what is the integral from 0 to infinity of (k^2/(sqrt(k^2+m^2)) * e^(-i*t*sqrt(k^2+m^2))?

Question

where m and t are positive integers and k is the variable. (* denotes multiplication)

Answer 1

Sh|t Howdy. I'd try it by parts. The first factor (which is a standard form) integrates (against dk) as:
k√(k²+m²)/2 - (m²/2)In(k + √(k² + m²))

Try it from there, but it's gonna be messy ☺


Doug

Answer 2

The integral diverges. It can be simplified a bit by setting r^2=k^2+m^2 which gives the integral from m to infinity of
sqrt(r^2-m^2)/(2r) exp(itr) dr.
This clearly diverges through oscillation since the square root is about the same as r as r->infty.
However, it may still be possible to interpret it as a distribution. If you double it, it is the Fourier transform of a bounded function (except for the integral from -m to m) so it looks like there will be a Dirac delta at the origin plus a sinc function plus some other function.