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| | ; Notation : [[21.1|21.1 Special Notation]]<br> |
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| | ; Properties : [[21.2|21.2 Definitions]]<br>[[21.3|21.3 Symmetry and Quasi-Periodicity]]<br>[[21.4|21.4 Graphics]]<br>[[21.5|21.5 Modular Transformations]]<br>[[21.6|21.6 Products]]<br> |
| ! scope="col" style="position: sticky; top: 0;" | DLMF
| | ; Applications : [[21.7|21.7 Riemann Surfaces]]<br>[[21.8|21.8 Abelian Functions]]<br>[[21.9|21.9 Integrable Equations]]<br> |
| ! scope="col" style="position: sticky; top: 0;" | Formula
| | ; Computation : [[21.10|21.10 Methods of Computation]]<br>[[21.11|21.11 Software]]<br> |
| ! scope="col" style="position: sticky; top: 0;" | Constraints
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| ! scope="col" style="position: sticky; top: 0;" | Maple
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| ! scope="col" style="position: sticky; top: 0;" | Mathematica
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| ! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
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| ! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
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| ! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
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| ! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| | [https://dlmf.nist.gov/21.2.E1 21.2.E1] || [[Item:Q6857|<math>\Riemanntheta@{\mathbf{z}}{\boldsymbol{{\Omega}}} = \sum_{\mathbf{n}\in\Integers^{g}}e^{2\pi i\left(\frac{1}{2}\mathbf{n}\cdot\boldsymbol{{\Omega}}\cdot\mathbf{n}+\mathbf{n}\cdot\mathbf{z}\right)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Riemanntheta@{\mathbf{z}}{\boldsymbol{{\Omega}}} = \sum_{\mathbf{n}\in\Integers^{g}}e^{2\pi i\left(\frac{1}{2}\mathbf{n}\cdot\boldsymbol{{\Omega}}\cdot\mathbf{n}+\mathbf{n}\cdot\mathbf{z}\right)}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>RiemannTheta(z, Omega) = sum(exp(2*Pi*I*((1)/(2)*n * Omega * n + n * z)), = ..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Missing Macro Error || Missing Macro Error || - || -
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| | [https://dlmf.nist.gov/21.2.E8 21.2.E8] || [[Item:Q6864|<math>\Riemanntheta@{z}{\Omega} = \Jacobithetatau{3}@{\pi z}{\Omega}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Riemanntheta@{z}{\Omega} = \Jacobithetatau{3}@{\pi z}{\Omega}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>RiemannTheta(z, Omega) = JacobiTheta3(Pi*z,exp(I*Pi*Omega))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Missing Macro Error || Missing Macro Error || - || -
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| | [https://dlmf.nist.gov/21.3.E1 21.3.E1] || [[Item:Q6869|<math>\Riemanntheta@{-\mathbf{z}}{\boldsymbol{{\Omega}}} = \Riemanntheta@{\mathbf{z}}{\boldsymbol{{\Omega}}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Riemanntheta@{-\mathbf{z}}{\boldsymbol{{\Omega}}} = \Riemanntheta@{\mathbf{z}}{\boldsymbol{{\Omega}}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>RiemannTheta(- z, Omega) = RiemannTheta(z, Omega)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Missing Macro Error || Missing Macro Error || - || -
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| | [https://dlmf.nist.gov/21.3.E2 21.3.E2] || [[Item:Q6870|<math>\Riemanntheta@{\mathbf{z}+\mathbf{m}_{1}}{\boldsymbol{{\Omega}}} = \Riemanntheta@{\mathbf{z}}{\boldsymbol{{\Omega}}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Riemanntheta@{\mathbf{z}+\mathbf{m}_{1}}{\boldsymbol{{\Omega}}} = \Riemanntheta@{\mathbf{z}}{\boldsymbol{{\Omega}}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>RiemannTheta(z + m[1], Omega) = RiemannTheta(z, Omega)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Missing Macro Error || Missing Macro Error || - || -
