I'm currently working on this paper: http://web.calstatela.edu/faculty/rcooper2/article.pdf and I want to proof Lemma 3.0 in the case of $n=2$, on page 441. It seems that the Ph.D. thesis the author refers to is unavailable.
Thus, I have to calculate the Inverse Fourier Transform of $\Gamma^{-1}(\xi_o+i\tau,\xi_1)$, where $\Gamma(\xi_o+i\tau,\xi_1)=\sqrt{c_s^{-2}(\xi_o+i\tau)^2-\xi_1^2}$ and $\xi_0$ is the Fourier variable in time and $\xi_1$ is the Fourier variable in space i.e. I have to calculate:
$\mathcal{F}^{-1}((c_s^{-2}(\xi_o+i\tau)^2-\xi_1^2)^{-\frac{1}{2}})(x_0,x_1)=\frac{1}{(2\pi)^2}\int \limits_{\mathbb{R^2}} e^{ix_o\xi_0+ix_1\xi_1}(c_s^{-2}(\xi_o+i\tau)^2-\xi_1^2)^{-\frac{1}{2}} d\xi_0d\xi_1$
Any help would be greatly appreciated.