Lecture Notes in Differential Equations - Bruce E. Shapiro

344 LESSON 32. THE LAPLACE TRANSFORM
a stairway. It is possible to define more complicated piecewise continuous
functions in terms of step functions.
Definition 32.11. The Unit Step Function with step at origin, U(t)
is defined by
{ 0, t < 0
U(t) =
(32.106)
1, t ≥ 0
The Unit Step Function with Step at t 0 , U(t − t 0 ) is then given by
{
0, t < t0
U(t − t 0 ) =
(32.107)
1, t ≥ t 0
The second function is obtained from the first by recalling that subtraction
of t 0 from the argument of a function translates it to the right by t 0 units
along the x-axis.
Figure 32.3: A Unit Step Function U(t − t 0 ) with step at t = t 0 .
1
t 0
Unit step functions make it convenient for us to define stepwise continuous
functions as simple formulas with the need to list separate cases. For
example, the function
⎧
⎪⎨ 0, t < 1
f(t) = 1, 0 ≤ t < 2
(32.108)
⎪⎩
0, t ≥ 2
can also be written as (figure 32.3)
while the function
f(t) = U(t − 1) − U(t − 2) (32.109)
⎧
⎪⎨ 0 t < −1
f(t) = e
⎪⎩
−t2 −1 ≤ t < 1
0 t > 1
(32.110)