DLMF:10.22.E72 (Q3446)

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DLMF:10.22.E72
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    Defining formula
    ∫ 0 ∞ J μ ⁡ ( a ⁢ t ) ⁢ J ν ⁡ ( b ⁢ t ) ⁢ J ν ⁡ ( c ⁢ t ) ⁢ t 1 - μ ⁢ d t = ( b ⁢ c ) μ - 1 ⁢ sin ⁡ ( ( μ - ν ) ⁢ π ) ⁢ ( sinh ⁡ χ ) μ - 1 2 ( 1 2 ⁢ π 3 ) 1 2 ⁢ a μ ⁢ e ( μ - 1 2 ) ⁢ i ⁢ π ⁢ Q ν - 1 2 1 2 - μ ⁡ ( cosh ⁡ χ ) , superscript subscript 0 Bessel-J 𝜇 𝑎 𝑡 Bessel-J 𝜈 𝑏 𝑡 Bessel-J 𝜈 𝑐 𝑡 superscript 𝑡 1 𝜇 𝑡 superscript 𝑏 𝑐 𝜇 1 𝜇 𝜈 superscript 𝜒 𝜇 1 2 superscript 1 2 superscript 𝜋 3 1 2 superscript 𝑎 𝜇 𝜇 1 2 imaginary-unit Legendre-Q-second-kind 1 2 𝜇 𝜈 1 2 𝜒 {\displaystyle{\displaystyle\int_{0}^{\infty}J_{\mu}\left(at\right)J_{\nu}% \left(bt\right)J_{\nu}\left(ct\right)t^{1-\mu}\mathrm{d}t=\frac{(bc)^{\mu-1}% \sin\left((\mu-\nu)\pi\right)(\sinh\chi)^{\mu-\frac{1}{2}}}{(\frac{1}{2}\pi^{3% })^{\frac{1}{2}}a^{\mu}}{\mathrm{e}^{(\mu-\frac{1}{2})\mathrm{i}\pi}}Q^{\frac{% 1}{2}-\mu}_{\nu-\frac{1}{2}}\left(\cosh\chi\right),}}
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    DLMF id
    DLMF:10.22.E72
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    constraint
    ℜ ⁡ μ > - 1 2 , ℜ ⁡ ν > - 1 , a > b + c , cosh ⁡ χ = ( a 2 - b 2 - c 2 ) / ( 2 ⁢ b ⁢ c ) formulae-sequence 𝜇 1 2 formulae-sequence 𝜈 1 formulae-sequence 𝑎 𝑏 𝑐 𝜒 superscript 𝑎 2 superscript 𝑏 2 superscript 𝑐 2 2 𝑏 𝑐 {\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2},\Re\nu>-1,a>b+c,\cosh\chi=(a^% {2}-b^{2}-c^{2})/(2bc)}}
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