Fundamentals of Plasma Physics

5.2 The Landau problem 157

(a)

(b)

(c)

Im p

Im p

Im p

Rep

Rep

Rep

complex

pplane

complex

pplane

p

defined

thisregion

only

p

defined

thisregion

only

p

defined

thisregion

only

analytic

continuation

of

p

leastdamped

mode pj

−i

i

Figure 5.2: Contours in complex p-plane

For this choice of path, Eq.(5.35) becomes

g(t) =

∫∞

0

dt′

β+i∫∞

β−i∞

d(pr+ipi)ψ(t′)e(pr+ipi)(t−t

′)

= i

∫∞

0

dt′eβ(t−t

′)

ψ(t′)

∫∞

−∞

dpieipi(t−t

′)

= 2πi

∫∞

0

dt′eβ(t−t

′)

ψ(t′)δ(t−t′)

= 2πiψ(t) (5.37)

where Eq.(5.36) has been used. Thus,ψ(t) = (2πi)−^1 g(t) and so the inverse of the
Laplace transform is

ψ(t)=

1

2 πi

∫β+i∞

β−i∞

dpψ(p)ept, β>γ. (5.38)