DLMF:18.17.E17 (Q5758)

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DLMF:18.17.E17
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    Statements

    test
    ∫ 0 1 ( 1 - x 2 ) λ - 1 2 ⁢ C 2 ⁢ n ( λ ) ⁡ ( x ) ⁢ cos ⁡ ( x ⁢ y ) ⁢ d x = ( - 1 ) n ⁢ π ⁢ Γ ⁡ ( 2 ⁢ n + 2 ⁢ λ ) ⁢ J λ + 2 ⁢ n ⁡ ( y ) ( 2 ⁢ n ) ! ⁢ Γ ⁡ ( λ ) ⁢ ( 2 ⁢ y ) λ , superscript subscript 0 1 superscript 1 superscript 𝑥 2 𝜆 1 2 ultraspherical-Gegenbauer-polynomial 𝜆 2 𝑛 𝑥 𝑥 𝑦 𝑥 superscript 1 𝑛 𝜋 Euler-Gamma 2 𝑛 2 𝜆 Bessel-J 𝜆 2 𝑛 𝑦 2 𝑛 Euler-Gamma 𝜆 superscript 2 𝑦 𝜆 {\displaystyle{\displaystyle\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{(% \lambda)}_{2n}\left(x\right)\cos\left(xy\right)\mathrm{d}x=\frac{(-1)^{n}\pi% \Gamma\left(2n+2\lambda\right)J_{\lambda+2n}\left(y\right)}{(2n)!\Gamma\left(% \lambda\right)(2y)^{\lambda}},}}
    0 references
    DLMF id
    DLMF:18.17.E17
    0 references
    Symbols used
    Bessel function of the first kind
    test
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2adec
    0 references
     
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