Solving or approximating a special class of fourier transforms

[mattmisk]

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:

$\int exp(j*θ(x))exp(-j*ω*x)dx$

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:

$\int exp(j*cos(x))exp(-j*ω*x)dx$

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.

 

[Mute]

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.

 

[mattmisk]

I will look into it - thanks for the suggestion!

 

[laurent711]

http://www.infoocean.info/avatar1.jpg You may be able to do some asymptotic approximations.