} = 2^{s}(1-2^{-s})\pi^{-s/2}\EulerGamma@{\tfrac{1}{2}s}\Riemannzeta@{s}</math>]] || <math>\realpart@@{\tfrac{1}{2}s} > 0</math> || <syntaxhighlight lang=mathematica>int((x)^(s - 1)* JacobiTheta2(0,exp(I*Pi*I*(x)^(2))), x = 0..infinity) = (2)^(s)*(1 - (2)^(- s))*(Pi)^(- s/2)* GAMMA((1)/(2)*s)*Zeta(s)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(x)^(s - 1)* EllipticTheta[2, 0, Exp[I*Pi*(I*(x)^(2))]], {x, 0, Infinity}, GenerateConditions->None] == (2)^(s)*(1 - (2)^(- s))*(Pi)^(- s/2)* Gamma[Divide[1,2]*s]*Zeta[s]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[1.6473133477045354, NIntegrate[Times[Power[x, -0.5], EllipticTheta[2, 0, Power[E, Times[-1, Pi, Power[x, 2]]]]]
Test Values: {x, 0, DirectedInfinity[1]}]], {Rule[s, 0.5]}</syntaxhighlight><br></div></div>
|-
| [https://dlmf.nist.gov/20.10.E2 20.10.E2] || [[Item:Q6838|<math>\int_{0}^{\infty}x^{s-1}(\Jacobithetatau{3}@{0}{ix^{2}}-1)\diff{x} = \pi^{-s/2}\EulerGamma@{\tfrac{1}{2}s}\Riemannzeta@{s}</math>]] || <math>\realpart@@{\tfrac{1}{2}s} > 0</math> || <syntaxhighlight lang=mathematica>int((x)^(s - 1)*(JacobiTheta3(0,exp(I*Pi*I*(x)^(2)))- 1), x = 0..infinity) = (Pi)^(- s/2)* GAMMA((1)/(2)*s)*Zeta(s)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(x)^(s - 1)*(EllipticTheta[3, 0, Exp[I*Pi*(I*(x)^(2))]]- 1), {x, 0, Infinity}, GenerateConditions->None] == (Pi)^(- s/2)* Gamma[Divide[1,2]*s]*Zeta[s]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/20.10.E3 20.10.E3] || [[Item:Q6839|<math>\int_{0}^{\infty}x^{s-1}(1-\Jacobithetatau{4}@{0}{ix^{2}})\diff{x} = (1-2^{1-s})\pi^{-s/2}\EulerGamma@{\tfrac{1}{2}s}\Riemannzeta@{s}</math>]] || <math>\realpart@@{\tfrac{1}{2}s} > 0</math> || <syntaxhighlight lang=mathematica>int((x)^(s - 1)*(1 - JacobiTheta4(0,exp(I*Pi*I*(x)^(2)))), x = 0..infinity) = (1 - (2)^(1 - s))*(Pi)^(- s/2)* GAMMA((1)/(2)*s)*Zeta(s)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(x)^(s - 1)*(1 - EllipticTheta[4, 0, Exp[I*Pi*(I*(x)^(2))]]), {x, 0, Infinity}, GenerateConditions->None] == (1 - (2)^(1 - s))*(Pi)^(- s/2)* Gamma[Divide[1,2]*s]*Zeta[s]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/20.10.E4 20.10.E4] || [[Item:Q6840|<math>\int_{0}^{\infty}e^{-st}\Jacobithetatau{1}@{\frac{\beta\pi}{2\ell}}{\frac{i\pi t}{\ell^{2}}}\diff{t} = \int_{0}^{\infty}e^{-st}\Jacobithetatau{2}@{\frac{(1+\beta)\pi}{2\ell}}{\frac{i\pi t}{\ell^{2}}}\diff{t}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>int(exp(- s*t)*JacobiTheta1((beta*Pi)/(2*ell),exp(I*Pi*(I*Pi*t)/((ell)^(2)))), t = 0..infinity) = int(exp(- s*t)*JacobiTheta2(((1 + beta)*Pi)/(2*ell),exp(I*Pi*(I*Pi*t)/((ell)^(2)))), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[- s*t]*EllipticTheta[1, Divide[\[Beta]*Pi,2*\[ScriptL]], Exp[I*Pi*(Divide[I*Pi*t,\[ScriptL]^(2)])]], {t, 0, Infinity}, GenerateConditions->None] == Integrate[Exp[- s*t]*EllipticTheta[2, Divide[(1 + \[Beta])*Pi,2*\[ScriptL]], Exp[I*Pi*(Divide[I*Pi*t,\[ScriptL]^(2)])]], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 9]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Times[-1.0, stIntegrate[Times[2.718281828459045, EllipticTheta[1, 2.356194490192345, Power[2.718281828459045, Times[-9.869604401089358, t]]]]
Test Values: {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], stIntegrate[Times[2.718281828459045, EllipticTheta[2, 3.9269908169872414, Power[2.718281828459045, Times[-9.869604401089358, t]]]], {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[ℓ, 1], Rule[β, Rational[3, 2]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Times[-1.0, stIntegrate[Times[2.718281828459045, EllipticTheta[1, 1.1780972450961724, Power[2.718281828459045, Times[-2.4674011002723395, t]]]]
