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| |- | | ; Notation : [[20.1|20.1 Special Notation]]<br> |
| ! DLMF !! Formula !! 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>]] || <code>JacobiTheta1(z,exp(I*Pi*tau)) = JacobiTheta1(z, q)</code> || <code>EllipticTheta[1, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[1, z, q]</code> || 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> |
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| | </div> |
| | [https://dlmf.nist.gov/20.2.E2 20.2.E2] || [[Item:Q6742|<math>\Jacobithetatau{2}@{z}{\tau} = \Jacobithetaq{2}@{z}{q}</math>]] || <code>JacobiTheta2(z,exp(I*Pi*tau)) = JacobiTheta2(z, q)</code> || <code>EllipticTheta[2, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[2, z, q]</code> || Failure || Failure || Error || Successful [Tested: 300]
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| | [https://dlmf.nist.gov/20.2.E3 20.2.E3] || [[Item:Q6743|<math>\Jacobithetatau{3}@{z}{\tau} = \Jacobithetaq{3}@{z}{q}</math>]] || <code>JacobiTheta3(z,exp(I*Pi*tau)) = JacobiTheta3(z, q)</code> || <code>EllipticTheta[3, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[3, z, q]</code> || Failure || Failure || Error || Successful [Tested: 300]
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| | [https://dlmf.nist.gov/20.2.E4 20.2.E4] || [[Item:Q6744|<math>\Jacobithetatau{4}@{z}{\tau} = \Jacobithetaq{4}@{z}{q}</math>]] || <code>JacobiTheta4(z,exp(I*Pi*tau)) = JacobiTheta4(z, q)</code> || <code>EllipticTheta[4, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[4, z, q]</code> || Failure || Failure || Error || Successful [Tested: 300]
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| | [https://dlmf.nist.gov/20.2.E5 20.2.E5] || [[Item:Q6745|<math>z_{m,n} = (m+n\tau)\pi</math>]] || <code>z[m , n] = (m + n*tau)* Pi</code> || <code>Subscript[z, m , n] == (m + n*\[Tau])* Pi</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [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>]] || <code>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))</code> || <code>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])]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[-18.62843952+6.320473139*I <- {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}</code><br><code>-4187.991134+3174.249087*I <- {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}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [72 / 300]<div class="mw-collapsible-content"><code>{Complex[-18.628439525286133, 6.320473094431787] <- {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]]]}</code><br><code>Complex[-4187.991166649552, 3174.249038247393] <- {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]]]}</code><br></div></div>
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| | [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>]] || <code>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))</code> || <code>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])]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[3.950576529-14.16574159*I <- {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}</code><br><code>194.3416227+3923.809342*I <- {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}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [72 / 300]<div class="mw-collapsible-content"><code>{Complex[3.9505765593957305, -14.165741580817551] <- {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]]]}</code><br><code>Complex[194.34158297403354, 3923.809350793304] <- {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]]]}</code><br></div></div>
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| | [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>]] || <code>JacobiTheta3(z +(m + n*tau)* Pi,exp(I*Pi*tau)) = (q)^(- (n)^(2))* exp(- 2*I*n*z)*JacobiTheta3(z,exp(I*Pi*tau))</code> || <code>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])]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[-8.181021151+18.44680448*I <- {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}</code><br><code>516.5479372-5365.925849*I <- {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}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [72 / 300]<div class="mw-collapsible-content"><code>{Complex[-8.181021187984683, 18.446804447343553] <- {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]]]}</code><br><code>Complex[516.547995329447, -5365.925840115722] <- {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]]]}</code><br></div></div>
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| | [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>]] || <code>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))</code> || <code>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])]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[-4.510694228-11.16801166*I <- {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}</code><br><code>-2085.869632-2449.864344*I <- {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}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [72 / 300]<div class="mw-collapsible-content"><code>{Complex[-4.5106942149502025, -11.168011665083736] <- {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]]]}</code><br><code>Complex[-2085.8696157878926, -2449.864367431773] <- {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]]]}</code><br></div></div>
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| | [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>]] || <code>JacobiTheta1(z,exp(I*Pi*tau)) = - JacobiTheta2(z +(1)/(2)*Pi,exp(I*Pi*tau))</code> || <code>EllipticTheta[1, z, Exp[I*Pi*(\[Tau])]] == - EllipticTheta[2, z +Divide[1,2]*Pi, Exp[I*Pi*(\[Tau])]]</code> || Successful || Failure || Skip - symbolical successful subtest || Successful [Tested: 70]
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| | [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>]] || <code>- JacobiTheta2(z +(1)/(2)*Pi,exp(I*Pi*tau)) = - I*M*JacobiTheta4(z +(1)/(2)*Pi*tau,exp(I*Pi*tau))</code> || <code>- 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])]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[-2.656130280+.8441101403*I <- {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}</code><br><code>2.726401771+.6812031274*I <- {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)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [60 / 300]<div class="mw-collapsible-content"><code>{Complex[-2.65613027348202, 0.8441101301235214] <- {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]]]}</code><br><code>Complex[-2.1127027753782777, -0.09362434622808774] <- {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]]]}</code><br></div></div>
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| | [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>]] || <code>- 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))</code> || <code>- 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])]]</code> || Successful || Failure || Skip - symbolical successful subtest || Successful [Tested: 300]
