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
where and denote the Sine Integral and the Cosine Integral functions, respectively.
Thus, the above integral exists. Note that in this case evaluating
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
what happens when is multiplied by ?
has the following properties
while its argument satisfies
The above observations suggest that
- the product is asymptotic to
a plot of the real and imaginary components (see the attached example, where the red line corresponds to , while the blue one represents ), would therefore show that:
- as , tends to overlap ,
- before that, the amplitude of appears "compressed" wrt
- appears "compressed" wrt also along the x axis (because of the phase contribution of )
- similar considerations apply to and
Thus, I would expect the following:
- also exists
- and perhaps
(but I am not at all sure, as it depends on how said positive and negative areas, both smaller for the product , 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, , characterised by the following properties:
- we only know (by other means) that exists
- and we also know that does NOT exist (so, such a convergence test would not help).
Anybody with suggestions about where I could find relevant literature on this subject?