Hi everyone,

I'm a physicist working on stochastic processes and I've come up against an integral that I'm not able to approximate using steepest descent (I don't have a large or small parameter), integration by parts, or any of the other common techniques. So I'd very much appreciate the input of any applied mathematicians!

The integral is $$f(x) = \int_{x}^{\infty} \frac{\Phi(t)}{t^{5}}dt$$ with $\Phi(t) = e^{i \pi t^{2} / 2}[C(t) + i S(t)]$. Here, $C(t)$ and $S(t)$ are the Fresnel integrals defined by $$C(t) + i S(t) = \int_{0}^{t} e^{i \pi u^{2} / 2} du\ .$$ What I really want is the behaviour of $f(x)$ for small $x$. But, the integral is formally divergent if $x = 0$.

Made a little progress with integration by parts, but I wasn't able to entirely separate my integral into convergent pieces.