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| | [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>]] || <code>JacobiTheta2(z,exp(I*Pi*tau)) = JacobiTheta1(z +(1)/(2)*Pi,exp(I*Pi*tau))</code> || <code>EllipticTheta[2, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[1, z +Divide[1,2]*Pi, Exp[I*Pi*(\[Tau])]]</code> || Successful || Failure || Skip - symbolical successful subtest || Successful [Tested: 70]
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| | [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>]] || <code>JacobiTheta1(z +(1)/(2)*Pi,exp(I*Pi*tau)) = M*JacobiTheta3(z +(1)/(2)*Pi*tau,exp(I*Pi*tau))</code> || <code>EllipticTheta[1, z +Divide[1,2]*Pi, Exp[I*Pi*(\[Tau])]] == M*EllipticTheta[3, z +Divide[1,2]*Pi*\[Tau], Exp[I*Pi*(\[Tau])]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[-.5985410657+1.995750316*I <- {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}</code><br><code>-.54909285e-2-5.605651596*I <- {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)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [60 / 300]<div class="mw-collapsible-content"><code>{Complex[-0.5985410729973577, 1.9957503125524838] <- {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]]]}</code><br><code>Complex[-1.3220776692839347, 1.382964384147599] <- {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]]]}</code><br></div></div>
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| | [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>]] || <code>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))</code> || <code>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])]]</code> || Successful || Failure || Skip - symbolical successful subtest || Successful [Tested: 300]
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| | [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>]] || <code>JacobiTheta3(z,exp(I*Pi*tau)) = JacobiTheta4(z +(1)/(2)*Pi,exp(I*Pi*tau))</code> || <code>EllipticTheta[3, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[4, z +Divide[1,2]*Pi, Exp[I*Pi*(\[Tau])]]</code> || Successful || Failure || Skip - symbolical successful subtest || Successful [Tested: 70]
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| | [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>]] || <code>JacobiTheta4(z +(1)/(2)*Pi,exp(I*Pi*tau)) = M*JacobiTheta2(z +(1)/(2)*Pi*tau,exp(I*Pi*tau))</code> || <code>EllipticTheta[4, z +Divide[1,2]*Pi, Exp[I*Pi*(\[Tau])]] == M*EllipticTheta[2, z +Divide[1,2]*Pi*\[Tau], Exp[I*Pi*(\[Tau])]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[-1.209558888+2.590545189*I <- {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}</code><br><code>-.8537358258+1.281173247*I <- {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)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [60 / 300]<div class="mw-collapsible-content"><code>{Complex[-1.2095588901959111, 2.5905451776573183] <- {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]]]}</code><br><code>Complex[-1.901447370098885, -0.21207958455265288] <- {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]]]}</code><br></div></div>
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| | [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>]] || <code>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))</code> || <code>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])]]</code> || Successful || Failure || Skip - symbolical successful subtest || Successful [Tested: 300]
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| | [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>]] || <code>JacobiTheta4(z,exp(I*Pi*tau)) = JacobiTheta3(z +(1)/(2)*Pi,exp(I*Pi*tau))</code> || <code>EllipticTheta[4, z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[3, z +Divide[1,2]*Pi, Exp[I*Pi*(\[Tau])]]</code> || Successful || Failure || Skip - symbolical successful subtest || Successful [Tested: 70]
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| | [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>]] || <code>JacobiTheta3(z +(1)/(2)*Pi,exp(I*Pi*tau)) = - I*M*JacobiTheta1(z +(1)/(2)*Pi*tau,exp(I*Pi*tau))</code> || <code>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])]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[.6082553523+1.594370406*I <- {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}</code><br><code>-3.995156823-3.872683361*I <- {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)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [60 / 300]<div class="mw-collapsible-content"><code>{Complex[0.6082553477594059, 1.594370409676146] <- {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]]]}</code><br><code>Complex[-2.5995966648178563, -0.1152010311326023] <- {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]]]}</code><br></div></div>
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| | [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>]] || <code>- 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))</code> || <code>- 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])]]</code> || Successful || Failure || Skip - symbolical successful subtest || Successful [Tested: 300]
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| | [https://dlmf.nist.gov/20.4.E1 20.4.E1] || [[Item:Q6755|<math>\Jacobithetaq{1}@{0}{q} = \Jacobithetaq{2}'@{0}{q}</math>]] || <code>JacobiTheta1(0, q) = diff( JacobiTheta2(0, q), 0$(1) )</code> || <code>EllipticTheta[1, 0, q] == D[EllipticTheta[2, 0, q], {0, 1}]</code> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><code>{Times[-1.0, D[EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]] <- {0.0, 1.0}]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Times[-1.0, D[EllipticTheta[2, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]] <- {0.0, 1.0}]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/20.4.E1 20.4.E1] || [[Item:Q6755|<math>\Jacobithetaq{2}'@{0}{q} = \Jacobithetaq{3}'@{0}{q}</math>]] || <code>diff( JacobiTheta2(0, q), 0$(1) ) = diff( JacobiTheta3(0, q), 0$(1) )</code> || <code>D[EllipticTheta[2, 0, q], {0, 1}] == D[EllipticTheta[3, 0, q], {0, 1}]</code> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><code>{Plus[D[EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]] <- {0.0, 1.0}], Times[-1.0, D[EllipticTheta[3, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], {0.0, 1.0}]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Plus[D[EllipticTheta[2, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]] <- {0.0, 1.0}], Times[-1.0, D[EllipticTheta[3, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]], {0.0, 1.0}]]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/20.4.E1 20.4.E1] || [[Item:Q6755|<math>\Jacobithetaq{3}'@{0}{q} = \Jacobithetaq{4}'@{0}{q}</math>]] || <code>diff( JacobiTheta3(0, q), 0$(1) ) = diff( JacobiTheta4(0, q), 0$(1) )</code> || <code>D[EllipticTheta[3, 0, q], {0, 1}] == D[EllipticTheta[4, 0, q], {0, 1}]</code> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><code>{Plus[D[EllipticTheta[3, 0.0, Complex[0.8660254037844387, 0.49999999999999994]] <- {0.0, 1.0}], Times[-1.0, D[EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], {0.0, 1.0}]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Plus[D[EllipticTheta[3, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]] <- {0.0, 1.0}], Times[-1.0, D[EllipticTheta[4, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]], {0.0, 1.0}]]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/20.4.E1 20.4.E1] || [[Item:Q6755|<math>\Jacobithetaq{4}'@{0}{q} = 0</math>]] || <code>diff( JacobiTheta4(0, q), 0$(1) ) = 0</code> || <code>D[EllipticTheta[4, 0, q], {0, 1}] == 0</code> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><code>{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]]]}</code><br><code>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]]]}</code><br></div></div>
