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| |- | | ; Notation : [[23.1|23.1 Special Notation]]<br> |
| ! DLMF !! Formula !! Maple !! Mathematica !! Symbolic<br>Maple !! Symbolic<br>Mathematica !! Numeric<br>Maple !! Numeric<br>Mathematica
| | ; Weierstrass Elliptic Functions : [[23.2|23.2 Definitions and Periodic Properties]]<br>[[23.3|23.3 Differential Equations]]<br>[[23.4|23.4 Graphics]]<br>[[23.5|23.5 Special Lattices]]<br>[[23.6|23.6 Relations to Other Functions]]<br>[[23.7|23.7 Quarter Periods]]<br>[[23.8|23.8 Trigonometric Series and Products]]<br>[[23.9|23.9 Laurent and Other Power Series]]<br>[[23.10|23.10 Addition Theorems and Other Identities]]<br>[[23.11|23.11 Integral Representations]]<br>[[23.12|23.12 Asymptotic Approximations]]<br>[[23.13|23.13 Zeros]]<br>[[23.14|23.14 Integrals]]<br> |
| |-
| | ; Modular Functions : [[23.15|23.15 Definitions]]<br>[[23.16|23.16 Graphics]]<br>[[23.17|23.17 Elementary Properties]]<br>[[23.18|23.18 Modular Transformations]]<br>[[23.19|23.19 Interrelations]]<br> |
| | [https://dlmf.nist.gov/23.2.E1 23.2.E1] || [[Item:Q7195|<math>\omega_{1}+\omega_{2}+\omega_{3} = 0</math>]] || <code>omega[1]+ omega[2]+ omega[3] = 0</code> || <code>Subscript[\[Omega], 1]+ Subscript[\[Omega], 2]+ Subscript[\[Omega], 3] == 0</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| | ; Applications : [[23.20|23.20 Mathematical Applications]]<br>[[23.21|23.21 Physical Applications]]<br> |
| |-
| | ; Computation : [[23.22|23.22 Methods of Computation]]<br>[[23.23|23.23 Tables]]<br>[[23.24|23.24 Software]]<br> |
| | [https://dlmf.nist.gov/23.2#Ex1 23.2#Ex1] || [[Item:Q7196|<math>\chi_{1} = a\omega_{1}+b\omega_{3}</math>]] || <code>chi[1] = a*omega[1]+ b*omega[3]</code> || <code>Subscript[\[Chi], 1] == a*Subscript[\[Omega], 1]+ b*Subscript[\[Omega], 3]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| | </div> |
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| | [https://dlmf.nist.gov/23.2#Ex2 23.2#Ex2] || [[Item:Q7197|<math>\chi_{3} = c\omega_{1}+d\omega_{3}</math>]] || <code>chi[3] = c*omega[1]+ d*omega[3]</code> || <code>Subscript[\[Chi], 3] == c*Subscript[\[Omega], 1]+ d*Subscript[\[Omega], 3]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/23.2.E3 23.2.E3] || [[Item:Q7198|<math>ad-bc = 1</math>]] || <code>a*d - b*c = 1</code> || <code>a*d - b*c == 1</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/23.2.E13 23.2.E13] || [[Item:Q7208|<math>\eta_{1}+\eta_{2}+\eta_{3} = 0</math>]] || <code>eta[1]+ eta[2]+ eta[3] = 0</code> || <code>Subscript[\[Eta], 1]+ Subscript[\[Eta], 2]+ Subscript[\[Eta], 3] == 0</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/23.2.E14 23.2.E14] || [[Item:Q7209|<math>\eta_{3}\omega_{2}-\eta_{2}\omega_{3} = \eta_{2}\omega_{1}-\eta_{1}\omega_{2}</math>]] || <code>eta[3]*omega[2]- eta[2]*omega[3] = eta[2]*omega[1]- eta[1]*omega[2]</code> || <code>Subscript[\[Eta], 3]*Subscript[\[Omega], 2]- Subscript[\[Eta], 2]*Subscript[\[Omega], 3] == Subscript[\[Eta], 2]*Subscript[\[Omega], 1]- Subscript[\[Eta], 1]*Subscript[\[Omega], 2]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/23.5.E2 23.5.E2] || [[Item:Q7228|<math>\eta_{1} = i\eta_{3}</math>]] || <code>eta[1] = I*eta[3]</code> || <code>Subscript[\[Eta], 1] == I*Subscript[\[Eta], 3]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/23.5#Ex7 23.5#Ex7] || [[Item:Q7233|<math>k^{2} = \tfrac{1}{2}</math>]] || <code>(k)^(2) = (1)/(2)</code> || <code>(k)^(2) == Divide[1,2]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/23.5#Ex8 23.5#Ex8] || [[Item:Q7234|<math>\compellintKk@{k} = \compellintKk'@{k}</math>]] || <code>EllipticK(k) = diff( EllipticK(k), k$(1) )</code> || <code>EllipticK[(k)^2] == D[EllipticK[(k)^2], {k, 1}]</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><code>{Indeterminate <- {Rule[k, 1]}</code><br><code>Complex[1.3320292471861073, -1.3934110303935494] <- {Rule[k, 2]}</code><br></div></div>
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| | [https://dlmf.nist.gov/23.5#Ex8 23.5#Ex8] || [[Item:Q7234|<math>\compellintKk'@{k} = \ifrac{\left(\EulerGamma@{\tfrac{1}{4}}\right)^{2}}{\left(4\sqrt{\pi}\right)}</math>]] || <code>diff( EllipticK(k), k$(1) ) = ((GAMMA((1)/(4)))^(2))/(4*sqrt(Pi))</code> || <code>D[EllipticK[(k)^2], {k, 1}] == Divide[(Gamma[Divide[1,4]])^(2),4*Sqrt[Pi]]</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><code>{Indeterminate <- {Rule[k, 1]}</code><br><code>Complex[-2.343228747081181, 0.3151532066437278] <- {Rule[k, 2]}</code><br></div></div>
