We have that, $$J_p(x) = \sqrt{\frac{2}{\pi x}} \sin \left( x - \frac{p\pi}{2} + \frac{\pi}{4}\right) + \frac{r_p(x)}{x\sqrt{x}}$$ We also know that there exists $M>0$ such that $|r_p(x)| \leq M$.
I am curious, can you tell me an estimate for $M$? You do not have to derive it, just as reference, or a link, with a bound on this $M$.