DLMF:18.17.E17 (Q5758)

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DLMF:18.17.E17
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    Statements

    Defining formula
    ∫ 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
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    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.m2adec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2ahdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2abdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1apdec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5abdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3apdec
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