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Math Help - Numerical Integration of Fourier Transform.

  1. #1
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    Numerical Integration of Fourier Transform.

    Hi Everyone,
    I have the complicated function in Fourier Domain. I want to find the inverse fourier of the complicated function. This function is a vector potential of the electrodynamic wheel for magnetic levitation in Fourier domain. I am trying to get the function in the rectangular coordinate system, f(x). F(w) is a function in fourier domain. I want to find the function f(x) which is a fourier inverse of F(w).
    I tried Mathematica, Maple and MATLAB to get the straight foruier inverse, but could not get the result.
    I am wondering if there is any way to get the inverse fourier of F(w). I did not get any idea , so I am thinking of integrating it by Numerical Technique. But, I still do not know which of the numerical technique works well for the highly oscillatory function like this (i.e. For the Integrand: 1/(2*pi)*F(w)*exp(i*w*x)).

    F(w)= [exp(t/2*w)*w^4*heaviside(-w)*cosh(r*y)] / [w*cosh(r*t/2)+r*sinh(r*t/2)]

    Where, r=sqrt(w^2+i*mu(we+v*w)), here r is a function of w as well.
    mu=permeability of the surface
    y=distance along y-axis, we=angular velocity of the wheel, t=thickness of the guide-way, v=translational velocity of the wheel, x=distance in x-axis (which is a variable)
    Now,
    f(x)=Integration(-Inf,Inf,1/(2*pi)*F(w)*exp(i*w*x)*dw)

    Could you please suggest me some ideas to find the Inverse Fourier of F(w) or Integrate by Numerical Technique (suggest me the best method for highly oscillatory Integrand).

    Thank you !!!
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  2. #2
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    Joined
    Aug 2008
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    You may wish to review the section "Advanced Numerical Integration" in Mathematica. One method is "DoubleExponentialOscillatory".

    If you post the numeric values of all the parameters, and a range for x, I'll try to numerically integrate it and create an interpolation function f(x) for the chosen range.
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