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| | [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>]] || <code>diff( JacobiTheta1(0, q), 0$(1) ) = JacobiTheta2(0, q)*JacobiTheta3(0, q)*JacobiTheta4(0, q)</code> || <code>D[EllipticTheta[1, 0, q], {0, 1}] == EllipticTheta[2, 0, q]*EllipticTheta[3, 0, q]*EllipticTheta[4, 0, q]</code> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><code>{Plus[D[0.0 <- {0.0, 1.0}], Times[-1.0, EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], EllipticTheta[3, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Plus[D[0.0 <- {0.0, 1.0}], Times[-1.0, EllipticTheta[2, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]], EllipticTheta[3, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]], EllipticTheta[4, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]]]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/20.4.E7 20.4.E7] || [[Item:Q6762|<math>\Jacobithetaq{1}''(0,q) = \Jacobithetaq{2}'''@{0}{q}</math>]] || <code>subs( temp=(0 , q) , diff( JacobiTheta1(temp, =), temp$(2) ) )*diff( JacobiTheta2(0, q), 0$(3) )</code> || <code>(D[EllipticTheta[1, temp, ==], {temp, 2}]/.temp-> (0 , q) )*D[EllipticTheta[2, 0, q], {0, 3}]</code> || Translation Error || Translation Error || - || Skip - symbolical successful subtest
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| | [https://dlmf.nist.gov/20.4.E7 20.4.E7] || [[Item:Q6762|<math>\Jacobithetaq{2}'''@{0}{q} = \Jacobithetaq{3}'''@{0}{q}</math>]] || <code>diff( JacobiTheta2(0, q), 0$(3) ) = diff( JacobiTheta3(0, q), 0$(3) )</code> || <code>D[EllipticTheta[2, 0, q], {0, 3}] == D[EllipticTheta[3, 0, q], {0, 3}]</code> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><code>{Plus[D[EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]] <- {0.0, 3.0}], Times[-1.0, D[EllipticTheta[3, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], {0.0, 3.0}]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Plus[D[EllipticTheta[2, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]] <- {0.0, 3.0}], Times[-1.0, D[EllipticTheta[3, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]], {0.0, 3.0}]]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/20.4.E7 20.4.E7] || [[Item:Q6762|<math>\Jacobithetaq{3}'''@{0}{q} = \Jacobithetaq{4}'''@{0}{q}</math>]] || <code>diff( JacobiTheta3(0, q), 0$(3) ) = diff( JacobiTheta4(0, q), 0$(3) )</code> || <code>D[EllipticTheta[3, 0, q], {0, 3}] == D[EllipticTheta[4, 0, q], {0, 3}]</code> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><code>{Plus[D[EllipticTheta[3, 0.0, Complex[0.8660254037844387, 0.49999999999999994]] <- {0.0, 3.0}], Times[-1.0, D[EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], {0.0, 3.0}]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Plus[D[EllipticTheta[3, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]] <- {0.0, 3.0}], Times[-1.0, D[EllipticTheta[4, 0.0, Complex[-0.4999999999999998, 0.8660254037844387]], {0.0, 3.0}]]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/20.4.E7 20.4.E7] || [[Item:Q6762|<math>\Jacobithetaq{4}'''@{0}{q} = 0</math>]] || <code>diff( JacobiTheta4(0, q), 0$(3) ) = 0</code> || <code>D[EllipticTheta[4, 0, q], {0, 3}] == 0</code> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><code>{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]]]}</code><br><code>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]]]}</code><br></div></div>
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| | [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>]] || <code>(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)</code> || <code>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]</code> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><code>{Plus[1.0, Times[Power[D[0.0 <- {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]]]}</code><br><code>Plus[1.0, Times[Power[D[0.0 <- {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]]]}</code><br></div></div>
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| | [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>]] || <code>(diff( JacobiTheta2(0, q), 0$(2) ))/(JacobiTheta2(0, q)) = - 1 - 8*sum(((q)^(2*n))/((1 + (q)^(2*n))^(2)), n = 1..infinity)</code> || <code>Divide[D[EllipticTheta[2, 0, q], {0, 2}],EllipticTheta[2, 0, q]] == - 1 - 8*Sum[Divide[(q)^(2*n),(1 + (q)^(2*n))^(2)], {n, 1, Infinity}, GenerateConditions->None]</code> || Error || Failure || - || Skipped - Because timed out
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| | [https://dlmf.nist.gov/20.4.E10 20.4.E10] || [[Item:Q6765|<math>\frac{\Jacobithetaq{3}''@{0}{q}}{\Jacobithetaq{3}@{0}{q}} = -8\sum_{n=1}^{\infty}\frac{q^{2n-1}}{(1+q^{2n-1})^{2}}</math>]] || <code>(diff( JacobiTheta3(0, q), 0$(2) ))/(JacobiTheta3(0, q)) = - 8*sum(((q)^(2*n - 1))/((1 + (q)^(2*n - 1))^(2)), n = 1..infinity)</code> || <code>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]</code> || Error || Aborted || - || Skipped - Because timed out
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| | [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>]] || <code>(diff( JacobiTheta4(0, q), 0$(2) ))/(JacobiTheta4(0, q)) = 8*sum(((q)^(2*n - 1))/((1 - (q)^(2*n - 1))^(2)), n = 1..infinity)</code> || <code>Divide[D[EllipticTheta[4, 0, q], {0, 2}],EllipticTheta[4, 0, q]] == 8*Sum[Divide[(q)^(2*n - 1),(1 - (q)^(2*n - 1))^(2)], {n, 1, Infinity}, GenerateConditions->None]</code> || Error || Aborted || - || Skipped - Because timed out
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| | [https://dlmf.nist.gov/20.4.E12 20.4.E12] || [[Item:Q6767|<math>\frac{\Jacobithetaq{1}'''@{0}{q}}{\Jacobithetaq{1}'@{0}{q}} = \frac{\Jacobithetaq{2}''@{0}{q}}{\Jacobithetaq{2}@{0}{q}}+\frac{\Jacobithetaq{3}''@{0}{q}}{\Jacobithetaq{3}@{0}{q}}+\frac{\Jacobithetaq{4}''@{0}{q}}{\Jacobithetaq{4}@{0}{q}}</math>]] || <code>(diff( JacobiTheta1(0, q), 0$(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))</code> || <code>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]]</code> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><code>{Plus[Times[Power[D[0.0 <- {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]]]}</code><br><code>Plus[Times[Power[D[0.0 <- {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, Complex[-0.4999999999999998, 0.8660254037844387]], -1]], Times[-1.0, D[EllipticTheta[3, 0.0, Complex[-0.</div></div>