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| | [https://dlmf.nist.gov/23.5.E6 23.5.E6] || [[Item:Q7235|<math>\eta_{1} = e^{\pi i/3}\eta_{3}</math>]] || <code>eta[1] = exp(Pi*I/ 3)*eta[3]</code> || <code>Subscript[\[Eta], 1] == Exp[Pi*I/ 3]*Subscript[\[Eta], 3]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/23.5#Ex11 23.5#Ex11] || [[Item:Q7239|<math>k^{2} = e^{\iunit\pi/3}</math>]] || <code>(k)^(2) = exp(I*Pi/ 3)</code> || <code>(k)^(2) == Exp[I*Pi/ 3]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><code>3/3]: [[.5000000000-.8660254040*I <- {k = 1}</code><br><code>3.500000000-.8660254040*I <- {k = 2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><code>{Complex[0.4999999999999999, -0.8660254037844386] <- {Rule[k, 1]}</code><br><code>Complex[3.5, -0.8660254037844386] <- {Rule[k, 2]}</code><br></div></div>
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| | [https://dlmf.nist.gov/23.5#Ex12 23.5#Ex12] || [[Item:Q7240|<math>\compellintKk@{k} = e^{\iunit\pi/6}\compellintKk'@{k}</math>]] || <code>EllipticK(k) = exp(I*Pi/ 6)*diff( EllipticK(k), k$(1) )</code> || <code>EllipticK[(k)^2] == Exp[I*Pi/ 6]*D[EllipticK[(k)^2], {k, 1}]</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><code>{Indeterminate <- {Rule[k, 1]}</code><br><code>Complex[1.4240716315220228, -1.1066114718975122] <- {Rule[k, 2]}</code><br></div></div>
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| | [https://dlmf.nist.gov/23.5#Ex12 23.5#Ex12] || [[Item:Q7240|<math>e^{\iunit\pi/6}\compellintKk'@{k} = e^{\iunit\pi/12}\frac{3^{1/4}\left(\EulerGamma@{\frac{1}{3}}\right)^{3}}{2^{7/3}\pi}</math>]] || <code>exp(I*Pi/ 6)*diff( EllipticK(k), k$(1) ) = exp(I*Pi/ 12)*((3)^(1/ 4)*(GAMMA((1)/(3)))^(3))/((2)^(7/ 3)* Pi)</code> || <code>Exp[I*Pi/ 6]*D[EllipticK[(k)^2], {k, 1}] == Exp[I*Pi/ 12]*Divide[(3)^(1/ 4)*(Gamma[Divide[1,3]])^(3),(2)^(7/ 3)* Pi]</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><code>{Indeterminate <- {Rule[k, 1]}</code><br><code>Complex[-2.1248830880335463, -0.38527593877730804] <- {Rule[k, 2]}</code><br></div></div>
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| | [https://dlmf.nist.gov/23.6#Ex1 23.6#Ex1] || [[Item:Q7241|<math>q = e^{i\pi\tau}</math>]] || <code>q = exp(I*Pi*tau)</code> || <code>q == Exp[I*Pi*\[Tau]]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/23.6#Ex2 23.6#Ex2] || [[Item:Q7242|<math>\tau = \omega_{3}/\omega_{1}</math>]] || <code>tau = omega[3]/ omega[1]</code> || <code>\[Tau] == Subscript[\[Omega], 3]/ Subscript[\[Omega], 1]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/23.6.E8 23.6.E8] || [[Item:Q7249|<math>\eta_{1} = -\frac{\pi^{2}}{12\omega_{1}}\frac{\Jacobithetaq{1}'''@{0}{q}}{\Jacobithetaq{1}'@{0}{q}}</math>]] || <code>eta[1] = -((Pi)^(2))/(12*omega[1])*(diff( JacobiTheta1(0, q), 0$(3) ))/(diff( JacobiTheta1(0, q), 0$(1) ))</code> || <code>Subscript[\[Eta], 1] == -Divide[(Pi)^(2),12*Subscript[\[Omega], 1]]*Divide[D[EllipticTheta[1, 0, q], {0, 3}],D[EllipticTheta[1, 0, q], {0, 1}]]</code> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Plus[Complex[0.8660254037844387, 0.49999999999999994], Times[Complex[0.712277344720507, -0.4112335167120565], Power[D[0.0 <- {0.0, 1.0}], -1], D[0.0, {0.0, 3.0}]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ω, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[η, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ω, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Plus[Complex[0.8660254037844387, 0.49999999999999994], Times[Complex[-0.4112335167120564, -0.712277344720507], Power[D[0.0 <- {0.0, 1.0}], -1], D[0.0, {0.0, 3.0}]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ω, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[η, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ω, 1], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><b</div></div>
