Fundamentals of Plasma Physics

5.2 The Landau problem 161

is simply

fˆ(p)=^1

p−q

, definedforallpprovidedfˆ(p)remainsanalytic. (5.58)

The Bromwich contour can now be deformed into the left hand plane as shown in Fig.
5.3. Becauseexp(pt)→ 0 for positivetand negativeRep,the integration contour can be
closed by an arc that goes to the left (cf. Fig.5.3) into the region whereRep→−∞. The
resulting contour encircles the pole atp=qand so the integral can be evaluated using the
method of residues as follows:

F(t)=

1

2 πi

∮

1

p−q

eptdp=lim

p→q

2 πi(p−q)

[

1

2 πi(p−q)

ept

]

=eqt. (5.59)

closure of

deformed contour

deformed contour

Im p original

Bromwich

contour

complex pplane

i

−i

f
p,f̂
p

both defined in this region

only f̂
p

defined in this region

Rep

Figure 5.3: Bromwich contour

This simple example shows that while the Bromwich contour formally givesthe inverse
Laplace inverse transform off ̃(p), the Bromwich contour by itself does not allow use of
the method of residues, since the poles of interest are located precisely in the left hand
complexpplane wheref ̃(p)is undefined. However, analytic continuation off(p)allows
deformation of the Bromwich contour into the formerly forbidden area, and then the inverse
transform may be easily evaluated using the method of residues.