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22.4 Residues and the Inverse Laplace Transform 747

y

σ + ib
Re(z) > σ
Re(z) < σ

σ x

σ – ib

FIGURE 22.4 σ in Theorem 22.7.

Theorem 22.7 can be proved using the version of the Cauchy integral formula given in
Section 20.3.7. Following a sketch of the argument and two examples computing inverse Laplace
transforms of functions, we will use the theorem to analyze heat diffusion in a homogeneous solid
cylinder.
Begin by writing, in the notation of this section,
1 σ+ib F(z)
F(s) =− lim dz.
2πi b→∞ σ−ib z − s
−1
Referring to Figure 22.4, take L through the integral (this is justified by hypotheses of the
theorem) to compute
1 σ+ib F(z)
−1 −1
L [F(s)](t) = lim L dz
2πi b→∞ σ−ib s − z
1 σ+ib
tz
= lim e F(z)dz
2πi b→∞ σ−ib
tz
= Res(e F(z), p),
p
tz
with this summation extending over all of the poles of e F(z). σ is chosen so that all of these
poles are to the right of σ.

EXAMPLE 22.21
2
2
Let a be a positive number. We will find the inverse Laplace transform of F(z) = 1/(a + z ).
We can do this using MAPLE, a table, or a method from Chapter 3. As an illustration of the
use of the theorem, note that F(z) has simple poles at ±ai. Compute the residues as
e ai e −ai
tz tz
Res(e F(z),ai) = and Res(e F(z),−ai) = .
2ai −2ai
This can be done easily by writing
e tz
tz
e F(z) =
(z − ai)(z + ai)
and using Corollary 22.1. By Theorem 22.7
1
−1 1 ai −ai sin(at).
L [F](t) = e − e =
2ai a

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October 14, 2010 15:37 THM/NEIL Page-747 27410_22_ch22_p729-750