DLMF:10.22.E58 (Q3432)

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DLMF:10.22.E58
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    Statements

    Defining formula
    ∫ 0 ∞ J ν ⁡ ( a ⁢ t ) ⁢ J ν ⁡ ( b ⁢ t ) t λ ⁢ d t = ( a ⁢ b ) ν ⁢ Γ ⁡ ( ν - 1 2 ⁢ λ + 1 2 ) 2 λ ⁢ ( a 2 + b 2 ) ν - 1 2 ⁢ λ + 1 2 ⁢ Γ ⁡ ( 1 2 ⁢ λ + 1 2 ) ⁢ 𝐅 ⁡ ( 2 ⁢ ν + 1 - λ 4 , 2 ⁢ ν + 3 - λ 4 ; ν + 1 ; 4 ⁢ a 2 ⁢ b 2 ( a 2 + b 2 ) 2 ) , superscript subscript 0 Bessel-J 𝜈 𝑎 𝑡 Bessel-J 𝜈 𝑏 𝑡 superscript 𝑡 𝜆 𝑡 superscript 𝑎 𝑏 𝜈 Euler-Gamma 𝜈 1 2 𝜆 1 2 superscript 2 𝜆 superscript superscript 𝑎 2 superscript 𝑏 2 𝜈 1 2 𝜆 1 2 Euler-Gamma 1 2 𝜆 1 2 scaled-hypergeometric-bold-F 2 𝜈 1 𝜆 4 2 𝜈 3 𝜆 4 𝜈 1 4 superscript 𝑎 2 superscript 𝑏 2 superscript superscript 𝑎 2 superscript 𝑏 2 2 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{J_{\nu}\left(at\right)J_{% \nu}\left(bt\right)}{t^{\lambda}}\mathrm{d}t=\frac{(ab)^{\nu}\Gamma\left(\nu-% \frac{1}{2}\lambda+\frac{1}{2}\right)}{2^{\lambda}(a^{2}+b^{2})^{\nu-\frac{1}{% 2}\lambda+\frac{1}{2}}\Gamma\left(\frac{1}{2}\lambda+\frac{1}{2}\right)}% \mathbf{F}\left(\frac{2\nu+1-\lambda}{4},\frac{2\nu+3-\lambda}{4};\nu+1;\frac{% 4a^{2}b^{2}}{(a^{2}+b^{2})^{2}}\right),}}
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    DLMF id
    DLMF:10.22.E58
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    constraint
    ℜ ⁡ ( 2 ⁢ ν + 1 ) > ℜ ⁡ λ > - 1 2 𝜈 1 𝜆 1 {\displaystyle{\displaystyle\Re(2\nu+1)>\Re\lambda>-1}}
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    a ≠ b 𝑎 𝑏 {\displaystyle{\displaystyle a\neq b}}
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    ℜ ⁡ ( 2 ⁢ ν + 1 ) > ℜ ⁡ λ > - 1 2 𝜈 1 𝜆 1 {\displaystyle{\displaystyle\Re(2\nu+1)>\Re\lambda>-1}}
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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.m2aardec
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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.m2asdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abedec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abedec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aaldec
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    $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function
    Defining formula
    𝐅 ⁡ ( a , b ; c ; z ) scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}
    xml-id
    C15.S2.E2.m2acdec
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    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1aqdec
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