Fourier Transform Step Function Info

Now, take the limit as ( \alpha \to 0^+ ):

[ \int_0^\infty e^-\alpha t e^-i\omega t dt = \int_0^\infty e^-(\alpha + i\omega) t dt = \frac1\alpha + i\omega ] fourier transform step function

[ \lim_\alpha \to 0^+ \frac1\alpha + i\omega = \frac1i\omega ] Now, take the limit as ( \alpha \to

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

confirming the result. | Function | Fourier Transform | |----------|------------------| | ( u(t) ) (unit step) | ( \pi\delta(\omega) + \frac1i\omega ) | | ( \textsgn(t) ) (sign) | ( \frac2i\omega ) | | Constant ( 1 ) | ( 2\pi\delta(\omega) ) |