DLMF:21.6.E3 (Q6891): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: number of elements (locally) / rank
 
Normal rank
Property / Symbols used: number of elements (locally) / qualifier
 
Defining formula:

𝒟 𝒟 {\displaystyle{\displaystyle\mathcal{D}}}

\mathcal{D}
Property / Symbols used: number of elements (locally) / qualifier
 
xml-id: C21.S6.XMD3.m1adec

Latest revision as of 14:50, 2 January 2020

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DLMF:21.6.E3
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    Statements

    Defining formula
    ∏ j = 1 h θ ⁡ ( ∑ k = 1 h T j ⁢ k ⁢ 𝐳 k | 𝛀 ) = 1 𝒟 g ⁢ ∑ 𝐀 ∈ 𝒦 ∑ 𝐁 ∈ 𝒦 e 2 ⁢ π ⁢ i ⁢ tr ⁡ [ 1 2 ⁢ 𝐀 T ⁢ 𝛀 ⁢ 𝐀 + 𝐀 T ⁢ [ 𝐙 + 𝐁 ] ] ⁢ ∏ j = 1 h θ ⁡ ( 𝐳 j + 𝛀 ⁢ 𝐚 j + 𝐛 j | 𝛀 ) , superscript subscript product 𝑗 1 ℎ Riemann-theta superscript subscript 𝑘 1 ℎ subscript 𝑇 𝑗 𝑘 subscript 𝐳 𝑘 𝛀 1 superscript 𝒟 𝑔 subscript 𝐀 𝒦 subscript 𝐁 𝒦 superscript 𝑒 2 𝜋 𝑖 trace delimited-[] 1 2 𝐀 𝛀 𝐀 𝐀 delimited-[] 𝐙 𝐁 superscript subscript product 𝑗 1 ℎ Riemann-theta subscript 𝐳 𝑗 𝛀 subscript 𝐚 𝑗 subscript 𝐛 𝑗 𝛀 {\displaystyle{\displaystyle\prod_{j=1}^{h}\theta\left(\sum_{k=1}^{h}T_{jk}% \mathbf{z}_{k}\middle|\boldsymbol{{\Omega}}\right)=\frac{1}{\mathcal{D}^{g}}% \sum_{\mathbf{A}\in\mathcal{K}}\sum_{\mathbf{B}\in\mathcal{K}}e^{2\pi i% \operatorname{tr}\left[\frac{1}{2}\mathbf{A}^{\mathrm{T}}\boldsymbol{{\Omega}}% \mathbf{A}+\mathbf{A}^{\mathrm{T}}[\mathbf{Z}+\mathbf{B}]\right]}\*\prod_{j=1}% ^{h}\theta\left(\mathbf{z}_{j}+\boldsymbol{{\Omega}}\mathbf{a}_{j}+\mathbf{b}_% {j}\middle|\boldsymbol{{\Omega}}\right),}}
    0 references
    DLMF id
    DLMF:21.6.E3
    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.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    Q10784
    Defining formula
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    Q11967
    Defining formula
    tr ⁡ 𝐀 trace 𝐀 {\displaystyle{\displaystyle\operatorname{tr}\NVar{\mathbf{A}}}}
    xml-id
    introduction.Sx4.p2.t1.r7.m3adec
    0 references
    Q11959
    Defining formula
    g 𝑔 {\displaystyle{\displaystyle g}}
    xml-id
    C21.S1.XMD1.m1adec
    0 references
    Q11966
    Defining formula
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C21.S1.XMD2.m1bdec
    0 references
    Q11960
    Defining formula
    𝛀 𝛀 {\displaystyle{\displaystyle\boldsymbol{{\Omega}}}}
    xml-id
    C21.S1.XMD3.m1dec
    0 references
    Q11968
    Defining formula
    T j ⁢ k subscript 𝑇 𝑗 𝑘 {\displaystyle{\displaystyle T_{jk}}}
    xml-id
    C21.S6.XMD1.m1dec
    0 references
    set of matrices (locally)
    Defining formula
    𝒦 𝒦 {\displaystyle{\displaystyle\mathcal{K}}}
    xml-id
    C21.S6.XMD2.m1adec
    0 references
    number of elements (locally)
    Defining formula
    𝒟 𝒟 {\displaystyle{\displaystyle\mathcal{D}}}
    xml-id
    C21.S6.XMD3.m1adec
    0 references
     
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