DLMF:18.17.E31 (Q5772)

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DLMF:18.17.E31
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    Statements

    Defining formula
    ∫ 0 ∞ e - a ⁢ x ⁢ x ν - 2 ⁢ n ⁢ L 2 ⁢ n - 1 ( ν - 2 ⁢ n ) ⁡ ( a ⁢ x ) ⁢ cos ⁡ ( x ⁢ y ) ⁢ d x = i ⁢ ( - 1 ) n ⁢ Γ ⁡ ( ν ) 2 ⁢ ( 2 ⁢ n - 1 ) ! ⁢ y 2 ⁢ n - 1 ⁢ ( ( a + i ⁢ y ) - ν - ( a - i ⁢ y ) - ν ) , superscript subscript 0 superscript 𝑒 𝑎 𝑥 superscript 𝑥 𝜈 2 𝑛 Laguerre-polynomial-L 𝜈 2 𝑛 2 𝑛 1 𝑎 𝑥 𝑥 𝑦 𝑥 𝑖 superscript 1 𝑛 Euler-Gamma 𝜈 2 2 𝑛 1 superscript 𝑦 2 𝑛 1 superscript 𝑎 𝑖 𝑦 𝜈 superscript 𝑎 𝑖 𝑦 𝜈 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-ax}x^{\nu-2n}L^{(\nu-2n)}_{2n% -1}\left(ax\right)\cos\left(xy\right)\mathrm{d}x=i\frac{(-1)^{n}\Gamma\left(% \nu\right)}{2(2n-1)!}y^{2n-1}\left((a+iy)^{-\nu}-(a-iy)^{-\nu}\right),}}
    0 references
    DLMF id
    DLMF:18.17.E31
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    constraint
    a > 0 𝑎 0 {\displaystyle{\displaystyle a>0}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2ajdec
    0 references
    Q11333
    Defining formula
    L n ( α ) ⁡ ( x ) Laguerre-polynomial-L 𝛼 𝑛 𝑥 {\displaystyle{\displaystyle L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}% \right)}}
    xml-id
    C18.S3.T1.t1.r12.m2afdec
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