Mathematical Methods for Physicists: A concise introduction - Site Map

THE LAPLACE TRANSFORMATION
This establishes not only the convergence but the absolute convergence of the
integral de®ning L‰ f …x†Š. Note that M=… p a† tends to zero as p !1. This
shows that
lim F… p† ˆ0 …9:3†
p!1
for all functions F…p† ˆL‰ f …x†Š such that f …x† satis®es the foregoing conditions
(1) and (2). It follows that if lim p!1 F… p† 6ˆ0, F…p† cannot be the Laplace transform
of any function f …x†.
It is obvious that functions of exponential order play a dominant role in the use
of Laplace transforms. One simple way of determining whether or not a speci®ed
function is of exponential order is the following one: if a constant b exists such
that
h i
lim
x!1 ebx jf …x† j
…9:4†
exists, the function f …x† is of exponential order (of the order of e bx †. To see this,
let the value of the above limit be K 6ˆ 0. Then, when x is large enough, je bx f …x†j
can be made as close to K as possible, so certainly
Thus, for suciently large x,
or
On the other hand, if
je bx f …x†j < 2K:
j f …x†j < 2Ke bx
j f …x†j < Me bx ; with M ˆ 2K:
lim
x!1 ‰ecx jf …x† jŠ ˆ1 …9:5†
for every ®xed c, the function f …x† is not of exponential order. To see this, let us
assume that b exists such that
from which it follows that
j f …x†j < Me bx for x X
je 2bx f …x†j < Me bx :
Then the choice of c ˆ 2b would give us je cx f …x†j < Me bx , and e cx f …x† !0as
x !1which contradicts Eq. (9.5).
Example 9.2
Show that x 3 is of exponential order as x !1.
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