Why is the following Fourier-operation, involving $\mathcal{F(\delta)}$ and $\mathcal{F(\frac{dq}{dt})}$ legitimate?

Question

How can we deduct $iwQ(w)+Q(w)=\frac{1}{\sqrt{2\pi}}$ ?

Answer 1

From the first equation you have

$$\mathcal{F}\{q(t)\}+\mathcal{F}\left\{\frac{dq(t)}{dt}\right\}=\mathcal{F}\{\delta(t)\}\tag{1}$$

With $\mathcal{F}\{q(t)\}=Q(\omega)$ and with the differentiation property of the Fourier transform you get

$$\mathcal{F}\left\{{\frac{dq(t)}{dt}}\right\}=i\omega Q(\omega)$$

Finally, with $\mathcal{F}\{\delta(t)\}=1/\sqrt{2\pi}$ (1) can be rewritten as

$$Q(\omega)+i\omega Q(\omega)=\frac{1}{\sqrt{2\pi}}$$