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
ofp
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)