Consider the following identity $$ \int_0^\infty \sin (\rho q) \, J_1 (r q) \, \mathrm{d}q = \frac{\rho}{r(r^2-\rho^2)^{1/2}} \, , $$ where $J_1$ denotes Bessel function of the first kind of order 1. This identity can easily be checked using computer algebra software such as Maple. By differentiating both members by the parameter $\rho$ and interchanging the order of differentiation and integration we readily obtain $$ \int_0^\infty q \cos (\rho q) \, J_1 (r q) \, \mathrm{d}q = \frac{r}{(r^2-\rho^2)^{3/2}} \, . $$
I would like to know whether performing such order interchange is appropriate. Or what conditions should be satisfied in order to proceed that way. Any hints / ideas would be highly appreciated. Thank you.
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