fourier transform and principal values

Question

Fourier transform and principal values

Can anyone tell me from how can i get the fouries transformation of prinicipal value of (1/x)

$$p.v\int \frac{1}{x}\Bigg(\int e^{-wix}\varphi(w)dw\Bigg)dx$$

Answer 1

Let for simplicity $T=p.v.(1/x)$. We know that $T$ solves - in the sense of distributions - the equation $$xT=1.$$ Take the Fourier trasform of both parts ($1$ is a tempered distribution, and so is $T$) to obtain

$$F[T]' = F[ixT] = F[i] = i\delta_0.$$ The antiderivative of both parts yield $$F[T]=C+iH$$ with $H$ being the Heaviside function. Now we can say that $T$ is an odd distribution, hence so is $F[T]$, therefore $$F[T] = \frac{i}{2}sgn(x).$$

Note that the actual multiplicative constant in front of $sgn(x)$ depends on your conventions for the Fourier transform.