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Analytic Approximation for an Oscillatory Integral

  1. Apr 4, 2014 #1
    I'm looking for a way to write down an analytic approximation for the following integral:

    [tex]\int_0^\infty \frac{k \sin(kr)}{\sqrt{1+v^2(k-k_F)^2}}dk[/tex]

    Let's assume that v kF >> 1, so that the the oscillating piece at large k doesn't contribute much uncertainty. Ideas? Thus far, Mathematica has failed me, though I have been able to generate some numerical solutions. Is there some way to take advantage of the fact that the integrand is peaked at kF?
  2. jcsd
  3. Apr 5, 2014 #2
    Code (Text):
    In[1]:= $Assumptions = v > 0 && kf > 0 && r > 0;
     Integrate[k /Sqrt[1 + v^2 (k - kf)^2], {k, 0, 2 kf}]

    Out[1]= (2 kf ArcSinh[kf v])/v
    Your integral will be strictly less than that, but at least it gives a closed form upper bound and it doesn't depend on substituting in some arbitrary values for constants. The only lower bound I can see from this would be negating the upper bound. That also doesn't integrate out to infinity, but from a few numerical examples it looks like by the time you are out to 2kf or some other appropriate multiple of kf or v kf that, as you have mentioned, you are far from the peak and the additional contribution may be modest.

    You can substitute a variety of reasonable constants and plot your expression and the expression without the Sin and see how these behave.
  4. Apr 7, 2014 #3
    The integral I am trying to solve is a version of a Fourier transform, so it would be better if the approximation were r-dependent.

    I was thinking that integrating by parts twice would sharpen the 1/√(...) piece and make it look like a delta-function. Then the integration would be easy if I constrained "r" to be sufficiently small (and if the boundary terms don't diverge). Still, I think I'm stuck in terms of providing an approximation at large "r," which is where I might be most interested in the integral's value...
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