Page 49 - Schaum's Outline of Theory and Problems of Signals and Systems

38 SIGNALS AND SYSTEMS [CHAP. 1

we obtain

This can be true only if

0
4(t)u(t) dt = 0 and kw4(t)[l - u(t)] dt = 0
1- a
These conditions imply that

b(t)u(t) = 0, t < 0 and c$(t)[l - u(t)] = 0, t > 0

Since 4(t) is arbitrary, we have
u(t)=O, t<O and 1 -u(t)=O,t>O

that is,

1.25. Verify Eqs. (1.23) and (1.24), that is,
1
(a) 6(at) = -6(t); (b) S(-t) = S(t)
la1
The proof will be based on the following equiualence property:
Let g,(t) and g2(t) be generalized functions. Then the equivalence property states that
g,(t) =gt(t) if and only if

for all suitably defined testing functions 4(t).
(a) With a change of variable, at = T, and hence t = r/a, dt = (l/a) dr, we obtain the
following equations:
If a > 0,

Thus, for any a