I am looking for suggestions, literature, etc., about techniques and theorems useful for comparing improper integrals of functions characterised by a damped oscillatory behaviour.

But let me use the following example to introduce in simple terms what I actually mean.
Consider the function
<br />
f(x)= \frac{e^{i(1+x)}}{1+x}dx \; ,<br />
we have

\int^{\infty}_{0} f(x)dx=\int^{\infty}_{0} \; \frac{cos(1+x)}{1+x}dx \; + \; i \; \int^{\infty}_{0} \; \frac{sin(1+x)}{1+x}dx \; =
=\; Ci(1)\; + \; i \left(Si(\infty)-Si(1)\right) \; = \; Ci(1)\; + \; i \left(\frac{\pi}{2}-Si(1)\right)

where Si(x) and Ci(x) denote the Sine Integral and the Cosine Integral functions, respectively.
Thus, the above integral exists. Note that in this case evaluating
\int^{\infty}_{0} \left|f(x)\right|dx
would not have helped in verifying convergence (as it diverges).

In other words, looking at a plot of the imaginary component,
the positive and negative areas add up to a finite value.
Positive and negative areas of the real component also add up to a finite value.

Let us now define

g(x) = \frac{a+ix}{a+t+ix} \; \; \; a, t > 0

what happens when f(x) is multiplied by g(x) ?

 g(x) has the following properties

 |g(x)| < 1
 |g(x)| \rightarrow 1 \; as \; x \rightarrow \infty

while its argument satisfies

 phase(g(x)) > 0
 phase(g(x)) \rightarrow 0 \; as \; x \rightarrow \infty

The above observations suggest that
- the product  f(x)g(x) is asymptotic to  f(x) \; as \; x \rightarrow \infty
- |f(x)g(x)| < |f(x)| \; \; \forall x \in [o, oo)

a plot of the real and imaginary components (see the attached example, where the red line corresponds to \Re(f(x)), while the blue one represents \Re(f(x)g(x)) ), would therefore show that:
- as x \rightarrow \infty, \Re(f(x)g(x)) tends to overlap \Re(f(x)),
- before that, the amplitude of \Re(f(x)g(x)) appears "compressed" wrt \Re(f(x))
- \Re(f(x)g(x)) appears "compressed" wrt \Re(f(x)) also along the x axis (because of the phase contribution of g(x))
- similar considerations apply to \Im(f(x)) and \Im(f(x)g(x))

Thus, I would expect the following:

- \int^{\infty}_{0} f(x)g(x)dx \; also exists

- and perhaps \left|\int^{\infty}_{0} f(x)g(x)dx\right| \; < \; \left|\int^{\infty}_{0} f(x)dx\right|
(but I am not at all sure, as it depends on how said positive and negative areas, both smaller for the product f(x)g(x) , actually add up together ...)

However, I do not possess sufficient experience and skills to formalise in a more rigorous way said observations, neither I know whether similar results might apply to a more general class of "damped oscillatory" functions, f(x)=u(x)+iv(x) , characterised by the following properties:

- we only know (by other means) that \int^{\infty}_{0} f(x)dx \; exists

- and we also know that \int^{\infty}_{0} |f(x)g(x)|dx does NOT exist (so, such a convergence test would not help).

Anybody with suggestions about where I could find relevant literature on this subject?

Thank you