I'm studying the behaviour of the Bessel function as $x \rightarrow \infty$ part of the assignment requires me to prove the following:
Prove the improper integrals $$\int_x^\infty sin(x-\xi)\frac{\mu}{\xi^2}z(\xi)d\xi,\ \ \ \int_x^\infty cos(x-\xi)\frac{\mu}{\xi^2}z(\xi)d\xi $$ with $$z(x)=z(a)cos(x-a)+z'(a)sin(x-a)+\int_a^x sin(x-\xi)\frac{\mu}{\xi^2}z(\xi)d\xi$$ converge.
And furthermore that the function $\zeta (x)$, defined by $$z(x)=\zeta(x)-\int^\infty_x sin(x-\zeta)\frac{\mu}{\zeta^2}z(\zeta)d\zeta$$ is a $C^2$ function that holds for the differential equation $\zeta''+\zeta=0$ (Which I think means that there must be $\rho \geq0$ and $\theta \in [0,2\pi[$, such that $\zeta(x)=\rho\ sin(x+\theta)$