DLMF:21.6.E6 (Q6894): Difference between revisions

From DRMF
Jump to navigation Jump to search
← Older editNewer edit →
‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
Defining formula:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2abdec

Revision as of 14:51, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:21.6.E6
No description defined

    Statements

    Defining formula
    θ ⁡ ( 𝐱 + 𝐲 + 𝐮 + 𝐯 2 | 𝛀 ) ⁢ θ ⁡ ( 𝐱 + 𝐲 - 𝐮 - 𝐯 2 | 𝛀 ) ⁢ θ ⁡ ( 𝐱 - 𝐲 + 𝐮 - 𝐯 2 | 𝛀 ) ⁢ θ ⁡ ( 𝐱 - 𝐲 - 𝐮 + 𝐯 2 | 𝛀 ) = 1 2 g ⁢ ∑ 𝜶 ∈ 1 2 ⁢ ℤ g / ℤ g ∑ 𝜷 ∈ 1 2 ⁢ ℤ g / ℤ g e 2 ⁢ π ⁢ i ⁢ ( 2 ⁢ 𝜶 ⋅ 𝛀 ⋅ 𝜶 + 𝜶 ⋅ [ 𝐱 + 𝐲 + 𝐮 + 𝐯 ] ) ⁢ θ ⁡ ( 𝐱 + 𝛀 ⁢ 𝜶 + 𝜷 | 𝛀 ) ⁢ θ ⁡ ( 𝐲 + 𝛀 ⁢ 𝜶 + 𝜷 | 𝛀 ) ⁢ θ ⁡ ( 𝐮 + 𝛀 ⁢ 𝜶 + 𝜷 | 𝛀 ) ⁢ θ ⁡ ( 𝐯 + 𝛀 ⁢ 𝜶 + 𝜷 | 𝛀 ) , Riemann-theta 𝐱 𝐲 𝐮 𝐯 2 𝛀 Riemann-theta 𝐱 𝐲 𝐮 𝐯 2 𝛀 Riemann-theta 𝐱 𝐲 𝐮 𝐯 2 𝛀 Riemann-theta 𝐱 𝐲 𝐮 𝐯 2 𝛀 1 superscript 2 𝑔 subscript 𝜶 1 2 𝑔 𝑔 subscript 𝜷 1 2 𝑔 𝑔 superscript 𝑒 2 𝜋 𝑖 ⋅ 2 𝜶 𝛀 𝜶 ⋅ 𝜶 delimited-[] 𝐱 𝐲 𝐮 𝐯 Riemann-theta 𝐱 𝛀 𝜶 𝜷 𝛀 Riemann-theta 𝐲 𝛀 𝜶 𝜷 𝛀 Riemann-theta 𝐮 𝛀 𝜶 𝜷 𝛀 Riemann-theta 𝐯 𝛀 𝜶 𝜷 𝛀 {\displaystyle{\displaystyle\theta\left(\frac{\mathbf{x}+\mathbf{y}+\mathbf{u}% +\mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)\theta\left(\frac{\mathbf{x% }+\mathbf{y}-\mathbf{u}-\mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)% \theta\left(\frac{\mathbf{x}-\mathbf{y}+\mathbf{u}-\mathbf{v}}{2}\middle|% \boldsymbol{{\Omega}}\right)\theta\left(\frac{\mathbf{x}-\mathbf{y}-\mathbf{u}% +\mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)=\frac{1}{2^{g}}\sum_{% \boldsymbol{{\alpha}}\in\frac{1}{2}{\mathbb{Z}^{g}}/{\mathbb{Z}^{g}}}\,\sum_{% \boldsymbol{{\beta}}\in\frac{1}{2}{\mathbb{Z}^{g}}/{\mathbb{Z}^{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)}\*\theta\left(\mathbf{x}+\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+% \boldsymbol{{\beta}}\middle|\boldsymbol{{\Omega}}\right)\theta\left(\mathbf{y}% +\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+\boldsymbol{{\beta}}\middle|% \boldsymbol{{\Omega}}\right)\theta\left(\mathbf{u}+\boldsymbol{{\Omega}}% \boldsymbol{{\alpha}}+\boldsymbol{{\beta}}\middle|\boldsymbol{{\Omega}}\right)% \theta\left(\mathbf{v}+\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+\boldsymbol{% {\beta}}\middle|\boldsymbol{{\Omega}}\right),}}
    0 references
    DLMF id
    DLMF:21.6.E6
    0 references
    Symbols used
    Riemann theta function
    Defining formula
    θ ⁡ ( 𝐳 | 𝛀 ) Riemann-theta 𝐳 𝛀 {\displaystyle{\displaystyle\theta\left(\NVar{\mathbf{z}}\middle|\NVar{% \boldsymbol{{\Omega}}}\right)}}
    xml-id
    C21.S2.E1.m2aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q6894&oldid=51548"