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| | [https://dlmf.nist.gov/23.6#Ex5 23.6#Ex5] || [[Item:Q7259|<math>\compellintKk^{2}@@{k} = (\compellintKk@{k})^{2}</math>]] || <code>(EllipticK(k))^(2) = (EllipticK(k))^(2)</code> || <code>(EllipticK[(k)^2])^(2) == (EllipticK[(k)^2])^(2)</code> || Successful || Successful || - || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/23.8.E5 23.8.E5] || [[Item:Q7287|<math>\eta_{1} = \frac{\pi^{2}}{2\omega_{1}}\left(\frac{1}{6}+\sum_{n=1}^{\infty}\csc^{2}@{\frac{n\pi\omega_{3}}{\omega_{1}}}\right)</math>]] || <code>eta[1] = ((Pi)^(2))/(2*omega[1])*((1)/(6)+ sum((csc((n*Pi*omega[3])/(omega[1])))^(2), n = 1..infinity))</code> || <code>Subscript[\[Eta], 1] == Divide[(Pi)^(2),2*Subscript[\[Omega], 1]]*(Divide[1,6]+ Sum[(Csc[Divide[n*Pi*Subscript[\[Omega], 3],Subscript[\[Omega], 1]]])^(2), {n, 1, Infinity}, GenerateConditions->None])</code> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
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| | [https://dlmf.nist.gov/23.9.E5 23.9.E5] || [[Item:Q7295|<math>c_{n} = \frac{3}{(2n+1)(n-3)}\sum_{m=2}^{n-2}c_{m}c_{n-m}</math>]] || <code>c[n] = (3)/((2*n + 1)*(n - 3))*sum(c[m]*c[n - m], m = 2..n - 2)</code> || <code>Subscript[c, n] == Divide[3,(2*n + 1)*(n - 3)]*Sum[Subscript[c, m]*Subscript[c, n - m], {m, 2, n - 2}, GenerateConditions->None]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/23.9.E8 23.9.E8] || [[Item:Q7298|<math>a_{m,n} = 3(m+1)a_{m+1,n-1}+\tfrac{16}{3}(n+1)a_{m-2,n+1}-\tfrac{1}{3}(2m+3n-1)(4m+6n-1)a_{m-1,n}</math>]] || <code>a[m , n] = 3*(m + 1)* a[m + 1 , n - 1]+(16)/(3)*(n + 1)* a[m - 2 , n + 1]-(1)/(3)*(2*m + 3*n - 1)*(4*m + 6*n - 1)* a[m - 1 , n]</code> || <code>Subscript[a, m , n] == 3*(m + 1)* Subscript[a, m + 1 , n - 1]+Divide[16,3]*(n + 1)* Subscript[a, m - 2 , n + 1]-Divide[1,3]*(2*m + 3*n - 1)*(4*m + 6*n - 1)* Subscript[a, m - 1 , n]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/23.10.E15 23.10.E15] || [[Item:Q7313|<math>A_{n} = \left(\frac{\pi^{2}G^{2}}{\omega_{1}}\right)^{n^{2}-1}\frac{q^{n(n-1)/2}}{i^{n-1}}\exp@{-\frac{(n-1)\eta_{1}}{3\omega_{1}}\left((2n-1)(\omega_{1}^{2}+\omega_{3}^{2})+3(n-1)\omega_{1}\omega_{3}\right)}</math>]] || <code>A[n] = exp(-((n - 1)* eta[1])/(3*omega[1])*((2*n - 1)*(omega(omega[1])^(2)+ omega(omega[3])^(2))+ 3*(n - 1)*omega[1]*omega[3]))</code> || <code>Subscript[A, n] == Exp[-Divide[(n - 1)* Subscript[\[Eta], 1],3*Subscript[\[Omega], 1]]*((2*n - 1)*(\[Omega](Subscript[\[Omega], 1])^(2)+ \[Omega](Subscript[\[Omega], 3])^(2))+ 3*(n - 1)*Subscript[\[Omega], 1]*Subscript[\[Omega], 3])]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[-.1339745960+.5000000000*I <- {G = 1/2*3^(1/2)+1/2*I, eta = 1/2*3^(1/2)+1/2*I, omega = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, A[n] = 1/2*3^(1/2)+1/2*I, eta[1] = 1/2*3^(1/2)+1/2*I, omega[1] = 1/2*3^(1/2)+1/2*I, omega[3] = 1/2*3^(1/2)+1/2*I, n = 1}</code><br><code>1.057001493+.6153915143*I <- {G = 1/2*3^(1/2)+1/2*I, eta = 1/2*3^(1/2)+1/2*I, omega = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, A[n] = 1/2*3^(1/2)+1/2*I, eta[1] = 1/2*3^(1/2)+1/2*I, omega[1] = 1/2*3^(1/2)+1/2*I, omega[3] = 1/2*3^(1/2)+1/2*I, n = 2}</code><br></div></div> || Skipped - Because timed out
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| | [https://dlmf.nist.gov/23.10#Ex1 23.10#Ex1] || [[Item:Q7314|<math>q = e^{\pi i\omega_{3}/\omega_{1}}</math>]] || <code>q = exp(Pi*I*omega[3]/ omega[1])</code> || <code>q == Exp[Pi*I*Subscript[\[Omega], 3]/ Subscript[\[Omega], 1]]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/23.10#Ex2 23.10#Ex2] || [[Item:Q7315|<math>G = \prod_{n=1}^{\infty}(1-q^{2n})</math>]] || <code>G = product(1 - (q)^(2*n), n = 1..infinity)</code> || <code>G == Product[1 - (q)^(2*n), {n, 1, Infinity}, GenerateConditions->None]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/23.11#Ex1 23.11#Ex1] || [[Item:Q7319|<math>f_{1}(s,\tau) = \frac{\cosh^{2}@{\tfrac{1}{2}\tau s}}{1-2e^{-s}\cosh@{\tau s}+e^{-2s}}</math>]] || <code>f[1]*(s , tau) = ((cosh((1)/(2)*tau*s))^(2))/(1 - 2*exp(- s)*cosh(tau*s)+ exp(- 2*s))</code> || <code>Subscript[f, 1]*(s , \[Tau]) == Divide[(Cosh[Divide[1,2]*\[Tau]*s])^(2),1 - 2*Exp[- s]*Cosh[\[Tau]*s]+ Exp[- 2*s]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[(.8660254040+.5000000000*I)*(-1.500000000, .8660254040+.5000000000*I)-.2283852288e-1-.9974318068e-1*I <- {s = -3/2, tau = 1/2*3^(1/2)+1/2*I, f[1] = 1/2*3^(1/2)+1/2*I}</code><br><code>(-.5000000000+.8660254040*I)*(-1.500000000, .8660254040+.5000000000*I)-.2283852288e-1-.9974318068e-1*I <- {s = -3/2, tau = 1/2*3^(1/2)+1/2*I, f[1] = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || Error
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| | [https://dlmf.nist.gov/23.11#Ex2 23.11#Ex2] || [[Item:Q7320|<math>f_{2}(s,\tau) = \frac{\cos^{2}@{\tfrac{1}{2}s}}{1-2e^{i\tau s}\cos@@{s}+e^{2i\tau s}}</math>]] || <code>f[2]*(s , tau) = ((cos((1)/(2)*s))^(2))/(1 - 2*exp(I*tau*s)*cos(s)+ exp(2*I*tau*s))</code> || <code>Subscript[f, 2]*(s , \[Tau]) == Divide[(Cos[Divide[1,2]*s])^(2),1 - 2*Exp[I*\[Tau]*s]*Cos[s]+ Exp[2*I*\[Tau]*s]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[(.8660254040+.5000000000*I)*(-1.500000000, .8660254040+.5000000000*I)+.1236929557-.8606824183e-1*I <- {s = -3/2, tau = 1/2*3^(1/2)+1/2*I, f[2] = 1/2*3^(1/2)+1/2*I}</code><br><code>(-.5000000000+.8660254040*I)*(-1.500000000, .8660254040+.5000000000*I)+.1236929557-.8606824183e-1*I <- {s = -3/2, tau = 1/2*3^(1/2)+1/2*I, f[2] = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || Error
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| | [https://dlmf.nist.gov/23.15.E1 23.15.E1] || [[Item:Q7330|<math>q = \exp@{-\pi\frac{\ccompellintKk@{k}}{\compellintKk@{k}}}</math>]] || <code>q = exp(- Pi*(EllipticCK(k))/(EllipticK(k)))</code> || <code>q == Exp[- Pi*Divide[EllipticK[1-(k)^2],EllipticK[(k)^2]]]</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><code>{Complex[-0.1339745962155613, 0.49999999999999994] <- {Rule[k, 1], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[0.9466424242240871, 0.7022944994770247] <- {Rule[k, 2], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/23.15#Ex1 23.15#Ex1] || [[Item:Q7331|<math>k = \frac{\Jacobithetaq{2}^{2}@{0}{q}}{\Jacobithetaq{3}^{2}@{0}{q}}</math>]] || <code>k = ((JacobiTheta2(0, q))^(2))/((JacobiTheta3(0, q))^(2))</code> || <code>k == Divide[(EllipticTheta[2, 0, q])^(2),(EllipticTheta[3, 0, q])^(2)]</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [5 / 30]<div class="mw-collapsible-content"><code>{Complex[1.0, -308.9309168668012] <- {Rule[k, 1], Rule[q, -0.5]}</code><br><code>Complex[2.0, -308.9309168668012] <- {Rule[k, 2], Rule[q, -0.5]}</code><br></div></div>
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| | [https://dlmf.nist.gov/23.15.E3 23.15.E3] || [[Item:Q7333|<math>\mathcal{A}\tau = \frac{a\tau+b}{c\tau+d}</math>]] || <code>A*tau = (a*tau + b)/(c*tau + d)</code> || <code>A*\[Tau] == Divide[a*\[Tau]+ b,c*\[Tau]+ d]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/23.15.E4 23.15.E4] || [[Item:Q7334|<math>ad-bc = 1</math>]] || <code>a*d - b*c = 1</code> || <code>a*d - b*c == 1</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/23.15.E6 23.15.E6] || [[Item:Q7336|<math>\modularlambdatau@{\tau} = \frac{\Jacobithetaq{2}^{4}@{0}{q}}{\Jacobithetaq{3}^{4}@{0}{q}}</math>]] || <code>Error</code> || <code>ModularLambda[\[Tau]] == Divide[(EllipticTheta[2, 0, q])^(4),(EllipticTheta[3, 0, q])^(4)]</code> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [4 / 100]<div class="mw-collapsible-content"><code>{Complex[95438.81139616246, 21.966995277463894] <- {Rule[q, -0.5], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[95438.81139616246, -0.8660254037844387] <- {Rule[q, -0.5], Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| |-
| |
| | [https://dlmf.nist.gov/23.15.E7 23.15.E7] || [[Item:Q7337|<math>\KleincompinvarJtau@{\tau} = \frac{\left(\Jacobithetaq{2}^{8}@{0}{q}+\Jacobithetaq{3}^{8}@{0}{q}+\Jacobithetaq{4}^{8}@{0}{q}\right)^{3}}{54\left(\Jacobithetaq{1}'@{0}{q}\right)^{8}}</math>]] || <code>Error</code> || <code>KleinInvariantJ[\[Tau]] == Divide[((EllipticTheta[2, 0, q])^(8)+ (EllipticTheta[3, 0, q])^(8)+ (EllipticTheta[4, 0, q])^(8))^(3),54*(D[EllipticTheta[1, 0, q], {0, 1}])^(8)]</code> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [100 / 100]<div class="mw-collapsible-content"><code>{Plus[Complex[-71.08223570333668, -2.1851275073468844*^-14], Times[-0.018518518518518517, Power[D[0.0 <- {0.0, 1.0}], -8], Power[Plus[Power[EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], 