Page 109 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

INTEGRAL TRANSFORMS: THE LAPLACE TRANSFORM 99

1

k t

FIGURE 6.1: The Heaviside step

6.2.7 Heaviside step
A frequently useful function is the Heaviside step function, defined as

U k (t) = 0 0 < t < k
(6.18)
= 1 k < t
It is shown in Fig. 6.1.
The Laplace transform is

∞
1
L[U k (t)] = e −st dt = e −ks (6.19)
s
t=k
The Heaviside step (sometimes called the unit step) is useful for finding the Laplace transforms
of periodic functions.

Example 6.4 (Periodic functions). For example, consider the periodic function shown in
Fig. 6.2.
It can be represented by an infinite series of shifted Heaviside functions as follows:

∞

n
f (t) = U 0 − 2U k + 2U 2k − 2U 3k + ··· = U 0 + (−1) 2U nk (6.20)
n=1

1

-1

FIGURE 6.2: A periodic square wave