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| | [https://dlmf.nist.gov/21.3.E3 21.3.E3] || [[Item:Q6871|<math>\Riemanntheta@{\mathbf{z}+\mathbf{m}_{1}+\boldsymbol{{\Omega}}\mathbf{m}_{2}}{\boldsymbol{{\Omega}}} = e^{-2\pi i\left(\frac{1}{2}\mathbf{m}_{2}\cdot\boldsymbol{{\Omega}}\cdot\mathbf{m}_{2}+\mathbf{m}_{2}\cdot\mathbf{z}\right)}\Riemanntheta@{\mathbf{z}}{\boldsymbol{{\Omega}}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Riemanntheta@{\mathbf{z}+\mathbf{m}_{1}+\boldsymbol{{\Omega}}\mathbf{m}_{2}}{\boldsymbol{{\Omega}}} = e^{-2\pi i\left(\frac{1}{2}\mathbf{m}_{2}\cdot\boldsymbol{{\Omega}}\cdot\mathbf{m}_{2}+\mathbf{m}_{2}\cdot\mathbf{z}\right)}\Riemanntheta@{\mathbf{z}}{\boldsymbol{{\Omega}}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>RiemannTheta(z + m[1]+ Omega*m[2], Omega) = exp(- 2*Pi*I*((1)/(2)*m[2] * Omega * m[2]+ m[2] * z))*RiemannTheta(z, Omega)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Missing Macro Error || Missing Macro Error || - || -
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| | [https://dlmf.nist.gov/21.6.E6 21.6.E6] || [[Item:Q6894|<math>\Riemanntheta@{\frac{\mathbf{x}+\mathbf{y}+\mathbf{u}+\mathbf{v}}{2}}{\boldsymbol{{\Omega}}}\Riemanntheta@{\frac{\mathbf{x}+\mathbf{y}-\mathbf{u}-\mathbf{v}}{2}}{\boldsymbol{{\Omega}}}\Riemanntheta@{\frac{\mathbf{x}-\mathbf{y}+\mathbf{u}-\mathbf{v}}{2}}{\boldsymbol{{\Omega}}}\Riemanntheta@{\frac{\mathbf{x}-\mathbf{y}-\mathbf{u}+\mathbf{v}}{2}}{\boldsymbol{{\Omega}}} = \frac{1}{2^{g}}\sum_{\boldsymbol{{\alpha}}\in\frac{1}{2}\Integers^{g}/\Integers^{g}}\,\sum_{\boldsymbol{{\beta}}\in\frac{1}{2}\Integers^{g}/\Integers^{g}}e^{2\pi i\left(2\boldsymbol{{\alpha}}\cdot\boldsymbol{{\Omega}}\cdot\boldsymbol{{\alpha}}+\boldsymbol{{\alpha}}\cdot[\mathbf{x}+\mathbf{y}+\mathbf{u}+\mathbf{v}]\right)}\*\Riemanntheta@{\mathbf{x}+\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+\boldsymbol{{\beta}}}{\boldsymbol{{\Omega}}}\Riemanntheta@{\mathbf{y}+\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+\boldsymbol{{\beta}}}{\boldsymbol{{\Omega}}}\Riemanntheta@{\mathbf{u}+\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+\boldsymbol{{\beta}}}{\boldsymbol{{\Omega}}}\Riemanntheta@{\mathbf{v}+\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+\boldsymbol{{\beta}}}{\boldsymbol{{\Omega}}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Riemanntheta@{\frac{\mathbf{x}+\mathbf{y}+\mathbf{u}+\mathbf{v}}{2}}{\boldsymbol{{\Omega}}}\Riemanntheta@{\frac{\mathbf{x}+\mathbf{y}-\mathbf{u}-\mathbf{v}}{2}}{\boldsymbol{{\Omega}}}\Riemanntheta@{\frac{\mathbf{x}-\mathbf{y}+\mathbf{u}-\mathbf{v}}{2}}{\boldsymbol{{\Omega}}}\Riemanntheta@{\frac{\mathbf{x}-\mathbf{y}-\mathbf{u}+\mathbf{v}}{2}}{\boldsymbol{{\Omega}}} = \frac{1}{2^{g}}\sum_{\boldsymbol{{\alpha}}\in\frac{1}{2}\Integers^{g}/\Integers^{g}}\,\sum_{\boldsymbol{{\beta}}\in\frac{1}{2}\Integers^{g}/\Integers^{g}}e^{2\pi i\left(2\boldsymbol{{\alpha}}\cdot\boldsymbol{{\Omega}}\cdot\boldsymbol{{\alpha}}+\boldsymbol{{\alpha}}\cdot[\mathbf{x}+\mathbf{y}+\mathbf{u}+\mathbf{v}]\right)}\*\Riemanntheta@{\mathbf{x}+\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+\boldsymbol{{\beta}}}{\boldsymbol{{\Omega}}}\Riemanntheta@{\mathbf{y}+\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+\boldsymbol{{\beta}}}{\boldsymbol{{\Omega}}}\Riemanntheta@{\mathbf{u}+\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+\boldsymbol{{\beta}}}{\boldsymbol{{\Omega}}}\Riemanntheta@{\mathbf{v}+\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+\boldsymbol{{\beta}}}{\boldsymbol{{\Omega}}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>RiemannTheta((x + y + u + v)/(2), Omega)*RiemannTheta((x + y - u - v)/(2), Omega)*RiemannTheta((x - y + u - v)/(2), Omega)*RiemannTheta((x - y - u + v)/(2), Omega) = (1)/((2)^(g))*sum(sum(exp(2*Pi*I*(2*alpha * Omega * alpha + alpha *(x + y + u + v)))* RiemannTheta(x + Omega*alpha + beta, Omega)*RiemannTheta(y + Omega*alpha + beta, Omega)*RiemannTheta(u + Omega*alpha + beta, Omega)*RiemannTheta(v + Omega*alpha + beta, Omega), = ..infinity), = ..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Missing Macro Error || Missing Macro Error || - || -
