I would like to find how this function decays as a function of $t$
$$\Re\left(\int_0^{\infty } w e^{-\frac{w^2}{2}}H_0^{(2)}(w r) e^{i t w} \, dw\right)$$
This is what it looks like for r=20. I would like to find out how it decays for large $t$ ($t>22$ for example). I do not know if this integral can be evaluated exactly, but regardless of that I would interested in learning how one might be able to figure out the solution for this asymptotically.