I try to solve the inverse Fourier transformation in two dimensional space of Gaussian function $e^{-tk^2}$
Started by the general form:
$\int \frac{d^2\vec{k}}{(2\pi)^2} e^{i\vec{k}\cdot\vec{r}} e^{-tk^2}$
where $t$ is a constant, and $k = \||\mathbf{\vec{k}}\||$
Then, I turn it into polar coordinate to obtain
$\frac{1}{(2\pi)^2} \int \int e^{ikr\cos \theta} e^{-tk^2} k dk d\theta$
However, I'm not sure how to solve this analytically since the exponent contains $\cos\theta$