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| | [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>]] || <code>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)</code> || <code>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]</code> || 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>]] || <code>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)</code> || <code>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]</code> || 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>]] || <code>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)</code> || <code>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]</code> || 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>]] || <code>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)</code> || <code>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]</code> || 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>]] || <code>JacobiTheta3(Pi*z,exp(I*Pi*tau)) = sum((p)^(2*n)* (q)^((n)^(2)), n = - infinity..infinity)</code> || <code>EllipticTheta[3, Pi*z, Exp[I*Pi*(\[Tau])]] == Sum[(p)^(2*n)* (q)^((n)^(2)), {n, - Infinity, Infinity}, GenerateConditions->None]</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [140 / 300]<div class="mw-collapsible-content"><code>{Indeterminate <- {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]]]}</code><br><code>Indeterminate <- {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]]]}</code><br></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>]] || <code>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)</code> || <code>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]</code> || Failure || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [20 / 100]<div class="mw-collapsible-content"><code>{DirectedInfinity[] <- {Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br><code>Indeterminate <- {Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[q, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}</code><br></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>]] || <code>(diff( JacobiTheta1(z, q), z$(1) ))/(JacobiTheta1(z, q))- cot(z) = 4*sin(2*z)*sum(((q)^(2*n))/(1 - 2*(q)^(2*n)* cos(2*z)+ (q)^(4*n)), n = 1..infinity)</code> || <code>Divide[D[EllipticTheta[1, z, q], {z, 1}],EllipticTheta[1, z, q]]- Cot[z] == 4*Sin[2*z]*Sum[Divide[(q)^(2*n),1 - 2*(q)^(2*n)* Cos[2*z]+ (q)^(4*n)], {n, 1, Infinity}, GenerateConditions->None]</code> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
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| | [https://dlmf.nist.gov/20.5.E10 20.5.E10] || [[Item:Q6777|<math>4\sin@{2z}\sum_{n=1}^{\infty}\frac{q^{2n}}{1-2q^{2n}\cos@{2z}+q^{4n}} = 4\sum_{n=1}^{\infty}\frac{q^{2n}}{1-q^{2n}}\sin@{2nz}</math>]] || <code>4*sin(2*z)*sum(((q)^(2*n))/(1 - 2*(q)^(2*n)* cos(2*z)+ (q)^(4*n)), n = 1..infinity) = 4*sum(((q)^(2*n))/(1 - (q)^(2*n))*sin(2*n*z), n = 1..infinity)</code> || <code>4*Sin[2*z]*Sum[Divide[(q)^(2*n),1 - 2*(q)^(2*n)* Cos[2*z]+ (q)^(4*n)], {n, 1, Infinity}, GenerateConditions->None] == 4*Sum[Divide[(q)^(2*n),1 - (q)^(2*n)]*Sin[2*n*z], {n, 1, Infinity}, GenerateConditions->None]</code> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
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| | [https://dlmf.nist.gov/20.5.E11 20.5.E11] || [[Item:Q6778|<math>\frac{\Jacobithetaq{2}'@{z}{q}}{\Jacobithetaq{2}@{z}{q}}+\tan@@{z} = -4\sin@{2z}\sum_{n=1}^{\infty}\frac{q^{2n}}{1+2q^{2n}\cos@{2z}+q^{4n}}</math>]] || <code>(diff( JacobiTheta2(z, q), z$(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)</code> || <code>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]</code> || 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>]] || <code>- 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)</code> || <code>- 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]</code> || 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>]] || <code>(diff( JacobiTheta3(z, q), z$(1) ))/(JacobiTheta3(z, q)) = - 4*sin(2*z)*sum(((q)^(2*n - 1))/(1 + 2*(q)^(2*n - 1)* cos(2*z)+ (q)^(4*n - 2)), n = 1..infinity)</code> || <code>Divide[D[EllipticTheta[3, z, q], {z, 1}],EllipticTheta[3, z, q]] == - 4*Sin[2*z]*Sum[Divide[(q)^(2*n - 1),1 + 2*(q)^(2*n - 1)* Cos[2*z]+ (q)^(4*n - 2)], {n, 1, Infinity}, GenerateConditions->None]</code> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| | [https://dlmf.nist.gov/20.5.E12 20.5.E12] || [[Item:Q6779|<math>-4\sin@{2z}\sum_{n=1}^{\infty}\frac{q^{2n-1}}{1+2q^{2n-1}\cos@{2z}+q^{4n-2}} = 4\sum_{n=1}^{\infty}(-1)^{n}\frac{q^{n}}{1-q^{2n}}\sin@{2nz}</math>]] || <code>- 4*sin(2*z)*sum(((q)^(2*n - 1))/(1 + 2*(q)^(2*n - 1)* cos(2*z)+ (q)^(4*n - 2)), n = 1..infinity) = 4*sum((- 1)^(n)*((q)^(n))/(1 - (q)^(2*n))*sin(2*n*z), n = 1..infinity)</code> || <code>- 4*Sin[2*z]*Sum[Divide[(q)^(2*n - 1),1 + 2*(q)^(2*n - 1)* Cos[2*z]+ (q)^(4*n - 2)], {n, 1, Infinity}, GenerateConditions->None] == 4*Sum[(- 1)^(n)*Divide[(q)^(n),1 - (q)^(2*n)]*Sin[2*n*z], {n, 1, Infinity}, GenerateConditions->None]</code> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
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| | [https://dlmf.nist.gov/20.5.E13 20.5.E13] || [[Item:Q6780|<math>\frac{\Jacobithetaq{4}'@{z}{q}}{\Jacobithetaq{4}@{z}{q}} = 4\sin@{2z}\sum_{n=1}^{\infty}\frac{q^{2n-1}}{1-2q^{2n-1}\cos@{2z}+q^{4n-2}}</math>]] || <code>(diff( JacobiTheta4(z, q), z$(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)</code> || <code>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]</code> || 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>]] || <code>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)</code> || <code>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]</code> || 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>]] || <code>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)</code> || <code>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]</code> || 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>]] || <code>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)</code> || <code>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]</code> || 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>]] || <code>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)</code> || <code>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]</code> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| | [https://dlmf.nist.gov/20.6.E2 20.6.E2] || [[Item:Q6786|<math>\Jacobithetatau{1}@{\pi z}{\tau} = \pi z\Jacobithetatau{1}'@{0}{\tau}\exp@{-\sum_{j=1}^{\infty}\frac{1}{2j}\delta_{2j}(\tau)z^{2j}}</math>]] || <code>JacobiTheta1(Pi*z,exp(I*Pi*tau)) = Pi*z*diff( JacobiTheta1(0,exp(I*Pi*tau)), 0$(1) )*exp(- sum((1)/(2*j)*delta[2*j]*(tau)* (z)^(2*j), j = 1..infinity))</code> || <code>EllipticTheta[1, Pi*z, Exp[I*Pi*(\[Tau])]] == Pi*z*D[EllipticTheta[1, 0, Exp[I*Pi*(\[Tau])]], {0, 1}]*Exp[- Sum[Divide[1,2*j]*Subscript[\[Delta], 2*j]*(\[Tau])* (z)^(2*j), {j, 1, Infinity}, GenerateConditions->None]]</code> || Translation Error || Translation Error || - || -
