Step & Impulse Functions

Real forcing is rarely smooth. A voltage gets switched on at a moment in time. A bell receives one sharp tap. Two idealized functions capture these: the unit step (Heaviside) u(t−a), which jumps from 0 to 1 at t = a, and the Dirac delta δ(t−a), an infinitely tall, infinitely narrow spike with area 1. Both have astonishingly clean Laplace transforms.