1. Inversion of the Laplace transform

The Laplace transform of a function f (t) of a function of exponential order is defined as

F(s) = ∫∞

0

e−st f (t) dt.

This transform had many applications in applied mathematics. It can be shown that the inverse

transform can be obtained as the Bromwich integral,

f (t) = 1

2πi ∫γ + i∞

γ − i∞

ezt F(z) dz.

The integrand here is over the line where is chosen sufficiently large that all

singularities of lie to the left of the line. In practice we evaluate the integral as the limit as

of a contour integral over the contour shown below. Note that the curved part is a

semicircle of radius centred on the point , not . We assume throughout that is

real.

Re (z) = γ γ

F(s)

R → ∞

R z = γ z = 0 t

R

−R

Re( z )

Im( z )

γ

The Bromwich contour

We are going to use this contour to evaluate the inverse transform of two functions. Both of

these can be done easily by elementary methods, but they serve to illustrate the method.

(i) Let

F(s) = 1

(s + 1)2 + 1

.

Since the poles of this are in the left half plane, we can chose γ = 0.

Use the Residue Theorem to find f (t).

Note that to obtain full marks you must give a proper proof that the integral around the curved

section of the Bromwich contour tends to zero as goes to infinity. In particular, you need to

find the maximum value of on the contour.

The Laplace transform of a function f (t) of a function of exponential order is defined as

F(s) = ∫∞

0

e−st f (t) dt.

This transform had many applications in applied mathematics. It can be shown that the inverse

transform can be obtained as the Bromwich integral,

f (t) = 1

2πi ∫γ + i∞

γ − i∞

ezt F(z) dz.

The integrand here is over the line where is chosen sufficiently large that all

singularities of lie to the left of the line. In practice we evaluate the integral as the limit as

of a contour integral over the contour shown below. Note that the curved part is a

semicircle of radius centred on the point , not . We assume throughout that is

real.

Re (z) = γ γ

F(s)

R → ∞

R z = γ z = 0 t

R

−R

Re( z )

Im( z )

γ

The Bromwich contour

We are going to use this contour to evaluate the inverse transform of two functions. Both of

these can be done easily by elementary methods, but they serve to illustrate the method.

(i) Let

F(s) = 1

(s + 1)2 + 1

.

Since the poles of this are in the left half plane, we can chose γ = 0.

Use the Residue Theorem to find f (t).

Note that to obtain full marks you must give a proper proof that the integral around the curved

section of the Bromwich contour tends to zero as goes to infinity. In particular, you need to

find the maximum value of on the contour.

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