8], Power[EllipticTheta[3, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], 8], Power[EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], 8]], 3]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Times[-0.018518518518518517, Power[D[0.0 <- {0.0, 1.0}], -8], Power[Plus[Power[EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], 8], Power[EllipticTheta[3, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], 8], Power[EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], 8]], 3]], {Rule[q, 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/23.15.E9 23.15.E9] || [[Item:Q7339|<math>\Dedekindeta@{\tau} = \left(\tfrac{1}{2}\Jacobithetaq{1}'@{0}{q}\right)^{1/3}</math>]] || <code>Error</code> || <code>DedekindEta[\[Tau]] == (Divide[1,2]*D[EllipticTheta[1, 0, q], {0, 1}])^(1/ 3)</code> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><code>{Plus[0.7682254223260567, Times[-0.7937005259840998, Power[D[0.0 <- {0.0, 1.0}], Rational[1, 3]]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Complex[0, 1]]}</code><br><code>Plus[0.7682254223260567, Times[-0.7937005259840998, Power[D[0.0 <- {0.0, 1.0}], Rational[1, 3]]]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[τ, Complex[0, 1]]}</code><br></div></div>
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| |-
| |
| | [https://dlmf.nist.gov/23.15.E9 23.15.E9] || [[Item:Q7339|<math>\left(\tfrac{1}{2}\Jacobithetaq{1}'@{0}{q}\right)^{1/3} = e^{i\pi\tau/12}\Jacobithetatau{3}@{\tfrac{1}{2}\pi(1+\tau)}{3\tau}</math>]] || <code>((1)/(2)*diff( JacobiTheta1(0, q), 0$(1) ))^(1/ 3) = exp(I*Pi*tau/ 12)*JacobiTheta3((1)/(2)*Pi*(1 + tau),exp(I*Pi*3*tau))</code> || <code>(Divide[1,2]*D[EllipticTheta[1, 0, q], {0, 1}])^(1/ 3) == Exp[I*Pi*\[Tau]/ 12]*EllipticTheta[3, Divide[1,2]*Pi*(1 + \[Tau]), Exp[I*Pi*(3*\[Tau])]]</code> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><code>{Plus[Complex[-0.7682254223260567, 1.7569052324234997*^-19], Times[0.7937005259840998, Power[D[0.0 <- {0.0, 1.0}], Rational[1, 3]]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Complex[0, 1]]}</code><br><code>Plus[Complex[-0.7682254223260567, 1.7569052324234997*^-19], Times[0.7937005259840998, Power[D[0.0 <- {0.0, 1.0}], Rational[1, 3]]]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[τ, Complex[0, 1]]}</code><br></div></div>
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| |-
| |
| | [https://dlmf.nist.gov/23.17#Ex1 23.17#Ex1] || [[Item:Q7340|<math>\modularlambdatau@{i} = \tfrac{1}{2}</math>]] || <code>Error</code> || <code>ModularLambda[I] == Divide[1,2]</code> || Missing Macro Error || Successful || - || Successful [Tested: 1]
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| |-
| |
| | [https://dlmf.nist.gov/23.17#Ex3 23.17#Ex3] || [[Item:Q7342|<math>\KleincompinvarJtau@{i} = 1</math>]] || <code>Error</code> || <code>KleinInvariantJ[I] == 1</code> || Missing Macro Error || Successful || - || Successful [Tested: 1]
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| |-
| |
| | [https://dlmf.nist.gov/23.17#Ex5 23.17#Ex5] || [[Item:Q7344|<math>\Dedekindeta@{i} = \frac{\EulerGamma@{\tfrac{1}{4}}}{2\pi^{3/4}}</math>]] || <code>Error</code> || <code>DedekindEta[I] == Divide[Gamma[Divide[1,4]],2*(Pi)^(3/ 4)]</code> || Missing Macro Error || Successful || - || Successful [Tested: 1]
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| |-
| |
| | [https://dlmf.nist.gov/23.17.E6 23.17.E6] || [[Item:Q7348|<math>\Dedekindeta@{\tau} = \sum_{n=-\infty}^{\infty}(-1)^{n}q^{(6n+1)^{2}/12}</math>]] || <code>Error</code> || <code>DedekindEta[\[Tau]] == Sum[(- 1)^(n)* (q)^((6*n + 1)^(2)/ 12), {n, - Infinity, Infinity}, GenerateConditions->None]</code> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><code>{Plus[0.7682254223260567, Times[-1.0, NSum[Times[Power[-1, n], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Times[Rational[1, 12], Power[Plus[1, Times[6, n]], 2]]]] <- {n, DirectedInfinity[-1], DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Complex[0, 1]]}</code><br><code>Plus[0.7682254223260567, Times[-1.0, NSum[Times[Power[-1, n], Power[Power[E, Times[Complex[0, Rational[2, 3]], Pi]], Times[Rational[1, 12], Power[Plus[1, Times[6, n]], 2]]]] <- {n, DirectedInfinity[-1], DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[τ, Complex[0, 1]]}</code><br></div></div>
