Complex Fourier Transform of $1/z^2$

Question

I try to compute explicitly the Fourier transform of $f: z \mapsto 1/z^2 \in \mathbb{C} $ as a transform in $\mathbb{C}$, i.e.
$$\mathcal{F}(f)(z)=\int_{\mathbb{C}}e^{-i2\pi\xi z}\frac{1}{\xi^2}d\xi$$ my attempts using polar coordinates and splitting the integral in real and imaginary part failed so far. I only treated Fourier transforms of functions defined on $\mathbb{R}^d$ and I did not find any suitable literature treating complex cases. Any help appreciated!