Inverse Laplace Transformation L−^1
The symbol L−^1 is used to denote the inverse Laplace transform operator with
the property that L−^1 undoes what Ldoes. That is, if the inverse Laplace transform
operator is applied to both sides of the equation L{ f(t)}=F(s)then
L−^1 L{f(t)}=f(t) = L−^1 {F(s)} (12.176)
This indicates that the table of Laplace transforms given above is to be interpreted
in either of two ways. Reading the above table left to right indicates L{ f(t)}=F(s)
and reading the table from right to left indicates f(t) = L−^1 {F(s)}.In general, one
can say
L{f(t)}=F(s) if and only if f(t) = L−^1 {F(s)} (12.177)
One use of the Laplace transform operator is to take a difficult problem in the
t−domain and transform it to an easier problem in the s−domain. Solve the easier
problem in the s−domain and convert the answer back to the t−domain. There are
two ways by which one can convert a Laplace transform back to the t−domain. One
conversion method is to have an extensive table of Laplace transforms so that one
can use table lookup to convert a function F(s)back to the correct function f(t)by
using the property (12.176) or (12.177). Table lookup is the preferred method for
now.
Another method used to find the inverse Laplace transform is more advanced
and requires knowledge of complex variable theory. This more advanced method is
expressed in the language of complex variables as
f(t) = L−^1 {F(s)}=L−^1 {F(s); s→t}=
1
2 πi Tlim→∞
∫γ+iT
γ−iT
estF(s)ds
where the integration is part of a line integral in the complex plane. Once you learn
all the theory involved, the inverse Laplace transform is really simple to use. The
difficulty in using the complex form for the inverse transform is that you will need
to take a complete course in complex variable theory to use and understand how to
employ it. The inverse Laplace transformation techniques often used are illustrated
in the figure 12-9.
