DLMF:10.22.E74 (Q3449)

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DLMF:10.22.E74
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    Statements

    Defining formula
    ∫ 0 ∞ J ν ⁡ ( a ⁢ t ) ⁢ J ν ⁡ ( b ⁢ t ) ⁢ J ν ⁡ ( c ⁢ t ) ⁢ t 1 - ν ⁢ d t = { 2 ν - 1 ⁢ A ν - 1 2 π 1 2 ⁢ ( a ⁢ b ⁢ c ) ν ⁢ Γ ⁡ ( ν + 1 2 ) , A > 0 , 0 , A ≤ 0 . superscript subscript 0 Bessel-J 𝜈 𝑎 𝑡 Bessel-J 𝜈 𝑏 𝑡 Bessel-J 𝜈 𝑐 𝑡 superscript 𝑡 1 𝜈 𝑡 cases superscript 2 𝜈 1 superscript 𝐴 𝜈 1 2 superscript 𝜋 1 2 superscript 𝑎 𝑏 𝑐 𝜈 Euler-Gamma 𝜈 1 2 𝐴 0 0 𝐴 0 {\displaystyle{\displaystyle\int_{0}^{\infty}J_{\nu}\left(at\right)J_{\nu}% \left(bt\right)J_{\nu}\left(ct\right)t^{1-\nu}\mathrm{d}t=\begin{cases}\dfrac{% 2^{\nu-1}A^{\nu-\frac{1}{2}}}{\pi^{\frac{1}{2}}(abc)^{\nu}\Gamma\left(\nu+% \frac{1}{2}\right)},&A>0,\\ 0,&A\leq 0.\end{cases}}}
    0 references
    DLMF id
    DLMF:10.22.E74
    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.m2abfdec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2avdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aaedec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abtdec
    0 references
     
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