Fourier Transform Step Function < EXTENDED - 2024 >

[ u(t) = \begincases 0, & t < 0 \ 1, & t > 0 \endcases ]

At first glance, finding its Fourier transform seems impossible. The Fourier transform of a function ( f(t) ) is: fourier transform step function

[ \lim_\alpha \to 0^+ \frac1\alpha + i\omega = \frac1i\omega ] [ u(t) = \begincases 0, & t &lt;

[ u(t) = \lim_\alpha \to 0^+ e^-\alpha t u(t), \quad \alpha > 0 ] [ u(t) = \begincases 0