Fourier Transform Step Function (FAST ⚡)

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

[ F(\omega) = \int_-\infty^\infty f(t) e^-i\omega t dt ] [ u(t) = \lim_\alpha \to 0^+ e^-\alpha t

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

[ \boxed\mathcalFu(t) = \pi \delta(\omega) + \frac1i\omega ]