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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>]] || <code>JacobiTheta2(Pi*z,exp(I*Pi*tau)) = JacobiTheta2(0,exp(I*Pi*tau))*exp(- sum((1)/(2*j)*alpha[2*j]*(tau)* (z)^(2*j), j = 1..infinity))</code> || <code>EllipticTheta[2, Pi*z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[2, 0, Exp[I*Pi*(\[Tau])]]*Exp[- Sum[Divide[1,2*j]*Subscript[\[Alpha], 2*j]*(\[Tau])* (z)^(2*j), {j, 1, Infinity}, GenerateConditions->None]]</code> || Aborted || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| | [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>]] || <code>JacobiTheta3(Pi*z,exp(I*Pi*tau)) = JacobiTheta3(0,exp(I*Pi*tau))*exp(- sum((1)/(2*j)*beta[2*j]*(tau)* (z)^(2*j), j = 1..infinity))</code> || <code>EllipticTheta[3, Pi*z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[3, 0, Exp[I*Pi*(\[Tau])]]*Exp[- Sum[Divide[1,2*j]*Subscript[\[Beta], 2*j]*(\[Tau])* (z)^(2*j), {j, 1, Infinity}, GenerateConditions->None]]</code> || Aborted || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| | [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>]] || <code>JacobiTheta4(Pi*z,exp(I*Pi*tau)) = JacobiTheta4(0,exp(I*Pi*tau))*exp(- sum((1)/(2*j)*gamma[2*j]*(tau)* (z)^(2*j), j = 1..infinity))</code> || <code>EllipticTheta[4, Pi*z, Exp[I*Pi*(\[Tau])]] == EllipticTheta[4, 0, Exp[I*Pi*(\[Tau])]]*Exp[- Sum[Divide[1,2*j]*Subscript[\[Gamma], 2*j]*(\[Tau])* (z)^(2*j), {j, 1, Infinity}, GenerateConditions->None]]</code> || Aborted || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| | [https://dlmf.nist.gov/20.6#Ex1 20.6#Ex1] || [[Item:Q6794|<math>\alpha_{2j}(\tau) = 2^{2j}\delta_{2j}(2\tau)-\delta_{2j}(\tau)</math>]] || <code>alpha[2*j]*(tau) = (2)^(2*j)* delta[2*j]*(2*tau)- delta[2*j]*(tau)</code> || <code>Subscript[\[Alpha], 2*j]*(\[Tau]) == (2)^(2*j)* Subscript[\[Delta], 2*j]*(2*\[Tau])- Subscript[\[Delta], 2*j]*(\[Tau])</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [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>]] || <code>beta[2*j]*(tau) = (2)^(2*j)* gamma[2*j]*(2*tau)- gamma[2*j]*(tau)</code> || <code>Subscript[\[Beta], 2*j]*(\[Tau]) == (2)^(2*j)* Subscript[\[Gamma], 2*j]*(2*\[Tau])- Subscript[\[Gamma], 2*j]*(\[Tau])</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/20.7.E1 20.7.E1] || [[Item:Q6796|<math>\Jacobithetaq{3}^{2}@{0}{q}\Jacobithetaq{3}^{2}@{z}{q} = \Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{4}^{2}@{z}{q}+\Jacobithetaq{2}^{2}@{0}{q}\Jacobithetaq{2}^{2}@{z}{q}</math>]] || <code>(JacobiTheta3(0, q))^(2)* (JacobiTheta3(z, q))^(2) = (JacobiTheta4(0, q))^(2)* (JacobiTheta4(z, q))^(2)+ (JacobiTheta2(0, q))^(2)* (JacobiTheta2(z, q))^(2)</code> || <code>(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)</code> || Failure || Failure || Error || Successful [Tested: 70]
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| | [https://dlmf.nist.gov/20.7.E2 20.7.E2] || [[Item:Q6797|<math>\Jacobithetaq{3}^{2}@{0}{q}\Jacobithetaq{4}^{2}@{z}{q} = \Jacobithetaq{2}^{2}@{0}{q}\Jacobithetaq{1}^{2}@{z}{q}+\Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{3}^{2}@{z}{q}</math>]] || <code>(JacobiTheta3(0, q))^(2)* (JacobiTheta4(z, q))^(2) = (JacobiTheta2(0, q))^(2)* (JacobiTheta1(z, q))^(2)+ (JacobiTheta4(0, q))^(2)* (JacobiTheta3(z, q))^(2)</code> || <code>(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)</code> || Failure || Failure || Error || Successful [Tested: 70]
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| | [https://dlmf.nist.gov/20.7.E3 20.7.E3] || [[Item:Q6798|<math>\Jacobithetaq{2}^{2}@{0}{q}\Jacobithetaq{4}^{2}@{z}{q} = \Jacobithetaq{3}^{2}@{0}{q}\Jacobithetaq{1}^{2}@{z}{q}+\Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{2}^{2}@{z}{q}</math>]] || <code>(JacobiTheta2(0, q))^(2)* (JacobiTheta4(z, q))^(2) = (JacobiTheta3(0, q))^(2)* (JacobiTheta1(z, q))^(2)+ (JacobiTheta4(0, q))^(2)* (JacobiTheta2(z, q))^(2)</code> || <code>(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)</code> || Failure || Failure || Error || Successful [Tested: 70]
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| | [https://dlmf.nist.gov/20.7.E4 20.7.E4] || [[Item:Q6799|<math>\Jacobithetaq{2}^{2}@{0}{q}\Jacobithetaq{3}^{2}@{z}{q} = \Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{1}^{2}@{z}{q}+\Jacobithetaq{3}^{2}@{0}{q}\Jacobithetaq{2}^{2}@{z}{q}</math>]] || <code>(JacobiTheta2(0, q))^(2)* (JacobiTheta3(z, q))^(2) = (JacobiTheta4(0, q))^(2)* (JacobiTheta1(z, q))^(2)+ (JacobiTheta3(0, q))^(2)* (JacobiTheta2(z, q))^(2)</code> || <code>(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)</code> || Failure || Failure || Error || Successful [Tested: 70]
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| | [https://dlmf.nist.gov/20.7.E5 20.7.E5] || [[Item:Q6800|<math>\Jacobithetaq{3}^{4}@{0}{q} = \Jacobithetaq{2}^{4}@{0}{q}+\Jacobithetaq{4}^{4}@{0}{q}</math>]] || <code>(JacobiTheta3(0, q))^(4) = (JacobiTheta2(0, q))^(4)+ (JacobiTheta4(0, q))^(4)</code> || <code>(EllipticTheta[3, 0, q])^(4) == (EllipticTheta[2, 0, q])^(4)+ (EllipticTheta[4, 0, q])^(4)</code> || Successful || Failure || - || Successful [Tested: 10]
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| | [https://dlmf.nist.gov/20.7.E6 20.7.E6] || [[Item:Q6801|<math>\Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{1}@{w+z}{q}\Jacobithetaq{1}@{w-z}{q} = \Jacobithetaq{3}^{2}@{w}{q}\Jacobithetaq{2}^{2}@{z}{q}-\Jacobithetaq{2}^{2}@{w}{q}\Jacobithetaq{3}^{2}@{z}{q}</math>]] || <code>(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)</code> || <code>(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)</code> || Failure || Failure || Error || Successful [Tested: 300]
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| | [https://dlmf.nist.gov/20.7.E7 20.7.E7] || [[Item:Q6802|<math>\Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{2}@{w+z}{q}\Jacobithetaq{2}@{w-z}{q} = \Jacobithetaq{4}^{2}@{w}{q}\Jacobithetaq{2}^{2}@{z}{q}-\Jacobithetaq{1}^{2}@{w}{q}\Jacobithetaq{3}^{2}@{z}{q}</math>]] || <code>(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)</code> || <code>(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)</code> || Failure || Failure || Error || Successful [Tested: 300]
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| | [https://dlmf.nist.gov/20.7.E8 20.7.E8] || [[Item:Q6803|<math>\Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{3}@{w+z}{q}\Jacobithetaq{3}@{w-z}{q} = \Jacobithetaq{4}^{2}@{w}{q}\Jacobithetaq{3}^{2}@{z}{q}-\Jacobithetaq{1}^{2}@{w}{q}\Jacobithetaq{2}^{2}@{z}{q}</math>]] || <code>(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)</code> || <code>(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)</code> || Failure || Failure || Error || Successful [Tested: 300]
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| | [https://dlmf.nist.gov/20.7.E9 20.7.E9] || [[Item:Q6804|<math>\Jacobithetaq{4}^{2}@{0}{q}\Jacobithetaq{4}@{w+z}{q}\Jacobithetaq{4}@{w-z}{q} = \Jacobithetaq{3}^{2}@{w}{q}\Jacobithetaq{3}^{2}@{z}{q}-\Jacobithetaq{2}^{2}@{w}{q}\Jacobithetaq{2}^{2}@{z}{q}</math>]] || <code>(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)</code> || <code>(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)</code> || Failure || Failure || Error || Successful [Tested: 300]
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| | [https://dlmf.nist.gov/20.7.E10 20.7.E10] || [[Item:Q6805|<math>\Jacobithetaq{1}@{2z}{q} = 2\frac{\Jacobithetaq{1}@{z}{q}\Jacobithetaq{2}@{z}{q}\Jacobithetaq{3}@{z}{q}\Jacobithetaq{4}@{z}{q}}{\Jacobithetaq{2}@{0}{q}\Jacobithetaq{3}@{0}{q}\Jacobithetaq{4}@{0}{q}}</math>]] || <code>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))</code> || <code>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]]</code> || Failure || Failure || Error || Successful [Tested: 70]
