I'm having a problem with the inverse Fourier transform (IFT) of $\dfrac{g(\omega)}{j\omega}$, where $j=\sqrt{-1}$, and $\omega$ is the angular frequency.
It seems that the IFT is convolution of $g(t)$ and $0.5\operatorname{sign}(t)$, but the answer looks dependent on the value $g(0)$.
for example, if $g(\omega)\sim C\omega$ where $w\sim 0$ ($C$ is constant), my answer looks correct.
but if $g(\omega)=C$ where $w\sim 0$, some consideration of adding delta function seems to be needed.
Really thank you for answer.