Concluding remarks

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 of f(t) = t sin bt.
Although we could calculate the Laplace transform directly, we can use (13.62) to give
b
2bs
¯ = (−1) d L [sin bt] = − d
= 2
,
for s > 0. f(s)
2
2
ds
ds s + b
(s + b2 )2
13.3 Concluding remarks
In this chapter we have discussed Fourier and Laplace transforms in some detail.
Both are examples of integral transforms, which can be considered in a more
general context.
A general integral transform of a function f(t) takes the form
b
K(α, t)f(t) dt,
(13.65)
F(α) =
a
where F(α) is the transform of f(t) with respect to the kernel K(α, t), and α is
the transform variable. For example, in the Laplace transform case K(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(x)Jn (kx)x dx,
F(k) =
0 ∞
f(x) =
F(k)Jn (kx)k dk,
0
where the Jn are Bessel functions of order n, and
(ii) the Mellin transform
∞
tz−1 f(t) dt,
F(z) =
0
i∞
1
t−z F(z) dz.
f(t) =
2πi −i∞
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.
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