Lecture Notes in Differential Equations - Bruce E. Shapiro

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can be written as
as illustrated in figure 32.4.
f(t) = e −t2 (U(t + 1) − U(t − 1) (32.111)
Figure 32.4: A piecewise continuous function defined with step functions.
1.
0.75
0.5
0.25
2 1 1 2
We can also use the combination to produce piecewise continuous translations,
for example,
{
e −(x−3)2 , x ≥ 2
f(t) =
= U(t − 2)f(t − 3) (32.112)
0, x < 2
The second factor (f(t−3)) translates the bell curve to the right by 3 units,
while the first factor cuts off the part to the left of t = 2 (figure 32.5).
Figure 32.5: A piecewise continuous function defined by a translation multiplied
by a step functions.
1.
0.75
0.5
0.25
2 1 1 2 3 4 5 6