Mathematical Methods for Physics and Engineering : A Comprehensive Guide

13.2 LAPLACE TRANSFORMS

We may now consider the effect of multiplying the Laplace transformf ̄(s)by

e−bs(b>0). From the definition (13.53),

e−bsf ̄(s)=

∫∞

0

e−s(t+b)f(t)dt

=

∫∞

0

e−szf(z−b)dz,

on puttingt+b=z. Thuse−bsf ̄(s) is the Laplace transform of a functiong(t)

defined by

g(t)=

{

0for0<t≤b,

f(t−b)fort>b.

In other words, the functionfhas been translated to ‘later’t(larger values oft)

by an amountb.

Further properties of Laplace transforms can be proved in similar ways and

are listed below.

(i) L[f(at)]=

1

a

f ̄

(s

a

)

, (13.61)

(ii) L[tnf(t)]=(−1)n

dnf ̄(s)

dsn

, forn=1, 2 , 3 ,..., (13.62)

(iii) L

[

f(t)

t

]

=

∫∞

s

f ̄(u)du, (13.63)

provided limt→ 0 [f(t)/t] exists.

Related results may be easily proved.

Find an expression for the Laplace transform oftd^2 f/dt^2.

From the definition of the Laplace transform we have

L

[

t

d^2 f

dt^2

]

=

∫∞

0

e−stt

d^2 f

dt^2

dt

=−

d

ds

∫∞

0

e−st

d^2 f

dt^2

dt

=−

d

ds

[s^2 f ̄(s)−sf(0)−f′(0)]

=−s^2

df ̄

ds

− 2 sf ̄+f(0).

Finally we mention the convolution theorem for Laplace transforms (which is

analogous to that for Fourier transforms discussed in subsection 13.1.7). If the

functionsfandghave Laplace transformsf ̄(s)and ̄g(s)then

L

[∫t

0

f(u)g(t−u)du

]

=f ̄(s) ̄g(s), (13.64)