DLMF:21.6.E4 (Q6892): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11959 / rank
 
Normal rank
Property / Symbols used: Q11959 / qualifier
 
Defining formula:

g 𝑔 {\displaystyle{\displaystyle g}}

g
Property / Symbols used: Q11959 / qualifier
 
xml-id: C21.S1.XMD1.m1bdec
Property / Symbols used
 
Property / Symbols used: Q11966 / rank
 
Normal rank
Property / Symbols used: Q11966 / qualifier
 
Defining formula:

h ℎ {\displaystyle{\displaystyle h}}

h
Property / Symbols used: Q11966 / qualifier
 
xml-id: C21.S1.XMD2.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11960 / rank
 
Normal rank
Property / Symbols used: Q11960 / qualifier
 
Defining formula:

𝛀 𝛀 {\displaystyle{\displaystyle\boldsymbol{{\Omega}}}}

\boldsymbol{{\Omega}}
Property / Symbols used: Q11960 / qualifier
 
xml-id: C21.S1.XMD3.m1adec
Property / Symbols used
 
Property / Symbols used: Q11968 / rank
 
Normal rank
Property / Symbols used: Q11968 / qualifier
 
Defining formula:

T j ⁢ k subscript 𝑇 𝑗 𝑘 {\displaystyle{\displaystyle T_{jk}}}

T_{jk}
Property / Symbols used: Q11968 / qualifier
 
xml-id: C21.S6.XMD1.m1adec
Property / Symbols used
 
Property / Symbols used: set of matrices (locally) / rank
 
Normal rank
Property / Symbols used: set of matrices (locally) / qualifier
 
Defining formula:

𝒦 𝒦 {\displaystyle{\displaystyle\mathcal{K}}}

\mathcal{K}
Property / Symbols used: set of matrices (locally) / qualifier
 
xml-id: C21.S6.XMD2.m1bdec
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.m1bdec

Latest revision as of 14:51, 2 January 2020

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DLMF:21.6.E4
No description defined

    Statements

    Defining formula
    ∏ j = 1 h θ ⁢ [ ∑ k = 1 h T j ⁢ k ⁢ 𝐜 k ∑ k = 1 h T j ⁢ k ⁢ 𝐝 k ] ⁡ ( ∑ k = 1 h T j ⁢ k ⁢ 𝐳 k | 𝛀 ) = 1 𝒟 g ⁢ ∑ 𝐀 ∈ 𝒦 ∑ 𝐁 ∈ 𝒦 e - 2 ⁢ π ⁢ i ⁢ ∑ j = 1 h 𝐛 j ⋅ 𝐜 j ⁢ ∏ j = 1 h θ ⁢ [ 𝐚 j + 𝐜 j 𝐛 j + 𝐝 j ] ⁡ ( 𝐳 j | 𝛀 ) , superscript subscript product 𝑗 1 ℎ Riemann-theta-characterstics superscript subscript 𝑘 1 ℎ subscript 𝑇 𝑗 𝑘 subscript 𝐜 𝑘 superscript subscript 𝑘 1 ℎ subscript 𝑇 𝑗 𝑘 subscript 𝐝 𝑘 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 superscript subscript 𝑗 1 ℎ ⋅ subscript 𝐛 𝑗 subscript 𝐜 𝑗 superscript subscript product 𝑗 1 ℎ Riemann-theta-characterstics subscript 𝐚 𝑗 subscript 𝐜 𝑗 subscript 𝐛 𝑗 subscript 𝐝 𝑗 subscript 𝐳 𝑗 𝛀 {\displaystyle{\displaystyle\prod_{j=1}^{h}\theta\genfrac{[}{]}{0.0pt}{}{\sum_% {k=1}^{h}T_{jk}\mathbf{c}_{k}}{\sum_{k=1}^{h}T_{jk}\mathbf{d}_{k}}\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\sum_{j=1}^{h}\mathbf{b}_{j}\cdot\mathbf{c}_{j}}\prod_{j=1}^{h}% \theta\genfrac{[}{]}{0.0pt}{}{\mathbf{a}_{j}+\mathbf{c}_{j}}{\mathbf{b}_{j}+% \mathbf{d}_{j}}\left(\mathbf{z}_{j}\middle|\boldsymbol{{\Omega}}\right),}}
    0 references
    DLMF id
    DLMF:21.6.E4
    0 references
    Symbols used
    Riemann theta function with characteristics
    Defining formula
    θ ⁢ [ 𝜶 𝜷 ] ⁡ ( | missing ) Riemann-theta-characterstics 𝜶 𝜷 missing {\displaystyle{\displaystyle\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{% \alpha}}}}{\NVar{\boldsymbol{{\beta}}}}\left(\middle|missing\right)\)\@add@PDF@RDFa@triples\end{document}}}
    xml-id
    C21.S2.E5.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.m2aadec
    0 references
    Q10784
    Defining formula
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2aadec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
    0 references
    Q11959
    Defining formula
    g 𝑔 {\displaystyle{\displaystyle g}}
    xml-id
    C21.S1.XMD1.m1bdec
    0 references
    Q11966
    Defining formula
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C21.S1.XMD2.m1cdec
    0 references
    Q11960
    Defining formula
    𝛀 𝛀 {\displaystyle{\displaystyle\boldsymbol{{\Omega}}}}
    xml-id
    C21.S1.XMD3.m1adec
    0 references
    Q11968
    Defining formula
    T j ⁢ k subscript 𝑇 𝑗 𝑘 {\displaystyle{\displaystyle T_{jk}}}
    xml-id
    C21.S6.XMD1.m1adec
    0 references
    set of matrices (locally)
    Defining formula
    𝒦 𝒦 {\displaystyle{\displaystyle\mathcal{K}}}
    xml-id
    C21.S6.XMD2.m1bdec
    0 references
    number of elements (locally)
    Defining formula
    𝒟 𝒟 {\displaystyle{\displaystyle\mathcal{D}}}
    xml-id
    C21.S6.XMD3.m1bdec
    0 references
     
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