DLMF:21.3.E3 (Q6871)

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DLMF:21.3.E3
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    Statements

    test
    θ ⁡ ( 𝐳 + 𝐦 1 + 𝛀 ⁢ 𝐦 2 | 𝛀 ) = e - 2 ⁢ π ⁢ i ⁢ ( 1 2 ⁢ 𝐦 2 ⋅ 𝛀 ⋅ 𝐦 2 + 𝐦 2 ⋅ 𝐳 ) ⁢ θ ⁡ ( 𝐳 | 𝛀 ) , Riemann-theta 𝐳 subscript 𝐦 1 𝛀 subscript 𝐦 2 𝛀 superscript 𝑒 2 𝜋 𝑖 ⋅ 1 2 subscript 𝐦 2 𝛀 subscript 𝐦 2 ⋅ subscript 𝐦 2 𝐳 Riemann-theta 𝐳 𝛀 {\displaystyle{\displaystyle\theta\left(\mathbf{z}+\mathbf{m}_{1}+\boldsymbol{% {\Omega}}\mathbf{m}_{2}\middle|\boldsymbol{{\Omega}}\right)=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)}\theta\left(\mathbf{z}\middle|\boldsymbol{{% \Omega}}\right),}}
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    DLMF id
    DLMF:21.3.E3
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    Symbols used
    Riemann theta function
    test
    θ ⁡ ( 𝐳 | 𝛀 ) Riemann-theta 𝐳 𝛀 {\displaystyle{\displaystyle\theta\left(\NVar{\mathbf{z}}\middle|\NVar{% \boldsymbol{{\Omega}}}\right)}}
    xml-id
    C21.S2.E1.m2abdec
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    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
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    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
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    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
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