5Transform theory

IB Complex Methods

5.6 The convolution theorem for Laplace transforms

Finally, we recall from IB Methods that the Fourier transforms turn convolutions

into products, and vice versa. We will now prove an analogous result for Laplace

transforms.

Definition

(Convolution)

.

The convolution of two functions

f

and

g

is defined

as

(f ∗ g)(t) =

Z

∞

−∞

f(t −t

0

)g(t

0

) dt

0

.

When f and g vanish for negative t, this simplifies to

(f ∗ g)(t) =

Z

t

0

f(t − t

0

)g(t) dt.

Theorem

(Convolution theorem)

.

The Laplace transform of a convolution is

given by

L(f ∗ g)(p) =

ˆ

f(p)ˆg(p).

Proof.

L(f ∗ g)(p) =

Z

∞

0

Z

t

0

f(t − t

0

)g(t

0

) dt

0

e

−pt

dt

We change the order of integration in the (

t, t

0

) plane, and adjust the limits

accordingly (see picture below)

=

Z

∞

0

Z

∞

t

0

f(t − t

0

)g(t

0

)e

−pt

dt

dt

0

We substitute u = t − t

0

to get

=

Z

∞

0

Z

∞

0

f(u)g(t

0

)e

−pu

e

−pt

0

du

dt

0

=

Z

∞

0

Z

∞

0

f(u)e

−pu

du

g(t

0

)e

−pt

0

dt

0

=

ˆ

f(p)ˆg(p).

Note the limits of integration are correct since they both represent the region

below

t

t

0