DLMF:20.5.E9 (Q6776)

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DLMF:20.5.E9
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    Statements

    Defining formula
    θ 3 ⁡ ( π ⁢ z | τ ) = ∑ n = - ∞ ∞ p 2 ⁢ n ⁢ q n 2 = ∏ n = 1 ∞ ( 1 - q 2 ⁢ n ) ⁢ ( 1 + q 2 ⁢ n - 1 ⁢ p 2 ) ⁢ ( 1 + q 2 ⁢ n - 1 ⁢ p - 2 ) , Jacobi-theta-tau 3 𝜋 𝑧 𝜏 superscript subscript 𝑛 superscript 𝑝 2 𝑛 superscript 𝑞 superscript 𝑛 2 superscript subscript product 𝑛 1 1 superscript 𝑞 2 𝑛 1 superscript 𝑞 2 𝑛 1 superscript 𝑝 2 1 superscript 𝑞 2 𝑛 1 superscript 𝑝 2 {\displaystyle{\displaystyle\theta_{3}\left(\pi z\middle|\tau\right)=\sum_{n=-% \infty}^{\infty}p^{2n}q^{n^{2}}\\ =\prod_{n=1}^{\infty}\left(1-q^{2n}\right)\left(1+q^{2n-1}p^{2}\right)\left(1+% q^{2n-1}p^{-2}\right),}}
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    DLMF id
    DLMF:20.5.E9
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    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.m1addec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
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    Q11947
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C20.S1.XMD2.m1hdec
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    Q11948
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C20.S1.XMD3.m1hdec
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