$$\int ^\pi _0 \frac{\sin(\frac{d k}{2}\cos{\psi})}{\frac{d k}{2}\cos{\psi}}\sin(\psi)e^{-ik\rho\sin(\psi-\alpha)} \, d\psi \sim C \frac{\sin(\frac{d k}{2}\alpha)}{\frac{d k}{2}\alpha}$$
Where $C$ is a complex constant. The relation is valid for $dk\gg1$ and $k\rho\gg1$ and $\alpha \ll 1$
Can anyone give any ideas as to how such a relation would be shown?
Numerical LHS:
Absolute value of LHS:
Absolute value of RHS:
