5Transform theory

IB Complex Methods 5.2 Laplace transform
The Fourier transform is a powerful tool for solving differential equations and
investigating physical systems, but it has two key restrictions:
(i)
Many functions of interest grow exponentially (e.g.
e
x
), and so do not have
Fourier transforms;
(ii)
There is no way of incorporating initial or boundary conditions in the
transform variable. When used to solve an ODE, the Fourier transform
merely gives a particular integral: there are no arbitrary constants produced
by the method.
So for solving differential equations, the Fourier transform is pretty limited.
Right into our rescue is the Laplace transform, which gets around these two
restrictions. However, we have to pay the price with a different restriction it
is only defined for functions f(t) which vanishes for t < 0 (by convention).
From now on, we shall make this assumption, so that if we refer to the
function
f
(
t
) =
e
t
for instance, we really mean
f
(
t
) =
e
t
H
(
t
), where
H
(
t
) is the
Heaviside step function,
H(t) =
(
1 t > 0
0 t < 0
.
Definition
(Laplace transform)
.
The Laplace transform of a function
f
(
t
) such
that f(t) = 0 for t < 0 is defined by
ˆ
f(p) =
Z
0
f(t)e
pt
dt.
This exists for functions that grow no more than exponentially fast.
There is no standard notation for the Laplace transform.
Notation. We sometimes write
ˆ
f = L(f ),
or
ˆ
f(p) = L(f(t)).
The variable p is also not standard. Sometimes, s is used instead.
Many functions (e.g.
t
and
e
t
) which do not have Fourier transforms do have
Laplace transforms.
Note that
ˆ
f
(
p
) =
˜
f
(
ip
), where
˜
f
is the Fourier transform, provided that
both transforms exist.
Example. Let’s do some exciting examples.
(i)
L(1) =
Z
0
e
pt
dt =
1
p
.
(ii) Integrating by parts, we find
L(t) =
1
p
2
.
(iii)
L(e
λt
) =
Z
0
e
(λp)t
dt =
1
p λ
.
(iv) We have
L(sin t) = L
1
2i
e
it
e
it
=
1
2i
1
p i
1
p + i
=
1
p
2
+ 1
.
Note that the integral only converges if
Re p
is sufficiently large. For example,
in (iii), we require
Re p > Re λ
. However, once we have calculated
ˆ
f
in this
domain, we can consider it to exist everywhere in the complete p-plane, except
at singularities (such as at
p
=
λ
in this example). This process of extending a
complex function initially defined in some part of the plane to a larger part is
known as analytic continuation.
So far, we haven’t done anything interesting with Laplace transform, and
this is going to continue in the next section!