DLMF:18.17.E32 (Q5773)

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DLMF:18.17.E32
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    Statements

    Defining formula
    ∫ 0 ∞ e - a ⁢ x ⁢ x ν - 1 - 2 ⁢ n ⁢ L 2 ⁢ n ( ν - 1 - 2 ⁢ n ) ⁡ ( a ⁢ x ) ⁢ cos ⁡ ( x ⁢ y ) ⁢ d x = ( - 1 ) n ⁢ Γ ⁡ ( ν ) 2 ⁢ ( 2 ⁢ n ) ! ⁢ y 2 ⁢ n ⁢ ( ( a + i ⁢ y ) - ν + ( a - i ⁢ y ) - ν ) , superscript subscript 0 superscript 𝑒 𝑎 𝑥 superscript 𝑥 𝜈 1 2 𝑛 Laguerre-polynomial-L 𝜈 1 2 𝑛 2 𝑛 𝑎 𝑥 𝑥 𝑦 𝑥 superscript 1 𝑛 Euler-Gamma 𝜈 2 2 𝑛 superscript 𝑦 2 𝑛 superscript 𝑎 𝑖 𝑦 𝜈 superscript 𝑎 𝑖 𝑦 𝜈 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-ax}x^{\nu-1-2n}L^{(\nu-1-2n)}% _{2n}\left(ax\right)\cos\left(xy\right)\mathrm{d}x=\frac{(-1)^{n}\Gamma\left(% \nu\right)}{2(2n)!}y^{2n}\left((a+iy)^{-\nu}+(a-iy)^{-\nu}\right),}}
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    DLMF id
    DLMF:18.17.E32
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