Long story short: I'm looking for a good way of showing that the Fourier transform of $1/x$ is a sign function.
Motivation and why this has been a problem:
I'm dealing with an equation similar to the Burgers equation in the form:
$$u_t = \varepsilon u_{xx} - u\cdot u_x$$
with initial conditions
$$u(0,x)=g'(x)$$
where $u_x$ denotes $\frac{\partial u}{\partial x}$.
In order to better understand my problem, I was looking at different ways of dealing with the Burgers equation.
It is well known that we can write the solutions of this equation as
$$u(t,x)= -2\varepsilon\sum_{j=1}^{2N}\frac{1}{x-z_j(t)}$$
where $z_j(t)$ are complex poles of the solution. In order to ensure the solutions will be real-valued, the poles are chosen such that
$$z_j = z^*_{j+N}$$ for $j=1,\dots,N$.
The problem arises when we look for periodic solutions of this equation. The literature I found about it suggests looking at periodic solutions of period $2\pi$ written as:
$$u(t,x)= -2\varepsilon\sum_{k=-\infty}^{\infty}\sum_{j=1}^{2N}\frac{1}{x-z_j(t)+2\pi k}$$
and then using the Poisson summation formula to rewrite this as a cotangent. This is the problem, I'm struggling to find literature about this representation.
The Poisson formula is:
$$\sum_{k=-\infty}^{\infty}f(x+2\pi k)=\sum_{n=-\infty}^{\infty}\hat{f}(n)e^{2\pi inx}$$
So the main problem is to define the Fourier transform of $\frac{1}{x}$, which all the literature indicates should be a sign function. The best hint I've found on the problem was to consider the function
$$G(x,\alpha)=\left\{\array{e^{-\alpha x}, \hspace{1cm} x>0 \\ -e^{\alpha x}, \hspace{1cm} x<0 }\right.$$
Compute the Fourier transform of the function $G$ and then take the limit for $\alpha\rightarrow 0$. This would show that the Fourier transform of the of the sign function is $\frac{1}{x}$ in a sense.
My problem with it is that I can't seem to justify that the transform of the limit is the limit of the transform in this case. Moreover, even if it is true, this calculation doesn't yield a good definition for $\hat{f}(0)=sign(0)$, which is needed for the Poisson formula.
I would have simply have given up on a strong definition of this transform if it weren't for the multiple sources claiming that it exists and equals the sign function (but lacking a thorough proof).
Does anyone know of a good reference where I could read about it?
