DLMF:18.17.E23 (Q5764)

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DLMF:18.17.E23
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    Statements

    Defining formula
    ∫ 0 ∞ e - 1 2 ⁢ x 2 ⁢ 𝐻𝑒 2 ⁢ n ⁡ ( x ) ⁢ cos ⁡ ( x ⁢ y ) ⁢ d x = ( - 1 ) n ⁢ 1 2 ⁢ π ⁢ y 2 ⁢ n ⁢ e - 1 2 ⁢ y 2 , superscript subscript 0 superscript 𝑒 1 2 superscript 𝑥 2 Hermite-polynomial-He 2 𝑛 𝑥 𝑥 𝑦 𝑥 superscript 1 𝑛 1 2 𝜋 superscript 𝑦 2 𝑛 superscript 𝑒 1 2 superscript 𝑦 2 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-\frac{1}{2}x^{2}}\mathit{He}_% {2n}\left(x\right)\cos\left(xy\right)\mathrm{d}x=(-1)^{n}\sqrt{\tfrac{1}{2}\pi% }y^{2n}e^{-\frac{1}{2}y^{2}},}}
    0 references
    DLMF id
    DLMF:18.17.E23
    0 references
    Symbols used
    Q11525
    Defining formula
    𝐻𝑒 n ⁡ ( x ) Hermite-polynomial-He 𝑛 𝑥 {\displaystyle{\displaystyle\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r14.m2aadec
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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.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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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1avdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2afdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3avdec
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    Q11743
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C18.S1.XMD1.m1rdec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1vdec
    0 references
     
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