How to compare integral of damped oscillatory functions

[Luca]

Hello,
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
[tex]
f(x)= \frac{e^{i(1+x)}}{1+x}dx \; ,
[/tex]
we have

[tex]\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 \; = [/tex]
[tex]=\; Ci(1)\; + \; i \left(Si(\infty)-Si(1)\right) \; = \; Ci(1)\; + \; i \left(\frac{\pi}{2}-Si(1)\right) [/tex]

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

In other words, looking at a plot of the imaginary component,
[tex]\frac{sin(1+x)}{1+x}[/tex],
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

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

what happens when [tex]f(x)[/tex] is multiplied by [tex]g(x)[/tex] ?

[tex] g(x) [/tex] has the following properties

[tex] |g(x)| < 1 [/tex]
[tex] |g(x)| \rightarrow 1 \; as \; x \rightarrow \infty [/tex]

while its argument satisfies

[tex] phase(g(x)) > 0 [/tex]
[tex] phase(g(x)) \rightarrow 0 \; as \; x \rightarrow \infty [/tex]

The above observations suggest that
- the product [tex] f(x)g(x) [/tex] is asymptotic to [tex] f(x) \; as \; x \rightarrow \infty [/tex]
- [tex]|f(x)g(x)| < |f(x)| \; \; \forall x \in [0, \infty ) [/tex]

a plot of the real and imaginary components (see the attached example, where the red line corresponds to [tex]\Re(f(x))[/tex], while the blue one represents [tex]\Re(f(x)g(x))[/tex] ), would therefore show that:

- as [tex]x \rightarrow \infty[/tex], [tex]\Re(f(x)g(x))[/tex] tends to overlap [tex]\Re(f(x))[/tex],
- before that, the amplitude of [tex]\Re(f(x)g(x))[/tex] appears "compressed" wrt [tex]\Re(f(x))[/tex]
- [tex]\Re(f(x)g(x))[/tex] appears "phase shifted" wrt [tex]\Re(f(x))[/tex] along the x-axis (because of the phase contribution of [tex]g(x)[/tex])
- similar considerations apply to [tex]\Im(f(x))[/tex] and [tex]\Im(f(x)g(x))[/tex]

Thus, I would expect the following:

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

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

However, I do not possesses 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, [tex]f(x)=u(x)+iv(x) [/tex], characterised by the following properties:

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

- and we also know that [tex]\int^{\infty}_{0} |f(x)g(x)|dx [/tex] 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
Luca

 

[jvicens]

Hello Luca,

Thank you for your post and for providing such a detailed example of the type of problem you are looking to solve. Comparing improper integrals of damped oscillatory functions can be a complex and challenging task, but there are certainly techniques and theorems that can be useful in this area.

One technique that may be helpful is the use of Fourier analysis. By decomposing your function into its Fourier components, you can gain a better understanding of its oscillatory behavior and how it may interact with other functions. This can also help in determining convergence or divergence of the integral. Additionally, there are specific theorems, such as the Riemann-Lebesgue lemma, that can be applied to improper integrals involving oscillatory functions.

Another approach is to use complex analysis. This involves looking at the complex plane and using techniques such as contour integration to evaluate integrals involving damped oscillatory functions. This can also be helpful in determining convergence or divergence, as well as in finding the exact value of the integral.

In terms of literature, I would recommend looking into books on Fourier analysis, complex analysis, and special functions. These topics are often covered in advanced calculus or mathematical analysis courses, and there are many textbooks available on these subjects. You may also find relevant research articles by searching for keywords such as "improper integrals," "damped oscillatory functions," and "Fourier analysis."

I hope this helps guide you in your research and I wish you the best of luck in your studies. Please feel free to follow up with any further questions or clarifications.