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| | [https://dlmf.nist.gov/20.7.E11 20.7.E11] || [[Item:Q6806|<math>\frac{\Jacobithetaq{1}@{z}{q}\Jacobithetaq{2}@{z}{q}}{\Jacobithetaq{1}@{2z}{q^{2}}} = \frac{\Jacobithetaq{3}@{z}{q}\Jacobithetaq{4}@{z}{q}}{\Jacobithetaq{4}@{2z}{q^{2}}}</math>]] || <code>(JacobiTheta1(z, q)*JacobiTheta2(z, q))/(JacobiTheta1(2*z, (q)^(2))) = (JacobiTheta3(z, q)*JacobiTheta4(z, q))/(JacobiTheta4(2*z, (q)^(2)))</code> || <code>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)]]</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 70]<div class="mw-collapsible-content"><code>{Complex[-0.5078048710711283, 0.5078048710711279] <- {Rule[q, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[-0.5078048710711284, 0.5078048710711281] <- {Rule[q, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/20.7.E11 20.7.E11] || [[Item:Q6806|<math>\frac{\Jacobithetaq{3}@{z}{q}\Jacobithetaq{4}@{z}{q}}{\Jacobithetaq{4}@{2z}{q^{2}}} = \Jacobithetaq{4}@{0}{q^{2}}</math>]] || <code>(JacobiTheta3(z, q)*JacobiTheta4(z, q))/(JacobiTheta4(2*z, (q)^(2))) = JacobiTheta4(0, (q)^(2))</code> || <code>Divide[EllipticTheta[3, z, q]*EllipticTheta[4, z, q],EllipticTheta[4, 2*z, (q)^(2)]] == EllipticTheta[4, 0, (q)^(2)]</code> || Failure || Failure || Error || Successful [Tested: 70]
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| | [https://dlmf.nist.gov/20.7.E12 20.7.E12] || [[Item:Q6807|<math>\frac{\Jacobithetaq{1}@{z}{q^{2}}\Jacobithetaq{4}@{z}{q^{2}}}{\Jacobithetaq{1}@{z}{q}} = \frac{\Jacobithetaq{2}@{z}{q^{2}}\Jacobithetaq{3}@{z}{q^{2}}}{\Jacobithetaq{2}@{z}{q}}</math>]] || <code>(JacobiTheta1(z, (q)^(2))*JacobiTheta4(z, (q)^(2)))/(JacobiTheta1(z, q)) = (JacobiTheta2(z, (q)^(2))*JacobiTheta3(z, (q)^(2)))/(JacobiTheta2(z, q))</code> || <code>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]]</code> || Failure || Failure || Error || Successful [Tested: 70]
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| | [https://dlmf.nist.gov/20.7.E12 20.7.E12] || [[Item:Q6807|<math>\frac{\Jacobithetaq{2}@{z}{q^{2}}\Jacobithetaq{3}@{z}{q^{2}}}{\Jacobithetaq{2}@{z}{q}} = \tfrac{1}{2}\Jacobithetaq{2}@{0}{q}</math>]] || <code>(JacobiTheta2(z, (q)^(2))*JacobiTheta3(z, (q)^(2)))/(JacobiTheta2(z, q)) = (1)/(2)*JacobiTheta2(0, q)</code> || <code>Divide[EllipticTheta[2, z, (q)^(2)]*EllipticTheta[3, z, (q)^(2)],EllipticTheta[2, z, q]] == Divide[1,2]*EllipticTheta[2, 0, q]</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 70]<div class="mw-collapsible-content"><code>{Complex[1.1102230246251565*^-16, -1.5053817239177183] <- {Rule[q, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[3.3306690738754696*^-16, -1.5053817239177185] <- {Rule[q, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/20.7.E13 20.7.E13] || [[Item:Q6808|<math>\Jacobithetaq{1}@{z}{q}\Jacobithetaq{1}@{w}{q} = \Jacobithetaq{3}@{z+w}{q^{2}}\Jacobithetaq{2}@{z-w}{q^{2}}-\Jacobithetaq{2}@{z+w}{q^{2}}\Jacobithetaq{3}@{z-w}{q^{2}}</math>]] || <code>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))</code> || <code>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)]</code> || Failure || Failure || Error || Successful [Tested: 300]
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| | [https://dlmf.nist.gov/20.7.E14 20.7.E14] || [[Item:Q6809|<math>\Jacobithetaq{3}@{z}{q}\Jacobithetaq{3}@{w}{q} = \Jacobithetaq{3}@{z+w}{q^{2}}\Jacobithetaq{3}@{z-w}{q^{2}}+\Jacobithetaq{2}@{z+w}{q^{2}}\Jacobithetaq{2}@{z-w}{q^{2}}</math>]] || <code>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))</code> || <code>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)]</code> || Failure || Failure || Error || Successful [Tested: 300]
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| | [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>]] || <code>JacobiTheta1(2*z,exp(I*Pi*2*tau)) = A*JacobiTheta1(z,exp(I*Pi*tau))*JacobiTheta2(z,exp(I*Pi*tau))</code> || <code>EllipticTheta[1, 2*z, Exp[I*Pi*(2*\[Tau])]] == A*EllipticTheta[1, z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[2, z, Exp[I*Pi*(\[Tau])]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[1.631641333-1.744983248*I <- {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}</code><br><code>-1.353330373+4.008308689*I <- {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)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [60 / 300]<div class="mw-collapsible-content"><code>{Complex[1.6316413333035786, -1.7449832486391479] <- {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]]]}</code><br><code>Complex[-0.25205232655780907, -0.3227610482702816] <- {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]]]}</code><br></div></div>
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| | [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>]] || <code>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))</code> || <code>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])]]</code> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [60 / 300]<div class="mw-collapsible-content"><code>{Complex[-1.4403734484961686, -1.1891981543571708] <- {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]]]}</code><br><code>Complex[-0.23150096143650367, 0.21570115304796234] <- {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]]]}</code><br></div></div>
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| | [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>]] || <code>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))</code> || <code>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])]]</code> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [60 / 300]<div class="mw-collapsible-content"><code>{Complex[0.3438479503598899, -0.39372543999621956] <- {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]]]}</code><br><code>Complex[0.12535543238516544, -0.5211900545642698] <- {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]]]}</code><br></div></div>
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| | [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>]] || <code>JacobiTheta4(2*z,exp(I*Pi*2*tau)) = A*JacobiTheta3(z,exp(I*Pi*tau))*JacobiTheta4(z,exp(I*Pi*tau))</code> || <code>EllipticTheta[4, 2*z, Exp[I*Pi*(2*\[Tau])]] == A*EllipticTheta[3, z, Exp[I*Pi*(\[Tau])]]*EllipticTheta[4, z, Exp[I*Pi*(\[Tau])]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[.88393938e-1-.6601554491*I <- {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}</code><br><code>-.5678871113-.5102031247*I <- {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)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [60 / 300]<div class="mw-collapsible-content"><code>{Complex[0.08839393747885427, -0.6601554493410663] <- {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]]]}</code><br><code>Complex[0.12758234205780994, -0.4874768056112989] <- {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]]]}</code><br></div></div>
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| | [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>]] || <code>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))</code> || <code>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])]]</code> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [60 / 300]<div class="mw-collapsible-content"><code>{Complex[-1.1596846442931608, -2.448595776474227] <- {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]]]}</code><br><code>Complex[0.3218907084595235, -0.36082838804303224] <- {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]]]}</code><br></div></div>
