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
f(x)= \frac{e^{i(1+x)}}{1+x}dx \; ,
we have

$\displaystyle \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 \; = $
$\displaystyle =\; Ci(1)\; + \; i \left(Si(\infty)-Si(1)\right) \; = \; Ci(1)\; + \; i \left(\frac{\pi}{2}-Si(1)\right) $

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

In other words, looking at a plot of the imaginary component,
$\displaystyle \frac{sin(1+x)}{1+x}$,
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

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

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

$\displaystyle g(x) $ has the following properties

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

while its argument satisfies

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

The above observations suggest that
- the product $\displaystyle f(x)g(x) $ is asymptotic to $\displaystyle f(x) \; as \; x \rightarrow \infty $
- $\displaystyle |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 $\displaystyle \Re(f(x))$, while the blue one represents $\displaystyle \Re(f(x)g(x))$ ), would therefore show that:
- as $\displaystyle x \rightarrow \infty$, $\displaystyle \Re(f(x)g(x))$ tends to overlap $\displaystyle \Re(f(x))$,
- before that, the amplitude of $\displaystyle \Re(f(x)g(x))$ appears "compressed" wrt $\displaystyle \Re(f(x))$
- $\displaystyle \Re(f(x)g(x))$ appears "compressed" wrt $\displaystyle \Re(f(x))$ also along the x axis (because of the phase contribution of $\displaystyle g(x)$)
- similar considerations apply to $\displaystyle \Im(f(x))$ and $\displaystyle \Im(f(x)g(x))$

Thus, I would expect the following:

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

- and perhaps $\displaystyle \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 $\displaystyle f(x)g(x)$ , 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, $\displaystyle f(x)=u(x)+iv(x) $, characterised by the following properties:

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

- and we also know that $\displaystyle \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