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| |-
| |
| | [https://dlmf.nist.gov/23.17.E7 23.17.E7] || [[Item:Q7349|<math>\modularlambdatau@{\tau} = 16q\prod_{n=1}^{\infty}\left(\frac{1+q^{2n}}{1+q^{2n-1}}\right)^{8}</math>]] || <code>Error</code> || <code>ModularLambda[\[Tau]] == 16*q*Product[(Divide[1 + (q)^(2*n),1 + (q)^(2*n - 1)])^(8), {n, 1, Infinity}, GenerateConditions->None]</code> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [24 / 100]<div class="mw-collapsible-content"><code>{Indeterminate <- {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Indeterminate <- {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| |-
| |
| | [https://dlmf.nist.gov/23.17.E8 23.17.E8] || [[Item:Q7350|<math>\Dedekindeta@{\tau} = q^{1/12}\prod_{n=1}^{\infty}(1-q^{2n})</math>]] || <code>Error</code> || <code>DedekindEta[\[Tau]] == (q)^(1/ 12)* Product[1 - (q)^(2*n), {n, 1, Infinity}, GenerateConditions->None]</code> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [4 / 10]<div class="mw-collapsible-content"><code>{DirectedInfinity[] <- {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[τ, Complex[0, 1]]}</code><br><code>DirectedInfinity[] <- {Rule[q, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]], Rule[τ, Complex[0, 1]]}</code><br></div></div>
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| |-
| |
| | [https://dlmf.nist.gov/23.18.E3 23.18.E3] || [[Item:Q7363|<math>\modularlambdatau@{\mathcal{A}\tau} = \modularlambdatau@{\tau}</math>]] || <code>Error</code> || <code>ModularLambda[A*\[Tau]] == ModularLambda[\[Tau]]</code> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 100]<div class="mw-collapsible-content"><code>{Complex[-5.551115123125783*^-17, -21.100969873679457] <- {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[-4.440892098500626*^-16, -21.100969873679432] <- {Rule[A, 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/23.18.E4 23.18.E4] || [[Item:Q7364|<math>\KleincompinvarJtau@{\mathcal{A}\tau} = \KleincompinvarJtau@{\tau}</math>]] || <code>Error</code> || <code>KleinInvariantJ[A*\[Tau]] == KleinInvariantJ[\[Tau]]</code> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [8 / 100]<div class="mw-collapsible-content"><code>{Complex[71.08223570333668, 2.1851275073468844*^-14] <- {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[-71.08223570333656, -1.2998925520285436*^-13] <- {Rule[A, 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/23.18.E5 23.18.E5] || [[Item:Q7365|<math>\Dedekindeta@{\mathcal{A}\tau} = \varepsilon(\mathcal{A})\left(-i(c\tau+d)\right)^{1/2}\Dedekindeta@{\tau}</math>]] || <code>Error</code> || <code>DedekindEta[A*\[Tau]] == \[CurlyEpsilon]*(A)*(- I*(c*\[Tau]+ d))^(1/ 2)* DedekindEta[\[Tau]]</code> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [180 / 300]<div class="mw-collapsible-content"><code>{Complex[0.11245781368984653, 0.4581664384510718] <- {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[c, -1.5], Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ε, 1], Rule[τ, Complex[0, 1]]}</code><br><code>Complex[-0.5688147076679476, 1.020829457922046] <- {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[c, -1.5], Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ε, 2], Rule[τ, Complex[0, 1]]}</code><br></div></div>
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| |-
| |
| | [https://dlmf.nist.gov/23.18.E6 23.18.E6] || [[Item:Q7366|<math>\varepsilon(\mathcal{A}) = \exp@{\pi i\left(\frac{a+d}{12c}+s(-d,c)\right)}</math>]] || <code>varepsilon*(A) = exp(Pi*I*((a + d)/(12*c)+ s*(- d , c)))</code> || <code>\[CurlyEpsilon]*(A) == Exp[Pi*I*(Divide[a + d,12*c]+ s*(- d , c))]</code> || Failure || Failure || Error || Error
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| |-
| |
| | [https://dlmf.nist.gov/23.18.E7 23.18.E7] || [[Item:Q7367|<math>s(d,c) = \sum_{r=1}^{c-1}\frac{r}{c}\left(\frac{dr}{c}-\floor{\frac{dr}{c}}-\frac{1}{2}\right),</math>]] || <code>s*(d , c) = sum((r)/(c)*((d*r)/(c)- floor((d*r)/(c))-(1)/(2)), r = 1..c - 1),</code> || <code>s*(d , c) == Sum[Divide[r,c]*(Divide[d*r,c]- Floor[Divide[d*r,c]]-Divide[1,2]), {r, 1, c - 1}, GenerateConditions->None],</code> || Skipped - Unable to analyze test case: Null || Skipped - Unable to analyze test case: Null || - || -
