Approximation to divergent integral

Question

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.

Answer 1

First, cut off the tail towards infinity:

$$f(x) = \int_{x}^1 \frac{\Phi(t)}{t^5} dt + \int_1^{\infty} \frac{\Phi(t)}{t^5} dt.$$

The second term is a constant, so you can compute it numerically once and for all.

Write $$e^{i \pi u^2/2} = 1 + \frac{i \pi}{2} u^2 +R(u)$$ and $$\int_{0}^t e^{i \pi u^2/2} du = t + \frac{i \pi}{6} t^3 + \int_{0}^t R(u) du.$$

So $$\frac{\Phi(t)}{t^5} = \left( t^{-4} + \frac{i \pi}{6} t^{-2} + t^{-5} \int_{0}^t R(u) du \right) \left( 1 + \frac{i \pi}{2} t^2 + R(t) \right)=$$ $$t^{-4} + \frac{2 \pi i}{3} t^{-2} + \left( t^{-4} R(t) - \frac{\pi^2}{12} + \int_{0}^t R(u) du \right).$$

So $$\int_{x}^1 \frac{\Phi(t)}{t^5} dt = \frac{1}{3}\left( x^{-3} - 1 \right) + \frac{2 \pi i}{3} \left( x^{-1} -1 \right) + \int_{x}^1 \left( t^{-4} R(t) - \frac{\pi^2}{12} + \int_{0}^t R(u) du \right) du.$$

The integrands in the last term are bounded functions, and they are being integrated over bounded domains, so there is no problem approximating them numerically.

If you want an asymptotic formula, instead of a numerical approximation, you should be able to keep taking more terms out to get a formula like $$f(x) = \frac{1}{3} x^{-3} + \frac{2 \pi i}{3} x^{-1} + C + a_1 x + a_2 x^2 + \cdots + a_n x^n + O(x^{n+1}) \quad \mathrm{as} \ x \to 0.$$ You probably won't be able to get the constant $C$ in closed form, because it involves all those convergent integrals. The other $a_i$ will be gettable in closed form, although they will get worse and worse as you compute more of them.