Test Values: {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], stIntegrate[Times[2.718281828459045, EllipticTheta[2, 1.9634954084936207, Power[2.718281828459045, Times[-2.4674011002723395, t]]]], {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[ℓ, 2</div></div>
|-
| [https://dlmf.nist.gov/20.10.E4 20.10.E4] || [[Item:Q6840|<math>\int_{0}^{\infty}e^{-st}\Jacobithetatau{2}@{\frac{(1+\beta)\pi}{2\ell}}{\frac{i\pi t}{\ell^{2}}}\diff{t} = -\frac{\ell}{\sqrt{s}}\sinh@{\beta\sqrt{s}}\sech@{\ell\sqrt{s}}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>int(exp(- s*t)*JacobiTheta2(((1 + beta)*Pi)/(2*ell),exp(I*Pi*(I*Pi*t)/((ell)^(2)))), t = 0..infinity) = -(ell)/(sqrt(s))*sinh(beta*sqrt(s))*sech(ell*sqrt(s))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[- s*t]*EllipticTheta[2, Divide[(1 + \[Beta])*Pi,2*\[ScriptL]], Exp[I*Pi*(Divide[I*Pi*t,\[ScriptL]^(2)])]], {t, 0, Infinity}, GenerateConditions->None] == -Divide[\[ScriptL],Sqrt[s]]*Sinh[\[Beta]*Sqrt[s]]*Sech[\[ScriptL]*Sqrt[s]]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [54 / 54]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[2.32235875408619, Times[-1.0, stIntegrate[Times[2.718281828459045, EllipticTheta[2, 3.9269908169872414, Power[2.718281828459045, Times[-9.869604401089358, t]]]]
Test Values: {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[s, Rational[-3, 2]], Rule[ℓ, 1], Rule[β, Rational[3, 2]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-2.046254548704581, Times[-1.0, stIntegrate[Times[2.718281828459045, EllipticTheta[2, 1.9634954084936207, Power[2.718281828459045, Times[-2.4674011002723395, t]]]]
Test Values: {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[s, Rational[-3, 2]], Rule[ℓ, 2], Rule[β, Rational[3, 2]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/20.10.E5 20.10.E5] || [[Item:Q6841|<math>\int_{0}^{\infty}e^{-st}\Jacobithetatau{3}@{\frac{(1+\beta)\pi}{2\ell}}{\frac{i\pi t}{\ell^{2}}}\diff{t} = \int_{0}^{\infty}e^{-st}\Jacobithetatau{4}@{\frac{\beta\pi}{2\ell}}{\frac{i\pi t}{\ell^{2}}}\diff{t}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>int(exp(- s*t)*JacobiTheta3(((1 + beta)*Pi)/(2*ell),exp(I*Pi*(I*Pi*t)/((ell)^(2)))), t = 0..infinity) = int(exp(- s*t)*JacobiTheta4((beta*Pi)/(2*ell),exp(I*Pi*(I*Pi*t)/((ell)^(2)))), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[- s*t]*EllipticTheta[3, Divide[(1 + \[Beta])*Pi,2*\[ScriptL]], Exp[I*Pi*(Divide[I*Pi*t,\[ScriptL]^(2)])]], {t, 0, Infinity}, GenerateConditions->None] == Integrate[Exp[- s*t]*EllipticTheta[4, Divide[\[Beta]*Pi,2*\[ScriptL]], Exp[I*Pi*(Divide[I*Pi*t,\[ScriptL]^(2)])]], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Times[-1.0, stIntegrate[Times[2.718281828459045, EllipticTheta[3, Times[3.9269908169872414, Power[ℓ, -1]], Power[2.718281828459045, Times[-9.869604401089358, t, Power[ℓ, -2]]]]]
Test Values: {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], stIntegrate[Times[2.718281828459045, EllipticTheta[4, Times[2.356194490192345, Power[ℓ, -1]], Power[2.718281828459045, Times[-9.869604401089358, t, Power[ℓ, -2]]]]], {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[β, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Times[-1.0, stIntegrate[Times[2.718281828459045, EllipticTheta[3, Times[2.356194490192345, Power[ℓ, -1]], Power[2.718281828459045, Times[-9.869604401089358, t, Power[ℓ, -2]]]]]
