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DLMF:18.17.E32 - MaRDI portal

DLMF:18.17.E32 (Q6470)

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DLMF:18.17.E32
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    Statements

    DLMF 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),}}
    0 references
    constraint
    a > 0 𝑎 0 {\displaystyle{\displaystyle a>0}}
    0 references
    Symbols used
    gamma function
    DLMF defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2akdec
    0 references
    Laguerre (or generalized Laguerre) polynomial
    DLMF 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.m2agdec
    0 references
    cosine function
    DLMF defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2ajdec
    0 references
    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aaedec
    0 references
    base of natural logarithm
    DLMF defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aodec
    0 references
    Q10729
    DLMF defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5ahdec
    0 references
    imaginary unit
    DLMF defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2addec
    0 references
    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aaedec
    0 references
    real variable
    DLMF defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C18.S1.XMD1.m1aadec
    0 references
    nonnegative integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1aedec
    0 references
    real variable
    DLMF defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1acdec
    0 references
     
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