Right now I'm solving a physical problem, but I faced with some mathematical difficulties. I'm having a problem with taking Fourier integral of $\frac{1}{|x|}$. It seems to me that I'm doing it too straightforward.
$$\int\limits_{-\infty}^{+\infty}\frac{e^{-ixk}}{|x|}dx = \int\limits_{0}^{+\infty}\frac{e^{ixk}}{x}dx + \int\limits_{0}^{+\infty}\frac{e^{-ixk}}{x}dx = 2\int\limits_{0}^{+\infty}\frac{\cos kx}{x}dx = -2\operatorname{Ci}(\xi = 0).$$
where $\xi = kx$. Here $k$ is a wave vector, $x$ is a cartesian coordinate. And cosine integral seemes to be appeared. I know that it is divergent in 0, but from some physical reasons one can put it $\xi = \xi_0 \ll1$ to make it converge.
I used Mathematica 10 to check it (I used InverseFourierTransform), but what I got is this: $$\int\limits_{-\infty}^{+\infty}\frac{e^{-ixk}}{|x|}dx = -2 (\gamma + \ln|k|).$$
The RHS looks similar to cosine integral but it's not.
P.S. The initial integral is divergent. What am I s'posed to do? Could you help me out?