DLMF:20.5.E14 (Q6781)

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DLMF:20.5.E14
No description defined

    Statements

    Defining formula
    θ 1 ⁡ ( z | τ ) = z ⁢ θ 1 ′ ⁡ ( 0 | τ ) ⁢ lim N → ∞ ⁡ ∏ n = - N N lim M → ∞ ⁡ ∏ m = - M | m | + | n | ≠ 0 M ( 1 + z ( m + n ⁢ τ ) ⁢ π ) , Jacobi-theta-tau 1 𝑧 𝜏 𝑧 diffop Jacobi-theta-tau 1 1 0 𝜏 subscript → 𝑁 superscript subscript product 𝑛 𝑁 𝑁 subscript → 𝑀 superscript subscript product 𝑚 𝑀 𝑚 𝑛 0 𝑀 1 𝑧 𝑚 𝑛 𝜏 𝜋 {\displaystyle{\displaystyle\theta_{1}\left(z\middle|\tau\right)=z\theta_{1}'% \left(0\middle|\tau\right)\*\lim_{N\to\infty}\prod_{n=-N}^{N}\lim_{M\to\infty}% \prod_{\begin{subarray}{c}m=-M\\ \left|m\right|+\left|n\right|\neq 0\end{subarray}}^{M}\left(1+\frac{z}{(m+n% \tau)\pi}\right),}}
    0 references
    DLMF id
    DLMF:20.5.E14
    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.m1aedec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
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    Q11951
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C20.S1.XMD1.m1dec
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    Q11947
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C20.S1.XMD2.m1mdec
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    Q11948
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C20.S1.XMD3.m1mdec
    0 references
    Q11949
    Defining formula
    τ 𝜏 {\displaystyle{\displaystyle\tau}}
    xml-id
    C20.S1.XMD4.m1edec
    0 references
     
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