392
F. Contour Integration
now being traversed in a clockwise sense around z = −i∈ (this just inverts the
limits in (F.21)). Thus as ∈ → 0,
F (t) → 1
for t > 0.
(F.27)
Summarizing these manoeuvres, for t < 0 we chose C 2 in (F.22) in the upper
half-plane (figure F.1(a)), and its contribution vanished as R → ∞. In this
case we have, as R → ∞,
∫ ∞ −izt
i
e
F (t) →
dz = 0
for t < 0.
(F.28)
2π
z + i∈
−∞
For t > 0 we chose C 2 in the lower half-plane (figure F.1(b)), when again its
contribution vanished as R → ∞. However, in this case F does not vanish,
but instead we have, as R → ∞,
∫ ∞ −izt
i
e
F (t) →
dz = 1
for t > 0.
(F.29)
2π
z + i∈
−∞
Equations (F.28) and (F.29) show that we may indeed write
∫ ∞ −izt
i
e
θ(t) = lim
dz
(F.30)
∊→0 2π
z + i∈
−∞
as claimed in section 6.3, equation (6.93).