 # Concluding remarks

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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.
459
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