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| | [https://dlmf.nist.gov/23.19.E1 23.19.E1] || [[Item:Q7368|<math>\modularlambdatau@{\tau} = 16\left(\frac{\Dedekindeta^{2}@{2\tau}\Dedekindeta@{\tfrac{1}{2}\tau}}{\Dedekindeta^{3}@{\tau}}\right)^{8}</math>]] || <code>Error</code> || <code>ModularLambda[\[Tau]] == 16*(Divide[(DedekindEta[2*\[Tau]])^(2)* DedekindEta[Divide[1,2]*\[Tau]],(DedekindEta[\[Tau]])^(3)])^(8)</code> || Missing Macro Error || Failure || - || Successful [Tested: 10]
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| |-
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| | [https://dlmf.nist.gov/23.19.E2 23.19.E2] || [[Item:Q7369|<math>\KleincompinvarJtau@{\tau} = \frac{4}{27}\frac{\left(1-\modularlambdatau@{\tau}+\modularlambdatau^{2}@{\tau}\right)^{3}}{\left(\modularlambdatau@{\tau}\left(1-\modularlambdatau@{\tau}\right)\right)^{2}}</math>]] || <code>Error</code> || <code>KleinInvariantJ[\[Tau]] == Divide[4,27]*Divide[(1 - ModularLambda[\[Tau]]+ (ModularLambda[\[Tau]])^(2))^(3),(ModularLambda[\[Tau]]*(1 - ModularLambda[\[Tau]]))^(2)]</code> || Missing Macro Error || Failure || - || Successful [Tested: 10]
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| | [https://dlmf.nist.gov/23.19.E4 23.19.E4] || [[Item:Q7371|<math>\Delta = (2\pi)^{12}\Dedekindeta^{24}@{\tau}</math>]] || <code>Error</code> || <code>\[CapitalDelta] == (2*Pi)^(12)* (DedekindEta[\[Tau]])^(24)</code> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [20 / 100]<div class="mw-collapsible-content"><code>{Complex[-8.27953934969212*^7, 0.49999990438754693] <- {Rule[Δ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[1.8191325291713696*^7, 0.49999997450648886] <- {Rule[Δ, 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/23.20#Ex1 23.20#Ex1] || [[Item:Q7374|<math>x_{3} = m^{2}-x_{1}-x_{2}</math>]] || <code>x[3] = (m)^(2)- x[1]- x[2]</code> || <code>Subscript[x, 3] == (m)^(2)- Subscript[x, 1]- Subscript[x, 2]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/23.20#Ex2 23.20#Ex2] || [[Item:Q7375|<math>y_{3} = -m(x_{3}-x_{1})-y_{1}</math>]] || <code>y[3] = - m*(x[3]- x[1])- y[1]</code> || <code>Subscript[y, 3] == - m*(Subscript[x, 3]- Subscript[x, 1])- Subscript[y, 1]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/23.20.E5 23.20.E5] || [[Item:Q7377|<math>v^{8}(1+u^{8}) = 4u^{4}</math>]] || <code>(v)^(8)*(1 + (u)^(8)) = 4*(u)^(4)</code> || <code>(v)^(8)*(1 + (u)^(8)) == 4*(u)^(4)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/23.20.E6 23.20.E6] || [[Item:Q7378|<math>u^{4}-v^{4}+2uv(1-u^{2}v^{2}) = 0</math>]] || <code>(u)^(4)- (v)^(4)+ 2*u*v*(1 - (u)^(2)* (v)^(2)) = 0</code> || <code>(u)^(4)- (v)^(4)+ 2*u*v*(1 - (u)^(2)* (v)^(2)) == 0</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |-
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| | [https://dlmf.nist.gov/23.20.E7 23.20.E7] || [[Item:Q7379|<math>u^{6}-v^{6}+5u^{2}v^{2}(u^{2}-v^{2})+4uv(1-u^{4}v^{4}) = 0</math>]] || <code>(u)^(6)- (v)^(6)+ 5*(u)^(2)* (v)^(2)*((u)^(2)- (v)^(2))+ 4*u*v*(1 - (u)^(4)* (v)^(4)) = 0</code> || <code>(u)^(6)- (v)^(6)+ 5*(u)^(2)* (v)^(2)*((u)^(2)- (v)^(2))+ 4*u*v*(1 - (u)^(4)* (v)^(4)) == 0</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/23.20.E8 23.20.E8] || [[Item:Q7380|<math>(1-u^{8})(1-v^{8}) = (1-uv)^{8}</math>]] || <code>(1 - (u)^(8))*(1 - (v)^(8)) = (1 - u*v)^(8)</code> || <code>(1 - (u)^(8))*(1 - (v)^(8)) == (1 - u*v)^(8)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/23.22.E2 23.22.E2] || [[Item:Q7389|<math>2\omega_{1} = -2i\omega_{3}</math>]] || <code>2*omega[1] = - 2*I*omega[3]</code> || <code>2*Subscript[\[Omega], 1] == - 2*I*Subscript[\[Omega], 3]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [288 / 300]<div class="mw-collapsible-content"><code>288/300]: [[.732050808+2.732050808*I <- {omega = 1/2*3^(1/2)+1/2*I, omega[1] = 1/2*3^(1/2)+1/2*I, omega[3] = 1/2*3^(1/2)+1/2*I}</code><br><code>3.464101616+2.000000000*I <- {omega = 1/2*3^(1/2)+1/2*I, omega[1] = 1/2*3^(1/2)+1/2*I, omega[3] = 1/2-1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [288 / 300]<div