Inverse Fourier transform of two variable function $F(k_x,k_y)=e^{ikz} e^{-ik_\rho ^2 z /2k}$

I am trying to find the inverse Fourier transform of: $$F(k_x,k_y)=e^{ikz} e^{-ik_\rho ^2 z /2k},$$ where $k^2 = k_x^2 +k_y^2 +k_z^2 = k_\rho ^2 +k_z^2$ is a constant.

I am getting confused as to how to go about this. Inverse fourier transform of $F(k_x)$ would be given by $\int F(k_x)e^{-ik_x x} \frac{dk_x}{2\pi}$? How do I change this when there is a function of two variables?

Furthermore, could anyone explain what is meant by the prefactor of the exponential?

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• This thesis chapter has a nice discussion on higher dimensional Fourier transforms. – Chris Mueller Feb 13 '14 at 20:59
• Is the $z$ in the exponential just a constant? – JeffDror Feb 13 '14 at 22:02
• I guess so but i'm not sure... the question doesn't state – user1887919 Feb 14 '14 at 9:30