DLMF:10.22.E56 (Q3430)

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DLMF:10.22.E56
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    Statements

    Defining formula
    ∫ 0 ∞ J μ ⁡ ( a ⁢ t ) ⁢ J ν ⁡ ( b ⁢ t ) t λ ⁢ d t = a μ ⁢ Γ ⁡ ( 1 2 ⁢ ν + 1 2 ⁢ μ - 1 2 ⁢ λ + 1 2 ) 2 λ ⁢ b μ - λ + 1 ⁢ Γ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ μ + 1 2 ⁢ λ + 1 2 ) ⁢ 𝐅 ⁡ ( 1 2 ⁢ ( μ + ν - λ + 1 ) , 1 2 ⁢ ( μ - ν - λ + 1 ) ; μ + 1 ; a 2 b 2 ) , superscript subscript 0 Bessel-J 𝜇 𝑎 𝑡 Bessel-J 𝜈 𝑏 𝑡 superscript 𝑡 𝜆 𝑡 superscript 𝑎 𝜇 Euler-Gamma 1 2 𝜈 1 2 𝜇 1 2 𝜆 1 2 superscript 2 𝜆 superscript 𝑏 𝜇 𝜆 1 Euler-Gamma 1 2 𝜈 1 2 𝜇 1 2 𝜆 1 2 scaled-hypergeometric-bold-F 1 2 𝜇 𝜈 𝜆 1 1 2 𝜇 𝜈 𝜆 1 𝜇 1 superscript 𝑎 2 superscript 𝑏 2 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{J_{\mu}\left(at\right)J_{% \nu}\left(bt\right)}{t^{\lambda}}\mathrm{d}t=\frac{a^{\mu}\Gamma\left(\frac{1}% {2}\nu+\frac{1}{2}\mu-\frac{1}{2}\lambda+\frac{1}{2}\right)}{2^{\lambda}b^{\mu% -\lambda+1}\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\lambda+\frac{% 1}{2}\right)}\*\mathbf{F}\left(\tfrac{1}{2}(\mu+\nu-\lambda+1),\tfrac{1}{2}(% \mu-\nu-\lambda+1);\mu+1;\frac{a^{2}}{b^{2}}\right),}}
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    DLMF id
    DLMF:10.22.E56
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    constraint
    ℜ ⁡ ( μ + ν + 1 ) > ℜ ⁡ λ > - 1 𝜇 𝜈 1 𝜆 1 {\displaystyle{\displaystyle\Re(\mu+\nu+1)>\Re\lambda>-1}}
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    0 < a < b 0 𝑎 𝑏 {\displaystyle{\displaystyle 0<a<b}}
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    ℜ ⁡ ( μ + ν + 1 ) > ℜ ⁡ λ > - 1 𝜇 𝜈 1 𝜆 1 {\displaystyle{\displaystyle\Re(\mu+\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.m2aapdec
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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.m2aqdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abcdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abcdec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aajdec
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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.m2abdec
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    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1aodec
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