DLMF:10.22.E57 (Q3431)

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DLMF:10.22.E57
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    Statements

    Defining formula
    ∫ 0 ∞ J μ ⁡ ( a ⁢ t ) ⁢ J ν ⁡ ( a ⁢ t ) t λ ⁢ d t = ( 1 2 ⁢ a ) λ - 1 ⁢ Γ ⁡ ( 1 2 ⁢ μ + 1 2 ⁢ ν - 1 2 ⁢ λ + 1 2 ) ⁢ Γ ⁡ ( λ ) 2 ⁢ Γ ⁡ ( 1 2 ⁢ λ + 1 2 ⁢ ν - 1 2 ⁢ μ + 1 2 ) ⁢ Γ ⁡ ( 1 2 ⁢ λ + 1 2 ⁢ μ - 1 2 ⁢ ν + 1 2 ) ⁢ Γ ⁡ ( 1 2 ⁢ λ + 1 2 ⁢ μ + 1 2 ⁢ ν + 1 2 ) , superscript subscript 0 Bessel-J 𝜇 𝑎 𝑡 Bessel-J 𝜈 𝑎 𝑡 superscript 𝑡 𝜆 𝑡 superscript 1 2 𝑎 𝜆 1 Euler-Gamma 1 2 𝜇 1 2 𝜈 1 2 𝜆 1 2 Euler-Gamma 𝜆 2 Euler-Gamma 1 2 𝜆 1 2 𝜈 1 2 𝜇 1 2 Euler-Gamma 1 2 𝜆 1 2 𝜇 1 2 𝜈 1 2 Euler-Gamma 1 2 𝜆 1 2 𝜇 1 2 𝜈 1 2 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{J_{\mu}\left(at\right)J_{% \nu}\left(at\right)}{t^{\lambda}}\mathrm{d}t=\frac{(\frac{1}{2}a)^{\lambda-1}% \Gamma\left(\frac{1}{2}\mu+\frac{1}{2}\nu-\frac{1}{2}\lambda+\frac{1}{2}\right% )\Gamma\left(\lambda\right)}{2\Gamma\left(\frac{1}{2}\lambda+\frac{1}{2}\nu-% \frac{1}{2}\mu+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\lambda+\frac{1}{2}\mu% -\frac{1}{2}\nu+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\lambda+\frac{1}{2}% \mu+\frac{1}{2}\nu+\frac{1}{2}\right)},}}
    0 references
    DLMF id
    DLMF:10.22.E57
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    constraint
    ℜ ⁡ ( μ + ν + 1 ) > ℜ ⁡ λ > 0 𝜇 𝜈 1 𝜆 0 {\displaystyle{\displaystyle\Re(\mu+\nu+1)>\Re\lambda>0}}
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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.m2aaqdec
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    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2ardec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abddec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abddec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aakdec
    0 references
    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1apdec
    0 references
     
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