DLMF:25.5.E14 (Q7627)

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DLMF:25.5.E14
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    Statements

    Defining formula
    ω ⁢ ( x ) ≡ ∑ n = 1 ∞ e - n 2 ⁢ π ⁢ x = 1 2 ⁢ ( θ 3 ⁡ ( 0 | i ⁢ x ) - 1 ) . equal-by definition 𝜔 𝑥 superscript subscript 𝑛 1 superscript 𝑒 superscript 𝑛 2 𝜋 𝑥 1 2 Jacobi-theta-tau 3 0 𝑖 𝑥 1 {\displaystyle{\displaystyle\omega(x)\equiv\sum_{n=1}^{\infty}e^{-n^{2}\pi x}=% \frac{1}{2}\left(\theta_{3}\left(0\middle|ix\right)-1\right).}}
    0 references
    DLMF id
    DLMF:25.5.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.m1adec
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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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    Q10769
    Defining formula
    ≡ equal-by definition {\displaystyle{\displaystyle\equiv}}
    xml-id
    introduction.Sx4.p1.t1.r9.m2adec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2ahdec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
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    Q12148
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C25.S1.XMD3.m1adec
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    Q12152
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C25.S1.XMD5.m1mdec
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