How to Calculate the Fourier transform of $\frac{1}{x-c}$?

Question

How to Calculate the Fourier transform of $\frac{1}{x-c}$? $c$ here is a complex number which does not lie on the real line. First I should figure out how to interpret this function as a tempered distribution. I suppose that it should be a $L^2$ function because its singularity is not on the real line and $\frac{1}{x^2}$ is integrable with $0$ removed. Is that correct? If so, how to calculate its Fourier transform?

Answer 1

Your correct that $\frac 1{x-c}\in L^2(\mathbb R)$. Also, notice that if $a\in \mathbb C- i\mathbb R$, with $\Re (a)>0$, $$\int _{0}^{+\infty}e^{2i\pi x\xi} e^{-a\xi}d\xi=-\frac {1}{2i\pi x -a}$$ So you should be able to finish :-)