Analytic Approximation for an Oscillatory Integral

[csmallw]

I'm looking for a way to write down an analytic approximation for the following integral:

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

Let's assume that v k_F >> 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 k_F?

 

[Bill Simpson]

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.

 

[csmallw]

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...