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),}}
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    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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