So here it is my problem: I need the Fourier Transform of the tangent function, namely:
$$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} \tan(x)\ e^{-ikx}\ dx$$
I tried for hours by hands and I got literally nothing.
I have W. Mathematica v. 11.01 on my computer, but after having run the command it took like forever (I let it run for $40$ minutes at least) then I aborted the operation (it was still running).
Probably my computer has not the necessary amount of power to compute it.
Could anyone of you try, please?
Unless there is some tricky trickilicious trick by which one can compute it by hands...
P.s. The integral is divergent, I need either a principal value treatise or a sort of regularization".
In terms of generalized functions, the tangent function has representation
$$\begin{align} \tan(x)\sim 2\sum_{n=1}^\infty (-1)^{n-1}\sin(2nx) \end{align}$$
Then, using the Fourier Transform
$$\mathscr{F}\{1\}=\frac1{\sqrt{2\pi}}\int_{-\infty}^\infty (1)e^{-ikx} \,dx=\sqrt{2\pi} \delta(k)$$
we find that
$$\mathscr{F}\{ \tan\}(k)=i\,\sqrt{2\pi } \sum_{n=1}^\infty (-1)^{n-1}\left(\delta(k+2n)-\delta(k-2n)\right)$$