Mathematical Methods for Physics and Engineering : A Comprehensive Guide

13.3 CONCLUDING REMARKS

The properties of the Laplace transform derived in this section can sometimes

be useful in finding the Laplace transforms of particular functions.

Find the Laplace transform off(t)=tsinbt.

Although we could calculate the Laplace transform directly, we can use (13.62) to give

f ̄(s)=(−1)d

ds

L[sinbt]=−

d

ds

(

b

s^2 +b^2

)

=

2 bs

(s^2 +b^2 )^2

, fors> 0 .

13.3 Concluding remarks

In this chapter we have discussed Fourier and Laplace transforms in some detail.

Both are examples ofintegral transforms, which can be considered in a more

general context.

A general integral transform of a functionf(t) takes the form

F(α)=

∫b

a

K(α, t)f(t)dt, (13.65)

whereF(α) is the transform off(t) with respect to thekernelK(α, t), andαis

the transform variable. For example, in the Laplace transform caseK(s, t)=e−st,

a=0,b=∞.

Very often the inverse transform can also be written straightforwardly and

we obtain a transform pair similar to that encountered in Fourier transforms.

Examples of such pairs are

(i) the Hankel transform

F(k)=

∫∞

0

f(x)Jn(kx)xdx,

f(x)=

∫∞

0

F(k)Jn(kx)kdk,

where theJnare Bessel functions of ordern,and

(ii) the Mellin transform

F(z)=

∫∞

0

tz−^1 f(t)dt,

f(t)=

1

2 πi

∫i∞

−i∞

t−zF(z)dz.

Although we do not have the space to discuss their general properties, the

reader should at least be aware of this wider class of integral transforms.