Often referred to as the
unit step function or just the
step function, the Heaviside function was dreamed up by the English
electrical engineer Oliver Heaviside while he was developing practical
Laplace transformation techniques. It is usually denoted as:
u(t-a) = { 1 for t > a (a >= 0)
{ 0 for t < a
Not only is this function useful for defining other
piecewise functions without using the space-consuming piecewise
notation (eg, if a function is equal to sin(t) for t between 1 and 2, but is zero elsewhere, we can write f(t) = (u(t-1)-u(t-2))*sin(x)), it also simplifies certain
Laplace transformations and
inverse transforms as a result of the
second-shifting theorem:
L{f(t-a)u(t-a)} = exp(-as)*F(s) (where F(s) = L{f(t)})
f(t-a)u(t-a) = L-1{exp(-as)*F(s)}