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| | [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>]] || <code>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))</code> || <code>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])]]</code> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [60 / 300]<div class="mw-collapsible-content"><code>{Complex[-2.54672123948714, 1.1372871673366372] <- {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]]]}</code><br><code>Complex[-0.36415557562453404, -0.3395547407401721] <- {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]]]}</code><br></div></div>
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| | [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>]] || <code>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))</code> || <code>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])]]</code> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [60 / 300]<div class="mw-collapsible-content"><code>{Complex[0.2353615104715142, -0.5335293147703523] <- {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]]]}</code><br><code>Complex[0.11871524589758675, -0.5091754766273449] <- {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]]]}</code><br></div></div>
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| | [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>]] || <code>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))</code> || <code>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])]]</code> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [60 / 300]<div class="mw-collapsible-content"><code>{Complex[0.3584730563399423, -0.5666107505620169] <- {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]]]}</code><br><code>Complex[0.11914720780154586, -0.5081951100786072] <- {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]]]}</code><br></div></div>
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| | [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>]] || <code>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))</code> || <code>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)]</code> || Failure || Failure || Error || Successful [Tested: 70]
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| | [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>]] || <code>JacobiTheta1(z,exp(I*Pi*tau + 1)) = exp(I*Pi/ 4)*JacobiTheta1(z,exp(I*Pi*tau))</code> || <code>EllipticTheta[1, z, Exp[I*Pi*(\[Tau]+ 1)]] == Exp[I*Pi/ 4]*EllipticTheta[1, z, Exp[I*Pi*(\[Tau])]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><code>70/70]: [[.7294764132+1.608567858*I <- {tau = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</code><br><code>1.107791050+1.561378050*I <- {tau = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 70]<div class="mw-collapsible-content"><code>{Complex[1.6985877827537141, -0.7949460182709149] <- {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[0.345921896794935, 1.4881712816971224] <- {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br></div></div>
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| | [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>]] || <code>JacobiTheta2(z,exp(I*Pi*tau + 1)) = exp(I*Pi/ 4)*JacobiTheta2(z,exp(I*Pi*tau))</code> || <code>EllipticTheta[2, z, Exp[I*Pi*(\[Tau]+ 1)]] == Exp[I*Pi/ 4]*EllipticTheta[2, z, Exp[I*Pi*(\[Tau])]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><code>70/70]: [[-.369621756e-1-.9012887423*I <- {tau = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</code><br><code>4.590414642+4.526034042*I <- {tau = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 70]<div class="mw-collapsible-content"><code>{Complex[0.22524015718924872, -1.3838317643459628] <- {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[2.711359141795916, -1.3916787489924032] <- {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/20.7.E28 20.7.E28] || [[Item:Q6823|<math>\Jacobithetatau{3}@{z}{\tau+1} = \Jacobithetatau{4}@{z}{\tau}</math>]] || <code>JacobiTheta3(z,exp(I*Pi*tau + 1)) = JacobiTheta4(z,exp(I*Pi*tau))</code> || <code>EllipticTheta[3, z, Exp[I*Pi*(\[Tau]+ 1)]] == EllipticTheta[4, z, Exp[I*Pi*(\[Tau])]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><code>70/70]: [[1.500564535+2.208881092*I <- {tau = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</code><br><code>-1.492914692-.5532090072*I <- {tau = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || Successful [Tested: 70]
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| | [https://dlmf.nist.gov/20.7.E29 20.7.E29] || [[Item:Q6824|<math>\Jacobithetatau{4}@{z}{\tau+1} = \Jacobithetatau{3}@{z}{\tau}</math>]] || <code>JacobiTheta4(z,exp(I*Pi*tau + 1)) = JacobiTheta3(z,exp(I*Pi*tau))</code> || <code>EllipticTheta[4, z, Exp[I*Pi*(\[Tau]+ 1)]] == EllipticTheta[3, z, Exp[I*Pi*(\[Tau])]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><code>70/70]: [[-.8770870366-.8516489897*I <- {tau = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</code><br><code>7.362801863+2.459098613*I <- {tau = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || Successful [Tested: 70]
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| | [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>]] || <code>(JacobiTheta1(z, (q)^(2))*JacobiTheta3(z, (q)^(2)))/(JacobiTheta1(z, I*q)) = (JacobiTheta2(z, (q)^(2))*JacobiTheta4(z, (q)^(2)))/(JacobiTheta2(z, I*q))</code> || <code>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]]</code> || Failure || Failure || Error || Successful [Tested: 70]
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| | [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>]] || <code>(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))</code> || <code>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]]</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 70]<div class="mw-collapsible-content"><code>{Complex[-1.1102230246251565*^-16, 0.47279727016045703] <- {Rule[q, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[4.440892098500626*^-16, 0.4727972701604571] <- {Rule[q, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| | [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>]] || <code>(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)</code> || <code>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]</code> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| | [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>]] || <code>EllipticK(k) = (1)/(2)*Pi*(JacobiTheta3(0,exp(I*Pi*tau)))^(2)</code> || <code>EllipticK[(k)^2] == Divide[1,2]*Pi*(EllipticTheta[3, 0, Exp[I*Pi*(\[Tau])]])^(2)</code> || Failure || Failure || Error || Successful [Tested: 10]
