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746 CHAPTER 22 Singularities and the Residue Theorem

to evaluate this integral, set this equal to the sum of y
the integrals on the sides of the rectangle, and take the
limit as R →∞. Assume the standard result that
√
∞ −x 2 π

e dx = .
0 2 π/4
x
16. Derive Fresnel’s integrals: R
∞ ∞ 1 π FIGURE 22.3 Path in

2 2
cos(x )dx = sin(x )dx = .
0 0 2 2 Problem 16, Section 22.3.
Hint:Integrate e iz 2 over the closed path bounding the
17. Let α and β be positive numbers. Show that
sector 0 ≤ x ≤ R,0 ≤ θ ≤ π/4, as in Figure 22.3. Use
∞

√
Cauchy’s theorem to evaluate this integral, then eval- x sin(αx) π −αβ/ 2 αβ
dx = e sin √ .
4
uate it as the sum of the integrals over the boundary 0 x + β 4 2β 2 2
segments of the sector. Show that the integral over
18. Let 0 <β <α. Show that
the circular arc tends to zero as R →∞, and use the
π
integrals over the line segments to obtain Fresnel’s 1 dθ = απ .
2 3/2
2
integrals. 0 (α + β cos(θ)) 2 (α − β )
22.4 Residues and the Inverse Laplace Transform
If f is a complex function defined at least for all z on [0,∞), then the Laplace transform of f is
∞

L[ f ](z) = e −zt f (t)dt
0
−1
for all z such that this integral converges. If L[ f ]= F, we write f = L [F]. In Chapter 3, we
−1
saw some techniques for manipulating L and L . The following theorem provides a formula
tz
for the inverse Laplace transform of F(s) in terms of residues e F(z), which we can sometimes
compute quite easily.
THEOREM 22.7 Inverse Laplace Transform

Let F be differentiable for all z except for poles z 1 ,··· , z n . Suppose for some real number σ, F
is differentiable for all z with Re(z) ≥ σ. Suppose there are numbers M and R such that

|zF(z)|≤ M for |z|≥ R.
For t ≥ 0, let
n

tz
f (t) = Res e F(z), z j .
j=1
Then
f = L [F].
−1
Because F is differentiable for Re(z)≥σ, F (z) exists at least for z to the right of the vertical

line x = σ. The condition that |zF(z)|≤ M for |z|≥ R means that zF(z) is bounded for all z on
and outside some sufficiently large circle. This condition is satisfied by any rational function
p(z)/q(z) if the degree of q(z) exceeds that of p(z).

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October 14, 2010 15:37 THM/NEIL Page-746 27410_22_ch22_p729-750