Unformatted text preview: 1 Solutions 47 But this last integral does not exist. Since the Laplace transform does not
exist at , for any , the Laplace transform does not exist.
43. is of exponential type because it is continuous and bounded. On the
other hand,
. Suppose there is a
and so that
for all
. We need only show that there are some for
which this inequality does not hold. Since
oscillates between
and let’s focus on those for which
. This happens when
is a multiple of
or
. If the inequality
is valid for all
it is valid for
for all
. We then get the
inequality
. Now divide by
, combine, complete the
square, and simplify to get the inequality
. Choose
so that
and
. Then this last inequality is not satisﬁed.
It follows that
is not of exponential type. Now consider the deﬁnite
integral
and compute by parts: We get Since is bounded and it follows that Taking limits as
in the equation above gives
. The righthand side exists because
is bounded.
(a) Show that
.
(b) Show that satisﬁes the recursion formula
.
(Hint : Integrate by parts.)
(c) Show that
when is a nonnegative integer.
44. (a)
(b)
The second equality is obtained by integration by parts using
.
(c) Repeated use of (b) gives .
, ...
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 Fall '08
 BELL,D
 Laplace, exponential type

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