Htykuut

Dear Math enthusiasts,

I'm trying to see if I can find an analytical expression for the Fourier transform of a Gaussian pulse $p(tau) = {rm e}^{-Btau^2}$ that is distorted by a hyperbolic distortion of the form $tau(t) = sqrt{t^2+a^2}-a$. What I mean is I look at the function $p(tau(t))$ and I try to transform this time $t$ to frequency.

This gives a Fourier integral of the following form $$G(omega) = int_{-infty}^infty {rm e}^{-Bleft(sqrt{t^2+a^2}-aright)^2} {rm e}^{-jmath omega t} {rm d}t.$$

Substituting the square-root doesn't help much, since it appears back when resubstituting $t$. I therefore tried to substitute $t = a sinh(x)$ instead. Assuming I didn't make a stupid mistake, this gives $$G(omega) = aint_{-infty}^infty {rm e}^{-Bleft(a cosh(x)-aright)^2} {rm e}^{-jmath omega a sinh(x)} cosh(x) {rm d}x.$$

From [*], eqn (2.3) I know that $int_{0}^infty {rm e}^{a cosh(x)} cosh(x) {rm d}x = K_0(a)$ where $K_0(a)$ is the modified Bessel function of the second kind. As the integrand is symmetric, it should be no problem to extend this to $(-infty,infty)$. But still I'm not sure this is enough to solve the integral. Mathematica has refused to give me anything, but it's possible I used it in a wrong way. I tried manipulating the exponent to bring everything to one hyperbolic trig function, but I failed.

Any hints how I can proceed?

edit: Note that since $p(tau(t))$ is even symmetric, $G(omega)$ is real-valued and even and we can get rid of complex numbers altogether, i.e., $$G(omega) = int_{-infty}^infty {rm e}^{-Bleft(sqrt{t^2+a^2}-aright)^2} cos( omega t) {rm d}t = 2 int_{0}^infty {rm e}^{-Bleft(sqrt{t^2+a^2}-aright)^2} cos( omega t) {rm d}t.$$ Does this make it easier? I'm not sure.