Test Values: {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], stIntegrate[Times[2.718281828459045, EllipticTheta[4, Times[0.7853981633974483, Power[ℓ, -1]], Power[2.718281828459045, Times[-9</div></div>
|-
| [https://dlmf.nist.gov/20.10.E5 20.10.E5] || [[Item:Q6841|<math>\int_{0}^{\infty}e^{-st}\Jacobithetatau{4}@{\frac{\beta\pi}{2\ell}}{\frac{i\pi t}{\ell^{2}}}\diff{t} = \frac{\ell}{\sqrt{s}}\cosh@{\beta\sqrt{s}}\csch@{\ell\sqrt{s}}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>int(exp(- s*t)*JacobiTheta4((beta*Pi)/(2*ell),exp(I*Pi*(I*Pi*t)/((ell)^(2)))), t = 0..infinity) = (ell)/(sqrt(s))*cosh(beta*sqrt(s))*csch(ell*sqrt(s))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[- s*t]*EllipticTheta[4, Divide[\[Beta]*Pi,2*\[ScriptL]], Exp[I*Pi*(Divide[I*Pi*t,\[ScriptL]^(2)])]], {t, 0, Infinity}, GenerateConditions->None] == Divide[\[ScriptL],Sqrt[s]]*Cosh[\[Beta]*Sqrt[s]]*Csch[\[ScriptL]*Sqrt[s]]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 18]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Times[Complex[-0.21488876057872602, 0.0], ℓ, Csc[Times[Complex[1.224744871391589, 0.0], ℓ]]], Times[-1.0, stIntegrate[Times[2.718281828459045, EllipticTheta[4, Times[2.356194490192345, Power[ℓ, -1]], Power[2.718281828459045, Times[-9.869604401089358, t, Power[ℓ, -2]]]]]
Test Values: {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[s, -1.5], Rule[β, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Times[Complex[0.668128228457918, 0.0], ℓ, Csc[Times[Complex[1.224744871391589, 0.0], ℓ]]], Times[-1.0, stIntegrate[Times[2.718281828459045, EllipticTheta[4, Times[0.7853981633974483, Power[ℓ, -1]], Power[2.718281828459045, Times[-9.869604401089358, t, Power[ℓ, -2]]]]]
Test Values: {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[s, -1.5], Rule[β, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/20.11.E5 20.11.E5] || [[Item:Q6846|<math>\genhyperF{2}{1}@{\tfrac{1}{2},\tfrac{1}{2}}{1}{k^{2}} = \Jacobithetatau{3}^{2}@{0}{\tau}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>hypergeom([(1)/(2),(1)/(2)], [1], ((JacobiTheta2(0,exp(I*Pi*tau)))^(2)/(JacobiTheta3(0,exp(I*Pi*tau)))^(2))^(2)) = (JacobiTheta3(0,exp(I*Pi*tau)))^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>HypergeometricPFQ[{Divide[1,2],Divide[1,2]}, {1}, ((EllipticTheta[2, 0, Exp[I*Pi*(\[Tau])]])^(2)/(EllipticTheta[3, 0, Exp[I*Pi*(\[Tau])]])^(2))^(2)] == (EllipticTheta[3, 0, Exp[I*Pi*(\[Tau])]])^(2)</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 10]
|-
| [https://dlmf.nist.gov/20.13.E2 20.13.E2] || [[Item:Q6852|<math>\ipderiv{\theta}{t} = \alpha\ipderiv[2]{\theta}{z}</math>]] || <math></math> || <syntaxhighlight lang=mathematica>diff(theta, t) = alpha*diff(theta, [z$(2)])</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[\[Theta], t] == \[Alpha]*D[\[Theta], {z, 2}]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 300]
|-
|}

Latest revision as of 17:35, 25 May 2021

Notation
20.1 Special Notation
Properties
20.2 Definitions and Periodic Properties
20.3 Graphics
20.4 Values at Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle z} = 0
20.5 Infinite Products and Related Results
20.6 Power Series
20.7 Identities
20.8 Watson’s Expansions
20.9 Relations to Other Functions
20.10 Integrals
20.11 Generalizations and Analogs
Applications
20.12 Mathematical Applications
20.13 Physical Applications
Computation
20.14 Methods of Computation
20.15 Tables
20.16 Software
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