DLMF:21.3.E5 (Q6873)

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DLMF:21.3.E5
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    Statements

    Defining formula
    θ ⁢ [ 𝜶 𝜷 ] ⁡ ( 𝐳 + 𝐦 1 + 𝛀 ⁢ 𝐦 2 | 𝛀 ) = e 2 ⁢ π ⁢ i ⁢ ( 𝜶 ⋅ 𝐦 1 - 𝜷 ⋅ 𝐦 2 - 1 2 ⁢ 𝐦 2 ⋅ 𝛀 ⋅ 𝐦 2 - 𝐦 2 ⋅ 𝐳 ) ⁢ θ ⁢ [ 𝜶 𝜷 ] ⁡ ( 𝐳 | 𝛀 ) . Riemann-theta-characterstics 𝜶 𝜷 𝐳 subscript 𝐦 1 𝛀 subscript 𝐦 2 𝛀 superscript 𝑒 2 𝜋 𝑖 ⋅ 𝜶 subscript 𝐦 1 ⋅ 𝜷 subscript 𝐦 2 ⋅ 1 2 subscript 𝐦 2 𝛀 subscript 𝐦 2 ⋅ subscript 𝐦 2 𝐳 Riemann-theta-characterstics 𝜶 𝜷 𝐳 𝛀 {\displaystyle{\displaystyle\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}% }}{\boldsymbol{{\beta}}}\left(\mathbf{z}+\mathbf{m}_{1}+\boldsymbol{{\Omega}}% \mathbf{m}_{2}\middle|\boldsymbol{{\Omega}}\right)=e^{2\pi i\left(\boldsymbol{% {\alpha}}\cdot\mathbf{m}_{1}-\boldsymbol{{\beta}}\cdot\mathbf{m}_{2}-\frac{1}{% 2}\mathbf{m}_{2}\cdot\boldsymbol{{\Omega}}\cdot\mathbf{m}_{2}-\mathbf{m}_{2}% \cdot\mathbf{z}\right)}\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{% \boldsymbol{{\beta}}}\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right).}}
    0 references
    DLMF id
    DLMF:21.3.E5
    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.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
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
    0 references
    Q11960
    Defining formula
    𝛀 𝛀 {\displaystyle{\displaystyle\boldsymbol{{\Omega}}}}
    xml-id
    C21.S1.XMD3.m1ddec
    0 references
    Q11961
    Defining formula
    𝜶 𝜶 {\displaystyle{\displaystyle\boldsymbol{{\alpha}}}}
    xml-id
    C21.S1.XMD4.m1adec
    0 references
    Q11962
    Defining formula
    𝜷 𝜷 {\displaystyle{\displaystyle\boldsymbol{{\beta}}}}
    xml-id
    C21.S1.XMD5.m1adec
    0 references
     
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