Unformatted text preview: 46 1 Solutions Solving for gives 36. For
,
type of order . . . Thus . So is of exponential 37. Suppose is on exponential type of order and is of exponential type
of order . Suppose
. Then there are numbers
and
so that
and
. Now
. If follows that
is of exponential type of
order .
38. Suppose is of exponential type of order and is of exponential type
of order . Then there are numbers
and so that
and
. Now
. If follows that
is of exponential type of order
.
39. If is bounded by
of order . , say, then . So is of exponential type 40. Suppose is of exponential type of order in the sense given in the text.
Then
can be chosen to be and satisﬁes the deﬁnition given in the
statement of the problem. Now suppose satisﬁes the deﬁnition given in
the statement of the problem. I.e. there is a
and
so that
for
. Since
is continuous on the interval
it has a
maximum,
, say. It follows that
on
and hence
, for all
. It follows that is of exponential type in
the sense given in the text.
41. Suppose and
by . But are real and . Then where
. As approaches inﬁnity so does . Since
it is clear that
, for all
, and hence
not bounded. It follows that
is not of exponential type. is bounded is 42. First of all, ﬁx
. Since
for all it follows that
by the comparison test. Thus
does not exist for
any real number . Now let be any real number. Then ...
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 Fall '08
 BELL,D
 Topology, Complex number, exponential type

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