We study the asymptotic behavior near $+ \infty$ of a solution $z \in E$ of the differential equation defined on $] 0 , + \infty [$, with $\lambda \in \mathbb { R } ^ { * }$ :
$$z ^ { \prime \prime } + \left( 1 + \frac { \lambda } { x ^ { 2 } } \right) z = 0 \tag{III.3}$$
Justify that
$$\int _ { x } ^ { + \infty } z ( u ) \sin ( u - x ) \frac { \mathrm { d } u } { u ^ { 2 } } = O \left( \frac { 1 } { x } \right)$$
near $+ \infty$.
Deduce the existence of constants $\alpha$ and $\beta$ such that near $+ \infty$,
$$z ( x ) = \alpha \cos ( x - \beta ) + O \left( \frac { 1 } { x } \right)$$