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Can someone tell me the reduction that the Fourier transform of $\frac{1}{x+iy}$ is $\frac{-2\pi i}{\omega_{x}+i\omega_{y}}$. I have tried rewriting $\frac{1}{x+iy}$ as $\frac{1}{x}(\sum_{k=0}^{\infty}(\frac{-iy}{x})_{k})$, but it is not a convergent sequence.

Saby123
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Ruin Riddle
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  • Note that this questions has been discussed (yet no accepted answer) here : https://math.stackexchange.com/questions/1533467/fourier-transform-of-frac1xiy – DominikS Oct 02 '23 at 09:37
  • Thank you very much! – Ruin Riddle Oct 02 '23 at 09:43
  • Are $x,y\in\mathbb R$? If so, you are essentially asking for $\mathcal F[1/z]$ with $z\in\mathbb C$, which would be a one-dimensional Fourier transform. – DominikS Oct 02 '23 at 09:47
  • Yes, $x, y \in \mathbb{R}$. Although I have little knowledge about the Fourier transform of a complex function, I have a urge to understand it. Would you mind giving me the definition of it? – Ruin Riddle Oct 02 '23 at 10:09
  • The definition of the Fourier transform naturally extends from real to complex arguments, see https://en.wikipedia.org/wiki/Fourier_transform#Complex_domain – DominikS Oct 02 '23 at 11:30
  • Another complication is that the aforementioned definition of the Fourier transform will not converge for your function, since the function has a non-integrable singularity at $(x, y) = (0, 0)$. To handle this, you will need to consider the extension of the F-Transform to tempered distributions, which will allow you to give meaning to the Fourier-transform of functions for which the integral does not converge. – DominikS Oct 02 '23 at 11:33
  • Tempered distributions is a relatively advanced topic (depending on your Maths education), if your are interested, this seems a fairly good overview of the topic: https://www.math.ucdavis.edu/~hunter/book/ch11.pdf – DominikS Oct 02 '23 at 11:40

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