DLMF:18.17.E24 (Q5765)

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DLMF:18.17.E24
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    Statements

    Defining formula
    ∫ 0 ∞ e - x 2 ⁢ 𝐻𝑒 2 ⁢ n ⁡ ( 2 ⁢ x ) ⁢ cos ⁡ ( x ⁢ y ) ⁢ d x = ( - 1 ) n ⁢ 1 2 ⁢ π ⁢ e - 1 4 ⁢ y 2 ⁢ 𝐻𝑒 2 ⁢ n ⁡ ( y ) . superscript subscript 0 superscript 𝑒 superscript 𝑥 2 Hermite-polynomial-He 2 𝑛 2 𝑥 𝑥 𝑦 𝑥 superscript 1 𝑛 1 2 𝜋 superscript 𝑒 1 4 superscript 𝑦 2 Hermite-polynomial-He 2 𝑛 𝑦 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-x^{2}}\mathit{He}_{2n}\left(2% x\right)\cos\left(xy\right)\mathrm{d}x=(-1)^{n}\tfrac{1}{2}\sqrt{\pi}e^{-\frac% {1}{4}y^{2}}\mathit{He}_{2n}\left(y\right).}}
    0 references
    DLMF id
    DLMF:18.17.E24
    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.m2abdec
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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.m2akdec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aedec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1awdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2agdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3awdec
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