Question

Find Fourier integral of 𝑓(𝑥) = 𝐻(𝑥 − 1) − 𝐻(𝑥 − 2)

Answer 1

$$\dfrac{1}{\sqrt{2\pi}}\displaystyle\int_{-\infin}^\infin(H(t-1)-H(t-2))e^{iwt}dt=$$

$$=\dfrac{1}{\sqrt{2\pi}}\delta(w)+\dfrac{ie^{iw}}{\sqrt{2\pi }w}-\big(\dfrac{1}{\sqrt{2\pi}}\delta(w)+\dfrac{ie^{2iw}}{\sqrt{2\pi }w}\big)=$$

$$=\dfrac{ie^{iw}(1-ie^{iw})}{\sqrt{2\pi }w}$$