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| | [https://dlmf.nist.gov/20.9#Ex2 20.9#Ex2] || [[Item:Q6833|<math>\compellintKk'@{k} = -i\tau\compellintKk@{k}</math>]] || <code>diff( EllipticK(k), k$(1) ) = - I*tau*EllipticK(k)</code> || <code>D[EllipticK[(k)^2], {k, 1}] == - I*\[Tau]*EllipticK[(k)^2]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><code>10/10]: [[-.6481210221+.3604335389*I <- {tau = 1/2*3^(1/2)+1/2*I}</code><br><code>-1.647990213-1.212940701*I <- {tau = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 10]<div class="mw-collapsible-content"><code>{Complex[2.220446049250313*^-16, 0.6473902356608235] <- {Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[-2.220446049250313*^-16, -0.8272591738499964] <- {Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| | [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>]] || <code>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</code> || <code>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</code> || Aborted || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><code>{Plus[Times[Complex[-0.8660254037844387, -0.49999999999999994], D[0.0 <- {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[EllipticTheta[4, 0.0, Complex[0.86602540378443</div></div>
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| | [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>]] || <code>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</code> || <code>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</code> || Aborted || Failure || Skipped - Because timed out || Successful [Tested: 10]
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| | [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>]] || <code>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)</code> || <code>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]</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 3]<div class="mw-collapsible-content"><code>{Plus[1.6473133477045354, NIntegrate[Times[Power[x, -0.5], EllipticTheta[2, 0, Power[E, Times[-1, Pi, Power[x, 2]]]]] <- {x, 0, DirectedInfinity[1]}]], {Rule[s, 0.5]}</code><br></div></div>
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| | [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>]] || <code>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)</code> || <code>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]</code> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
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| | [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>]] || <code>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)</code> || <code>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]</code> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| | [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>]] || <code>- stint(exp(1)*JacobiTheta1((beta*Pi)/(2*ell),exp(I*Pi*(I*Pi*t)/((ell)^(2)))), t = 0..infinity) = - stint(exp(1)*JacobiTheta2(((1 + beta)* Pi)/(2*ell),exp(I*Pi*(I*Pi*t)/((ell)^(2)))), t = 0..infinity)</code> || <code>- stIntegrate[E*EllipticTheta[1, Divide[\[Beta]*Pi,2*\[ScriptL]], Exp[I*Pi*(Divide[I*Pi*t,\[ScriptL]^(2)])]], {t, 0, Infinity}, GenerateConditions->None] == - stIntegrate[E*EllipticTheta[2, Divide[(1 + \[Beta])* Pi,2*\[ScriptL]], Exp[I*Pi*(Divide[I*Pi*t,\[ScriptL]^(2)])]], {t, 0, Infinity}, GenerateConditions->None]</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 9]<div class="mw-collapsible-content"><code>{Plus[Times[-1.0, stIntegrate[Times[2.718281828459045, EllipticTheta[1, 2.356194490192345, Power[2.718281828459045, Times[-9.869604401089358, t]]]] <- {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]]}</code><br><code>Plus[Times[-1.0, stIntegrate[Times[2.718281828459045, EllipticTheta[1, 1.1780972450961724, Power[2.718281828459045, Times[-2.4674011002723395, t]]]] <- {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], Rule[β, Rational[3, 2]]}</code><br></div></div>
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| | [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>]] || <code>- stint(exp(1)*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))</code> || <code>- stIntegrate[E*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]]</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [54 / 54]<div class="mw-collapsible-content"><code>{Plus[2.32235875408619, Times[-1.0, stIntegrate[Times[2.718281828459045, EllipticTheta[2, 3.9269908169872414, Power[2.718281828459045, Times[-9.869604401089358, t]]]] <- {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[s, Rational[-3, 2]], Rule[ℓ, 1], Rule[β, Rational[3, 2]]}</code><br><code>Plus[-2.046254548704581, Times[-1.0, stIntegrate[Times[2.718281828459045, EllipticTheta[2, 1.9634954084936207, Power[2.718281828459045, Times[-2.4674011002723395, t]]]] <- {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[s, Rational[-3, 2]], Rule[ℓ, 2], Rule[β, Rational[3, 2]]}</code><br></div></div>
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| | [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>]] || <code>- stint(exp(1)*JacobiTheta3(((1 + beta)* Pi)/(2*ell),exp(I*Pi*(I*Pi*t)/((ell)^(2)))), t = 0..infinity) = - stint(exp(1)*JacobiTheta4((beta*Pi)/(2*ell),exp(I*Pi*(I*Pi*t)/((ell)^(2)))), t = 0..infinity)</code> || <code>- stIntegrate[E*EllipticTheta[3, Divide[(1 + \[Beta])* Pi,2*\[ScriptL]], Exp[I*Pi*(Divide[I*Pi*t,\[ScriptL]^(2)])]], {t, 0, Infinity}, GenerateConditions->None] == - stIntegrate[E*EllipticTheta[4, Divide[\[Beta]*Pi,2*\[ScriptL]], Exp[I*Pi*(Divide[I*Pi*t,\[ScriptL]^(2)])]], {t, 0, Infinity}, GenerateConditions->None]</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><code>{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]]]]] <- {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]}</code><br><code>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]]]]] <- {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], stIntegrate[Times[2.718281828459045, EllipticTheta[4, Times[0.7853981633974483, Power[ℓ, -1]], Power[2.718281828459045, Times[-9.869604401089358, t, Power[ℓ, -2]]]]], {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]</div></div>
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| | [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>]] || <code>- stint(exp(1)*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))</code> || <code>- stIntegrate[E*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]]</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 18]<div class="mw-collapsible-content"><code>{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]]]]] <- {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[s, -1.5], Rule[β, 1.5]}</code><br><code>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]]]]] <- {t, 0.0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[s, -1.5], Rule[β, 0.5]}</code><br></div></div>
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| | [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>]] || <code>hypergeom([(1)/(2),(1)/(2)], [1], (k)^(2)) = (JacobiTheta3(0,exp(I*Pi*tau)))^(2)</code> || <code>HypergeometricPFQ[{Divide[1,2],Divide[1,2]}, {1}, (k)^(2)] == (EllipticTheta[3, 0, Exp[I*Pi*(\[Tau])]])^(2)</code> || Failure || Failure || Error || Successful [Tested: 10]
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| | [https://dlmf.nist.gov/20.13.E2 20.13.E2] || [[Item:Q6852|<math>\ipderiv{\theta}{t} = \alpha\ipderiv[2]{\theta}{z}</math>]] || <code>diff(theta, t) = alpha*diff(theta, [z$(2)])</code> || <code>D[\[Theta], t] == \[Alpha]*D[\[Theta], {z, 2}]</code> || Successful || Successful || - || Successful [Tested: 300]
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