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Solving or approximating a special class of fourier transforms

  1. Apr 17, 2012 #1
    I have a large class of fourier transform integrals that I would like to solve, approximate, or just get some insight into how they work. They take the form:

    [itex]\int exp(j*θ(x))exp(-j*ω*x)dx[/itex]

    with the restriction that θ(x) is real.

    Now this general class is very hard, but perhaps someone has some insight to it...?

    Let's consider a specific example which is of interest to me:

    [itex]\int exp(j*cos(x))exp(-j*ω*x)dx[/itex]

    So it's the fourier transform of a sinusoidally-rotating rotating exponential...anybody have any clues how to solve or get a functional approximation for the result?

    Anyways, just thought I'd post here and see if anyone has encountered similar problems.

    This is being used to compute the near-to-far-field transformation of a special kind of holograms.
     
  2. jcsd
  3. Apr 17, 2012 #2

    Mute

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    You may be able to do some asymptotic approximations. See the book by Bender and Orszag, "Advanced Mathematical methods for scientists and engineers - asymptotic methods and perturbation theory". Specifically, refer to chapter 6.
     
  4. Apr 17, 2012 #3
    I will look into it - thanks for the suggestion!
     
  5. Apr 17, 2012 #4
    http://www.infoocean.info/avatar1.jpg [Broken]You may be able to do some asymptotic approximations.
     
    Last edited by a moderator: May 5, 2017
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