Let the function $$f(\mathbf{r})=\int_{\Omega }e^{i\mathbf{k} \cdot \mathbf{r}}d^2\mathbf{k}$$, where $\mathbf{k} ,\mathbf{r}\in\mathbb{R}^2$, and $\Omega \subset \mathbb{R}^2$ is some finite region with a boundary $\partial \Omega$.
Without knowing the analytical form of $f(\mathbf{r})$, how do we analyze the asymptotic behaviour of $f(\mathbf{r})$ at large distance $\left | \mathbf{r} \right |\rightarrow \infty $? Does it behave like $f(\mathbf{r})\sim e^{-\left | \mathbf{r} \right |/L}$ or $f(\mathbf{r})\sim \left | \mathbf{r} \right |^{-a}$ at large distance, with $L$ the characteristic length and $a$ some positive number?
Thanks in advance.