I'm trying to evaluate the following inverse Fourier transform:
$$\frac{1}{4 \pi^2} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(\bar{k}, \bar{l}) e^{\iota( \bar{k} \bar{X}+\bar{l} \bar{Y})} d \bar{k} d \bar{l}$$
where $\bar{k}$ and $\bar{l}$ are real. I'm thinking of first integrating with respect to $\bar{k}$. To do that, I'm deforming the contour in complex $\bar{k}$ plane and then I'll apply the Residue theorem. When I'm finding the poles of $f(\bar{k}, \bar{l})$, what value I should choose for $\bar{l}$ ?