class="mw-collapsible-content"><code>{Complex[0.7320508075688775, 2.732050807568877] <- {Rule[ω, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ω, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ω, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[3.4641016151377544, 2.0] <- {Rule[ω, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ω, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ω, 3], Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/23.22.E2 23.22.E2] || [[Item:Q7389|<math>-2i\omega_{3} = \frac{\left(\EulerGamma@{\frac{1}{4}}\right)^{2}}{2\sqrt{\pi}c^{1/4}}</math>]] || <code>- 2*I*omega[3] = ((GAMMA((1)/(4)))^(2))/(2*sqrt(Pi)*(c)^(1/ 4))</code> || <code>- 2*I*Subscript[\[Omega], 3] == Divide[(Gamma[Divide[1,4]])^(2),2*Sqrt[Pi]*(c)^(1/ 4)]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[-1.369296462+.637245654*I <- {c = -3/2, omega = 1/2*3^(1/2)+1/2*I, omega[3] = 1/2*3^(1/2)+1/2*I}</code><br><code>-.637245654+3.369296462*I <- {c = -3/2, omega = 1/2*3^(1/2)+1/2*I, omega[3] = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[-1.3692964596386887, 0.6372456520698113] <- {Rule[c, -1.5], Rule[ω, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ω, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[-0.6372456520698113, 3.3692964596386883] <- {Rule[c, -1.5], Rule[ω, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ω, 3], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/23.22.E3 23.22.E3] || [[Item:Q7390|<math>2\omega_{1} = 2e^{-\pi i/3}\omega_{3}</math>]] || <code>2*omega[1] = 2*exp(- Pi*I/ 3)*omega[3]</code> || <code>2*Subscript[\[Omega], 1] == 2*Exp[- Pi*I/ 3]*Subscript[\[Omega], 3]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[0.+2.000000001*I <- {omega = 1/2*3^(1/2)+1/2*I, omega[1] = 1/2*3^(1/2)+1/2*I, omega[3] = 1/2*3^(1/2)+1/2*I}</code><br><code>.732050807-.732050808*I <- {omega = 1/2*3^(1/2)+1/2*I, omega[1] = 1/2*3^(1/2)+1/2*I, omega[3] = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[0.0, 1.9999999999999998] <- {Rule[ω, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ω, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ω, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[0.7320508075688772, -0.7320508075688773] <- {Rule[ω, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ω, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ω, 3], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/23.22.E3 23.22.E3] || [[Item:Q7390|<math>2e^{-\pi i/3}\omega_{3} = \frac{\left(\EulerGamma@{\frac{1}{3}}\right)^{3}}{2\pi d^{1/6}}</math>]] || <code>2*exp(- Pi*I/ 3)*omega[3] = ((GAMMA((1)/(3)))^(3))/(2*Pi*(d)^(1/ 6))</code> || <code>2*Exp[- Pi*I/ 3]*Subscript[\[Omega], 3] == Divide[(Gamma[Divide[1,3]])^(3),2*Pi*(d)^(1/ 6)]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[-1.316213396-.7333114397*I <- {d = 1/2*3^(1/2)+1/2*I, omega = 1/2*3^(1/2)+1/2*I, omega[3] = 1/2*3^(1/2)+1/2*I}</code><br><code>-2.048264203+1.998739369*I <- {d = 1/2*3^(1/2)+1/2*I, omega = 1/2*3^(1/2)+1/2*I, omega[3] = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[-1.3162133925119985, -0.7333114390610043] <- {Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ω, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ω, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[-2.048264200080876, 1.9987393685078727] <- {Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ω, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ω, 3], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/23.22#Ex1 23.22#Ex1] || [[Item:Q7391|<math>2\omega_{1} = 0.867568+i1.466607</math>]] || <code>2*omega[1] = 0.867568 + I*1.466607</code> || <code>2*Subscript[\[Omega], 1] == 0.867568 + I*1.466607</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/23.22#Ex2 23.22#Ex2] || [[Item:Q7392|<math>2\omega_{3} = -1.223741+i1.328694</math>]] || <code>2*omega[3] = - 1.223741 + I*1.328694</code> || <code>2*Subscript[\[Omega], 3] == - 1.223741 + I*1.328694</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/23.22#Ex3 23.22#Ex3] || [[Item:Q7393|<math>\tau = 0.305480+i1.015109</math>]] || <code>tau = 0.305480 + I*1.015109</code> || <code>\[Tau] == 0.305480 + I*1.015109</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |}
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