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How to compare integral of damped oscillatory functions

  1. Mar 30, 2010 #1
    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 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, [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
     

    Attached Files:

  2. jcsd
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