DLMF:18.17.E21 (Q5762)

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DLMF:18.17.E21
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    Statements

    Defining formula
    ∫ 0 1 P n ⁡ ( 1 - 2 ⁢ x 2 ) ⁢ sin ⁡ ( x ⁢ y ) ⁢ d x = 1 2 ⁢ π ⁢ ( J n + 1 2 ⁡ ( 1 2 ⁢ y ) ) 2 . superscript subscript 0 1 Legendre-spherical-polynomial 𝑛 1 2 superscript 𝑥 2 𝑥 𝑦 𝑥 1 2 𝜋 superscript Bessel-J 𝑛 1 2 1 2 𝑦 2 {\displaystyle{\displaystyle\int_{0}^{1}P_{n}\left(1-2x^{2}\right)\sin\left(xy% \right)\mathrm{d}x=\tfrac{1}{2}\pi\left(J_{n+\frac{1}{2}}\left(\tfrac{1}{2}y% \right)\right)^{2}.}}
    0 references
    DLMF id
    DLMF:18.17.E21
    0 references
    Symbols used
    Bessel function of the first kind
    Defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2addec
    0 references
    Q10920
    Defining formula
    P n ⁡ ( x ) Legendre-spherical-polynomial 𝑛 𝑥 {\displaystyle{\displaystyle P_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r10.m2addec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ahdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1atdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3atdec
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