I would like to compute the Fourier transform of the Heaviside function. To do this, I want to use the fact that : $$ F(vp \frac{1}{x}) = -2i\pi H+i\pi $$ where $F$ is the Fourier transform operator and $H$ is the Heaviside step function.
I know that this expression is correct but my problem is that I don't know how to find the correct expression of $F(H)$ from there.
Here is what I get : $$2i\pi H = F(vp\frac{1}{x})+i\pi \Leftrightarrow F(H)=-\frac{1}{2i\pi}F\bigg(F(vp\frac{1}{x})\bigg)+\frac{1}{2}\delta$$ where delta is the dirac distribution.
I know the correct final expression is: $$F(H)=\frac{1}{2i\pi}vp\frac{1}{x}+\frac{1}{2}\delta$$
But I don't really see how to obtain this final expression.
Take the Fourier transform of both sides of $F(vp \frac{1}{x}) = -2i\pi H+i\pi.$
Then use the fact that $F(F(f(x))) = [2\pi]f(-x)$ (whether the factor $2\pi$ is there or not depends on your exact definition of the Fourier transform).