Now, take the limit as ( \alpha \to 0^+ ):
The Fourier transform of the step function is a classic example of how generalized functions (distributions) like the delta function allow us to include non-convergent but physically meaningful signals into the frequency domain framework. fourier transform step function
[ u(t) = \begincases 0, & t < 0 \ 1, & t > 0 \endcases ] Now, take the limit as ( \alpha \to
[ u(t) = \frac12 + \frac12 \textsgn(t) ] & t <
[ \boxed\mathcalFu(t) = \pi \delta(\omega) + \frac1i\omega ]