Asymptotic expansion of $\int ^ {\pi /2}_ 0J^ 2_ \nu (\lambda \textrm {cos} \theta ) d\theta$

Abstract:

An asymptotic expansion is obtained, as $\lambda \to + \infty$, for the integral \[ I(\lambda ) = \int _0^{\pi /2} {J_v^2(\lambda \cos \theta )\;d\theta ,} \] where ${J_v}(t)$ is the Bessel function of the first kind and $v > - \tfrac {1}{2}$. This integral arises in studies of crystallography and diffraction theory. We show in particular that $I(\lambda ) \sim \ln \lambda /\lambda \pi$.