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| | [https://dlmf.nist.gov/21.7.E1 21.7.E1] || [[Item:Q6897|<math>P(\lambda,\mu) = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>P(\lambda,\mu) = 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">P(lambda , mu) = 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">P[\[Lambda], \[Mu]] == 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/21.7.E11 21.7.E11] || [[Item:Q6909|<math>\mu^{2} = Q(\lambda)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\mu^{2} = Q(\lambda)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(mu)^(2) = Q(lambda)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">\[Mu]^(2) == Q[\[Lambda]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/21.7.E13 21.7.E13] || [[Item:Q6911|<math>\boldsymbol{{\eta}}(T) = \boldsymbol{{\eta}}(T^{c})</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\boldsymbol{{\eta}}(T) = \boldsymbol{{\eta}}(T^{c})</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">eta(T) = eta((T)^(c))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">\[Eta][T] == \[Eta][(T)^(c)]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/21.9.E1 21.9.E1] || [[Item:Q6917|<math>4u_{t} = 6uu_{x}+u_{xxx}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>4u_{t} = 6uu_{x}+u_{xxx}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">4*u[t] = 6*u*u[x]+ u[x, x, x]</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">4*Subscript[u, t] == 6*u*Subscript[u, x]+ Subscript[u, x, x, x]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/21.9.E2 21.9.E2] || [[Item:Q6918|<math>iu_{t} = -\tfrac{1}{2}u_{xx}+|u|^{2}u</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>iu_{t} = -\tfrac{1}{2}u_{xx}+|u|^{2}u</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">I*u[t] = -(1)/(2)*u[x, x]+(abs(u))^(2)* u</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">I*Subscript[u, t] == -Divide[1,2]*Subscript[u, x, x]+(Abs[u])^(2)* u</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/21.9.E3 21.9.E3] || [[Item:Q6919|<math>(-4u_{t}+6uu_{x}+u_{xxx})_{x}+3u_{yy} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>(-4u_{t}+6uu_{x}+u_{xxx})_{x}+3u_{yy} = 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">- 4*u[t]+ 6*u*u[x]+ u[x, x, x][x]+ 3*u[y, y] = 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[- 4*Subscript[u, t]+ 6*u*Subscript[u, x]+ Subscript[u, x, x, x], x]+ 3*Subscript[u, y, y] == 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/21.9.E4 21.9.E4] || [[Item:Q6920|<math>u(x,y,t) = c+2\pderiv[2]{}{x}\ln@{\Riemanntheta@{\mathbf{k}x+\mathbf{l}y+\boldsymbol{{\omega}}t+\boldsymbol{{\phi}}}{\boldsymbol{{\Omega}}}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>u(x,y,t) = c+2\pderiv[2]{}{x}\ln@{\Riemanntheta@{\mathbf{k}x+\mathbf{l}y+\boldsymbol{{\omega}}t+\boldsymbol{{\phi}}}{\boldsymbol{{\Omega}}}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>u(x , y , t) = c + 2*diff(ln(RiemannTheta(k*x + l*y + omega*t + phi, Omega)), [x$(2)])</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Missing Macro Error || Missing Macro Error || - || -
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| |}
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| </div> | | </div> |