DLMF:20.5.E17 (Q6784)

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DLMF:20.5.E17
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    Statements

    Defining formula
    θ 4 ⁡ ( z | τ ) = θ 4 ⁡ ( 0 | τ ) ⁢ lim N → ∞ ⁡ ∏ n = 1 - N N lim M → ∞ ⁡ ∏ m = - M M ( 1 + z ( m + ( n - 1 2 ) ⁢ τ ) ⁢ π ) . Jacobi-theta-tau 4 𝑧 𝜏 Jacobi-theta-tau 4 0 𝜏 subscript → 𝑁 superscript subscript product 𝑛 1 𝑁 𝑁 subscript → 𝑀 superscript subscript product 𝑚 𝑀 𝑀 1 𝑧 𝑚 𝑛 1 2 𝜏 𝜋 {\displaystyle{\displaystyle\theta_{4}\left(z\middle|\tau\right)=\theta_{4}% \left(0\middle|\tau\right)\*\lim_{N\to\infty}\prod_{n=1-N}^{N}\lim_{M\to\infty% }\prod_{m=-M}^{M}\left(1+\frac{z}{(m+(n-\tfrac{1}{2})\tau)\pi}\right).}}
    0 references
    DLMF id
    DLMF:20.5.E17
    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.m1ahdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ahdec
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    Q11951
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C20.S1.XMD1.m1cdec
    0 references
    Q11947
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C20.S1.XMD2.m1pdec
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    Q11948
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C20.S1.XMD3.m1pdec
    0 references
    Q11949
    Defining formula
    τ 𝜏 {\displaystyle{\displaystyle\tau}}
    xml-id
    C20.S1.XMD4.m1hdec
    0 references
     
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