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I know that if the orders of the Bessel functions were the same, the following integral will give a representation for the dirac delta distribution

$$ k\int_0^\infty r J_0(kr)J_0(k'r)\,dr =\delta(k-k')$$

Is there a known, notable result for the analogous integral with Bessel functions of different orders, for example

$$ I_{02}(k,k')=k\int_0^\infty r J_2(kr)J_0(k'r)\,dr =?$$

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  • $\begingroup$ For general v is:$$k \int_0^{\infty } r J_v(k r) J_0(\text{kprime} r) \, dr=\frac{2 k G_{2,2}^{1,1}\left(\frac{k^2}{\text{kprime}^2}| \begin{array}{c} 0,0 \\ \frac{v}{2},-\frac{v}{2} \\ \end{array} \right)}{\text{kprime}^2}$$ where: G is MeijerG function. $\endgroup$ Commented Dec 2, 2023 at 13:24

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