Definite integral involving bessel functions of first and second kind

Question

Is there any standard solution of the integral: $\lim_{\epsilon \to 0} \int_{\epsilon}^{a} J_m(k_1\rho)Y_m(k_2\rho)\rho \, d\rho$. where the integer $m\geq0$ and $a<\infty$

Answer 1

For example, from DLMF, equation 10.22.4, if $k_1^2 \ne k_2^2$, $$ \int \rho\,J_m\left(k_1\rho\right)Y_m(k_2\rho)d\rho=\frac{\rho\left[k_1 J_{m+1}\left(k_1\rho\right)Y_m(k_2\rho)-k_2 J_m\left(k_1\rho\right)% Y_{m+1}(k_2\rho)\right]}{k_1^{2}-k_2^{2}}, $$