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

(λ−p)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!