Inverse Fourier transform of $\frac{jwL}{R+jwl}$

Question

I am trying to find the inverse Fourier transform of:

$$\frac{jwL}{R+jwl}$$

My current attempt is

$$\mathcal{F^{-1}(\frac{jwL}{R+jwl})}$$ $$\mathcal{F^{-1} ({jwL})} \oplus \mathcal{F^{-1}(\frac{1}{R+jwL}}) $$ $$L.\mathcal{F^{-1} ({jw}}) \oplus \frac{1}{L}.\mathcal{F^{-1}(\frac{1}{\frac{R}{L}+jw}}) $$ $$ L.\mathcal{F^{-1} ({jw}}) \oplus \frac{1}{L}.u(t).e^{\frac{Rt}{L}} $$

I get stuck with $$ \mathcal{F^{-1} ({jw}}) $$ as $$\frac{1}{2 \pi} \int_{-\infty}^\infty jwe^{jwt} $$ doesn't converge

Answer 1

Recall that:

1. $$\mathcal{F}(e^{-\alpha t}u(t)) = \frac{1}{\alpha + j\omega};$$
2. $$\mathcal{F}(\delta(t)) = 1.$$

Notice that: $$\frac{j\omega L}{R+j\omega L} = \frac{j\omega L + R - R}{R+j\omega L} = 1- \frac{R}{R+j\omega L}.$$

The inverse Fourier transform of is:

$$\delta(t) - \frac{R}{L}e^{-\frac{Rt}{L}}u(t).$$