Define
$\tilde U(\tau ,\omega ) = \frac{1}{{\Lambda (\tau ,\omega )}}\exp \left[ {i\int\limits_0^\tau {\left( {{{\left( {\frac{\omega }{{{\omega _h}}}} \right)}^{\frac{1}{{\pi Q(\tau ')}}}} - 1} \right)\omega } d\tau '} \right]$
$\Lambda (\tau ,\omega ) = \frac{{\beta (\tau ,\omega ) + {\sigma ^2}}}{{{{\left( {\beta (\tau ,\omega )} \right)}^2} + {\sigma ^2}}}$
$\beta (\tau ,\omega ) = \exp \left[ { - \int\limits_0^\tau {\frac{\omega }{{2Q(\tau ')}}{{\left( {\frac{\omega }{{{\omega _h}}}} \right)}^{\frac{{ - 1}}{{\pi Q(\tau ')}}}}} d\tau '} \right]$
In the above, $\tau$ is the time, and $\omega$ is the frequency, and $Q(\tau)$ is another function of time $\tau$.
The equations are for a filter applied to seismic data. The equation can be found on pg. 128 of this monograph.
I would like to take the inverse continuous Fourier transform of $\tilde U(\tau ,\omega )$ so that I get a function $\tilde U(\tau ,\tau )$ which is expressed only in terms of $\tau$.
This gets rid of $\omega$ so that the expression is only in the time domain.
I have tried using both a CAS and numerous attempts on paper, but I am uncertain as to whether this can be done using continuous mathematics. I am inclined to believe that it would be simpler to approximate $\tilde U(\tau ,\tau )$ using numerical methods. How to approach this problem?