DLMF:18.17.E20 (Q5761)

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DLMF:18.17.E20
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    Statements

    test
    ∫ 0 1 P n ⁡ ( 1 - 2 ⁢ x 2 ) ⁢ cos ⁡ ( x ⁢ y ) ⁢ d x = ( - 1 ) n ⁢ 1 2 ⁢ π ⁢ J n + 1 2 ⁡ ( 1 2 ⁢ y ) ⁢ J - n - 1 2 ⁡ ( 1 2 ⁢ y ) , superscript subscript 0 1 Legendre-spherical-polynomial 𝑛 1 2 superscript 𝑥 2 𝑥 𝑦 𝑥 superscript 1 𝑛 1 2 𝜋 Bessel-J 𝑛 1 2 1 2 𝑦 Bessel-J 𝑛 1 2 1 2 𝑦 {\displaystyle{\displaystyle\int_{0}^{1}P_{n}\left(1-2x^{2}\right)\cos\left(xy% \right)\mathrm{d}x=(-1)^{n}\tfrac{1}{2}\pi J_{n+\frac{1}{2}}\left(\tfrac{1}{2}% y\right)J_{-n-\frac{1}{2}}\left(\tfrac{1}{2}y\right),}}
    0 references
    DLMF id
    DLMF:18.17.E20
    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.m2acdec
    0 references
    Q10920
    test
    P n ⁡ ( x ) Legendre-spherical-polynomial 𝑛 𝑥 {\displaystyle{\displaystyle P_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r10.m2acdec
    0 references
     
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