DLMF:18.17.E30 (Q5771)

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DLMF:18.17.E30
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    Statements

    test
    ∫ 0 ∞ x 2 ⁢ n ⁢ e - 1 2 ⁢ x 2 ⁢ L n ( n - 1 2 ) ⁡ ( 1 2 ⁢ x 2 ) ⁢ cos ⁡ ( x ⁢ y ) ⁢ d x = 1 2 ⁢ π ⁢ y 2 ⁢ n ⁢ e - 1 2 ⁢ y 2 ⁢ L n ( n - 1 2 ) ⁡ ( 1 2 ⁢ y 2 ) . superscript subscript 0 superscript 𝑥 2 𝑛 superscript 𝑒 1 2 superscript 𝑥 2 Laguerre-polynomial-L 𝑛 1 2 𝑛 1 2 superscript 𝑥 2 𝑥 𝑦 𝑥 1 2 𝜋 superscript 𝑦 2 𝑛 superscript 𝑒 1 2 superscript 𝑦 2 Laguerre-polynomial-L 𝑛 1 2 𝑛 1 2 superscript 𝑦 2 {\displaystyle{\displaystyle\int_{0}^{\infty}x^{2n}e^{-\frac{1}{2}x^{2}}L^{(n-% \frac{1}{2})}_{n}\left(\tfrac{1}{2}x^{2}\right)\cos\left(xy\right)\mathrm{d}x=% \sqrt{\tfrac{1}{2}\pi}y^{2n}e^{-\frac{1}{2}y^{2}}L^{(n-\frac{1}{2})}_{n}\left(% \tfrac{1}{2}y^{2}\right).}}
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    DLMF id
    DLMF:18.17.E30
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    Symbols used
    Q11333
    test
    L n ( α ) ⁡ ( x ) Laguerre-polynomial-L 𝛼 𝑛 𝑥 {\displaystyle{\displaystyle L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}% \right)}}
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    C18.S3.T1.t1.r12.m2aedec
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    the ratio of the circumference of a circle to its diameter
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    π {\displaystyle{\displaystyle\pi}}
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    C3.S12.E1.m2aqdec
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    cosine function
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    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
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    C4.S14.E2.m2ahdec
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    Q10770
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    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
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    C1.S4.SS4.m1aacdec
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    base of natural logarithm
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    e {\displaystyle{\displaystyle\mathrm{e}}}
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    C4.S2.E11.m2amdec
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    Q10771
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    ∫ {\displaystyle{\displaystyle\int}}
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    C1.S4.SS4.m3aacdec
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    Q11743
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    y 𝑦 {\displaystyle{\displaystyle y}}
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    C18.S1.XMD1.m1ydec
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    Q11726
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    n 𝑛 {\displaystyle{\displaystyle n}}
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    C18.S1.XMD6.m1acdec
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    Q11727
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    x 𝑥 {\displaystyle{\displaystyle x}}
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    C18.S2.XMD3.m1aadec
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