DLMF:10.22.E46 (Q3420)

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DLMF:10.22.E46
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    Statements

    test
    ∫ 0 ∞ t ν + 1 ⁢ J ν ⁡ ( a ⁢ t ) ( t 2 + b 2 ) μ + 1 ⁢ d t = a μ ⁢ b ν - μ 2 μ ⁢ Γ ⁡ ( μ + 1 ) ⁢ K ν - μ ⁡ ( a ⁢ b ) , superscript subscript 0 superscript 𝑡 𝜈 1 Bessel-J 𝜈 𝑎 𝑡 superscript superscript 𝑡 2 superscript 𝑏 2 𝜇 1 𝑡 superscript 𝑎 𝜇 superscript 𝑏 𝜈 𝜇 superscript 2 𝜇 Euler-Gamma 𝜇 1 modified-Bessel-second-kind 𝜈 𝜇 𝑎 𝑏 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{t^{\nu+1}J_{\nu}\left(at% \right)}{(t^{2}+b^{2})^{\mu+1}}\mathrm{d}t=\frac{a^{\mu}b^{\nu-\mu}}{2^{\mu}% \Gamma\left(\mu+1\right)}K_{\nu-\mu}\left(ab\right),}}
    0 references
    DLMF id
    DLMF:10.22.E46
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    constraint
    ℜ ⁡ b > 0 𝑏 0 {\displaystyle{\displaystyle\Re b>0}}
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    - 1 < ℜ ⁡ ν < 2 ⁢ ℜ ⁡ μ + 3 2 1 𝜈 2 𝜇 3 2 {\displaystyle{\displaystyle-1<\Re\nu<2\Re\mu+\tfrac{3}{2}}}
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    a > 0 𝑎 0 {\displaystyle{\displaystyle a>0}}
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    ℜ ⁡ b > 0 𝑏 0 {\displaystyle{\displaystyle\Re b>0}}
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    - 1 < ℜ ⁡ ν < 2 ⁢ ℜ ⁡ μ + 3 2 1 𝜈 2 𝜇 3 2 {\displaystyle{\displaystyle-1<\Re\nu<2\Re\mu+\tfrac{3}{2}}}
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    Symbols used
    Bessel function of the first kind
    test
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2aaidec
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    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aldec
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    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aasdec
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    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aasdec
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    modified Bessel function of the second kind
    test
    K ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E3.m2adec
    0 references
    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aaadec
    0 references
     
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