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${\int }_{0}^{\infty }{x}^{\mu -v+1+2n}{J}_{\mu }\left(ax\right){J}_{\mu }\left(bx\right)\frac{\mathrm{d}x}{{x}^{2}+{c}^{2}}={\left(-1\right)}^{n}{c}^{\mu -v+2n}{I}_{v}\left(bc\right){\mathbit{K}}_{\mu }\left(ac\right)$

ET II 49(15)

$[b>0,??a>b,??Rec>0,?Recv−2n+2>Reμ>−n−1,??n≥0?an?integer]$

6.578

1.

$\begin{array}{cc}\underset{0}{\overset{\infty }{\int }}{x}^{\varrho -1}{J}_{\lambda }\left(ax\right){J}_{\mu }\left(bx\right){J}_{v}\left(cx\right)\mathrm{d}x& =\frac{{2}^{\varrho -1}{a}^{\lambda }{b}^{\mu }{c}^{-\lambda -\mu -\varrho }\Gamma \left(\frac{\lambda +\mu +v+\varrho }{2}\right)}{\Gamma \left(\lambda +1\right)\Gamma \left(\mu +1\right)\text{?}\Gamma \text{?}\left(1-\frac{\lambda +\mu +v+\varrho }{2}\right)}\\ ×{F}_{4}\left(\frac{\lambda +\mu -v+e}{2},\frac{\lambda +\mu +v+\varrho }{2};\lambda +1,\mu +1;\frac{{a}^{2}}{{c}^{2}},\frac{{b}^{2}}{{c}^{2}}\right)\end{array}$

ET II 351(9)

$[Re(λ+μ+v+ϱ)>0,?Reϱ<52,?a>0,?b>0,?c>0,?c>a+b]$

2.

$\begin{array}{c}\underset{0}{\overset{\infty }{\int }}{x}^{\varrho -1}{J}_{\lambda }\left(ax\right){J}_{\mu }\left(bx\end{array}$

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