DLMF:1.8.E16 (Q270)

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DLMF:1.8.E16
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    Statements

    Defining formula
    ∑ n = - ∞ ∞ e - ( n + x ) 2 ⁢ ω = π ω ⁢ ( 1 + 2 ⁢ ∑ n = 1 ∞ e - n 2 ⁢ π 2 / ω ⁢ cos ⁡ ( 2 ⁢ n ⁢ π ⁢ x ) ) , superscript subscript 𝑛 superscript 𝑒 superscript 𝑛 𝑥 2 𝜔 𝜋 𝜔 1 2 superscript subscript 𝑛 1 superscript 𝑒 superscript 𝑛 2 superscript 𝜋 2 𝜔 2 𝑛 𝜋 𝑥 {\displaystyle{\displaystyle\sum_{n=-\infty}^{\infty}e^{-(n+x)^{2}\omega}={% \sqrt{\frac{\pi}{\omega}}\*\left(1+2\sum_{n=1}^{\infty}e^{-n^{2}\pi^{2}/\omega% }\cos\left(2n\pi x\right)\right)},}}
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    DLMF id
    DLMF:1.8.E16
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    constraint
    ℜ ⁡ ω > 0 𝜔 0 {\displaystyle{\displaystyle\Re\omega>0}}
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    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ajdec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2addec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1adec
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    Q10757
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C1.S1.XMD7.m1mdec
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