DLMF:20.5.E7 (Q6774)

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DLMF:20.5.E7
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    Statements

    Defining formula
    θ 3 ⁡ ( z | τ ) = θ 3 ⁡ ( 0 | τ ) ⁢ ∏ n = 1 ∞ cos ⁡ ( ( n - 1 2 ) ⁢ π ⁢ τ + z ) ⁢ cos ⁡ ( ( n - 1 2 ) ⁢ π ⁢ τ - z ) cos 2 ⁡ ( ( n - 1 2 ) ⁢ π ⁢ τ ) , Jacobi-theta-tau 3 𝑧 𝜏 Jacobi-theta-tau 3 0 𝜏 superscript subscript product 𝑛 1 𝑛 1 2 𝜋 𝜏 𝑧 𝑛 1 2 𝜋 𝜏 𝑧 2 𝑛 1 2 𝜋 𝜏 {\displaystyle{\displaystyle\theta_{3}\left(z\middle|\tau\right)=\theta_{3}% \left(0\middle|\tau\right)\prod_{n=1}^{\infty}\frac{\cos\left((n-\tfrac{1}{2})% \pi\tau+z\right)\cos\left((n-\tfrac{1}{2})\pi\tau-z\right)}{{\cos^{2}}\left((n% -\tfrac{1}{2})\pi\tau\right)},}}
    0 references
    DLMF id
    DLMF:20.5.E7
    0 references
    Symbols used
    Q11946
    Defining formula
    θ j ⁡ ( z | τ ) Jacobi-theta-tau 𝑗 𝑧 𝜏 {\displaystyle{\displaystyle\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}% \right)}}
    xml-id
    C20.S2.SS1.m1abdec
    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
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aedec
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