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Some questions concerning asymptotic expansions of integrals

  1. Mar 24, 2012 #1
    I've started self-teaching asymptotic methods, and I have some theoretic questions (and lots of doubts!).

    1. Say I have the asymptotic expansion

    [itex]f(x) \asymp \alpha \sum_n a_n x^{-n}[/itex]

    for [itex]x[/itex] large, where [itex]\alpha[/itex] is some prefactor.

    How can I estimate the value of [itex]n[/itex] for the term of least magnitude?

    2. Suppose I have the integral

    [itex]I(\lambda) = \int_{a}^{b} f(t) \exp{(i\lambda \phi(t))} dt [/itex],

    for [itex]\lambda[/itex] large.
    In the stationary phase method, if the function [itex]\phi(t)[/itex] has no stationary point in the interval [itex][a,b][/itex], am I wrong to believe that then [itex]I(\lambda)[/itex] is small beyond all orders (as the rapid oscillations of the phase imply cancellations)? How to formally derive the order of magnitude of [itex]I(\lambda)[/itex]?


    Thanks!
     
  2. jcsd
  3. Mar 24, 2012 #2

    Redbelly98

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    1. Two ways that come to mind are
    • If an is defined explicitly, then treat n as if it were a continuous variable and use normal calculus methods to find the minima and maxima of anx-n.

    • Or, solve for n:
      anx-n = an+1x-(n+1)

    2. You're correct that the integral would be small. Somebody else will have to chime in on how to figure out the order of magnitude.
     
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