# Complex Fourier Transform of $1/z^2$

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!

• Are you trying to evaluation this integral $$\int_{-\infty}^\infty \int_{-\infty}^\infty \frac{e^{-i2\pi(k_xx+k_yy)}}{k_x^2+k_y^2}\,dk_x\,dk_y=2\pi \int_0^\infty \frac{J_0(2\pi k)}{k}\,dk$$It diverges. – Mark Viola May 27 '18 at 13:50
• @MarkViola, to be honest - Im not sure: the task simply said: compute the Fourier transform of $1/z^2$ for complex z. To define it as an integral over the complex plane was the only possibility that came to my mind! Do you think there could be another way ? (recently we treated basic fourier transform, distributions and the Hilbert transform) – Simonsays May 27 '18 at 14:03