1550078481-Ordinary_Differential_Equations__Roberts_

234 Ordinary Differential Equations

THEOREM 5.2 If f(x) and g(x) are defined and piecewise continuous

on [O, b] for all finite b > 0 and of exponential order a as x ---> +oo and if

.C[f(x)] = .C[g(x)], then f(x) = g(x) at all points x E [O, oo) where f(x) and

g(x) are both continuous.

A consequence of this theorem is that a given Laplace transform, F(s),
cannot have more than one continuous function f(x) defined on [O, oo) which
transforms into F(s). Although there may be many functions g(x) which
transform into F ( s), there is only one continuous function defined on [ 0 , oo)
which transforms into F(s). Therefore, by defining the inverse Laplace
transform as follows there is a unique inverse Laplace transform.

DEFINITION Inverse Laplace Transform

If there exists a continuous function f(x ) defined on the interval [O, oo)

such that .C[f(x)] = F(s), then f(x) is call ed the inverse Laplace trans-

form of F(s) and we write .c-^1 (F(s)) = f(x ).

Just as there are functions f(x) which do not have Laplace transforms (re-

call f(x) = ex

2

does not h ave a Laplace transform), there a re many functions

F(s) which do not h ave inverse Laplace transforms. For example, F(s) = 1

is a function which, by the definition above, does not h ave an inverse Laplace
transform, since there is no continuous function f(x) defined on the interval

[O, oo) such that .C[f(x)] = l. However, in section 5.5, we define and discuss

the Dirac delta function, 5(x). This "function" is not a function on [O, +oo)
in the classical sense. Moreover, it is not continuous on [ 0, oo). Nonetheless,

in distribution theory, the delta function has the property that .C[J(x)] = l.

Hence, 5 ( x) is called t he inverse Laplace transform of F ( s) = 1.

Table 5 .1 contains several functions, f(x), and their corresponding Laplace
transform, F(s). The left column of Table 5.1 contains functions, f(x), which
are continuous on [O, oo) and the corresponding entry in the right column
contains their Laplace transform, F(s). Since each function f(x) is an inverse
Laplace transform of the function F(s) appearing in the corresponding right
column, the left column is labell ed f(x) = .c-^1 (F(s)). A more extensive
table of Laplace transforms and a table summarizing the properties of Laplace
transforms appear in Appendix C.