Inverse Fourier Transform of $\sqrt {-iw} \mathcal F(w)$

Question

If I use the following set of Fourier transforms:

$$\mathcal F(w) = \int_{0}^T dtf(t)e^{iwt},$$

$$ f(t) = \frac{1}{2\pi} \int_{-w_N}^{w_N} dw\mathcal F(w)e^{-iwt},$$

where $f(t)$ is an arbitrary time domain signal with total duration $T$, $w= 2\pi f$ is angular frequency, and $w_N$ is the Nyquist angular frequency. The underlying assumption here is that the signal $f(t)$ is casual in the sense that it has a zero value for $t \leq 0$.

Now, what is the inverse Fourier Transform of $\sqrt {-iw} \mathcal F(w)$:

${}$ $$\mathcal F^{-1} (\sqrt {-iw} \mathcal F(w))=\quad?$$

${}$

Any help is appreciated.

Answer 1

Hint. Refer Table of Fourier/Inverse Fourier transform. And make it a good habit to write the differential after the integrand.