1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

12.10 • CONVOLUTION 575

y

100

80

60

40

20

y= o, (t)

ill

100

80

y y

100

80

40.

20.

0. 1

1= 01 <•!

100

Figure 12 .29 Graphs of y = o. (t) for a= 1 0, 40, and 100.

y y

y =J.<t)

a^2

0.1

Figure 12. 30 The integral of Oa (t) is fa (t), which becomes Uo (t) when a-+ O.

2

We now tum to the unit impulse function. First, we consider the function

fa (t) obtained by integrating 04 (t):

fa (t) = 1 o,. (r) dr = £, for 0 ~ t ~a;

t { 0, for t < O;

o 1, for a< t.

Hence Uo (t) = Jim fa (t), as illustrated in Figure 12.30.
a .... o
vVe demonstrate the response of a system to the unit impulse function in
Example 12 .32.

• EXAMPLE 12. 32 Solve the initial value problem

y" (t) + 4y' (t) + 13y (t) = 38 (t) J with y (0) = o and y' (o-) = o.