Heaviside step function
/ HEV-ee-syde /
How do you write down, in one symbol, the moment a switch is flipped — off before some instant, on ever after? The Heaviside step function is exactly that idealised switch: a function that is 0 for negative argument and 1 for positive argument, jumping instantaneously from one level to the other at the origin.
Written H(x) (or u(x)), it is defined by H(x) = 0 for x less than 0 and H(x) = 1 for x greater than 0, with the value at the jump x = 0 chosen by convention (often 1/2). A shifted copy H(x minus a) turns on at x = a, so combinations like H(x minus a) minus H(x minus b) cut out a clean rectangular pulse that is 1 only between a and b — the standard way to express a force, voltage, or source that acts only during a finite window. Although H has a jump and so is not differentiable in the ordinary sense, in the language of distributions its derivative is the Dirac delta: H-prime(x) = delta(x), capturing the idea that the instantaneous rate of a sudden jump is an infinite spike of unit total.
The step function is the everyday vocabulary of switching: it models a circuit turned on at t = 0, a load suddenly applied to a beam, a drug dose administered at a fixed time, or a boundary condition imposed abruptly. It is indispensable in the Laplace transform, where the second shifting theorem uses H to handle inputs that start at a later time, and it pairs with the Dirac delta as the integral-and-derivative duo at the foundation of generalised functions.
A force that switches on at t = 2 and off at t = 5 is written F(t) = F_0 times ( H(t minus 2) minus H(t minus 5) ): the two steps build a single rectangular pulse of height F_0 living only on the interval from 2 to 5.
Differences of shifted steps are the standard building blocks for forces and signals that act only over a finite window.
The value H(0) is a matter of convention and rarely affects an integral; the deeper point is that H is differentiable only in the distributional sense, where H-prime = delta.