17.4 Properties of the Laplace transform
Having seen how to obtain elementary Laplace transforms we now turn to the general
properties of the transform. These enable us to calculate more complicated transforms and
also to apply the Laplace transform to such things as differential equations. We will not
be very rigorous about the proofs, the emphasis being on getting across the key ideas.
- The Laplace transform is linear
The Laplace transform is alinear operation.Thatis,iff(t),g(t)are two functions with
Laplace transformsf(s) ̃ ,g(s) ̃ anda,bare constants then
L[af (t)+bg(t)]=aL[f(t)]+bL[g(t)]=af(s) ̃ +bg(s) ̃Or, briefly:
Laplace transform of sum=sum of Laplace transforms.
- The first shift theorem
The exponential functioneatis a mainstay of applied mathematics, and it plays a partic-
ularly special role in the theory of the Laplace transform, embodied in thefirst shift
theorem(in case you are wondering, there is a ‘second shift theorem’ but we won’t be
needing it in this book).
SupposeL[f(t)]=f(s) ̃ fors>s 0 ,andletabe any real number. Then
L[eatf(t)]=f(s ̃ −a) for s>s 0 +aThis can be proved by coupling theeatin the Laplace transform with theestand noting
that this effectively replacessbys−a.
Note that the results foreattn,eatsinωt,eatcosωtgiven in Table 17.1 can be obtained
using the shift theorem.
- The Laplace transform of the derivative
This is the crucial property for the application of Laplace transforms to the solution
of differential equations. Suppose thatf(t)and its derivativef′(t)have Laplace trans-
forms. Then
L[f′(t)]=sL[f(t)]−f( 0 )This follows on integrating by parts:
L[f′(t)]=∫∞0f′(t)e−stdt=[f(t)e−st]∞ 0 +s∫∞0f(t)e−stdt=−f( 0 )+sL[f(t)]on assuming that the limit at infinity vanishes.